Recent zbMATH articles in MSC 60Fhttps://zbmath.org/atom/cc/60F2021-05-28T16:06:00+00:00WerkzeugDonsker type theorem for fractional Poisson process.https://zbmath.org/1459.600742021-05-28T16:06:00+00:00"Araya, Héctor"https://zbmath.org/authors/?q=ai:araya.hector"Bahamonde, Natalia"https://zbmath.org/authors/?q=ai:bahamonde.natalia"Torres, Soledad"https://zbmath.org/authors/?q=ai:torres.soledad"Viens, Frederi"https://zbmath.org/authors/?q=ai:viens.frederi-gSummary: In this paper we study a Donsker type theorem for the fractional Poisson process (fPp). We present the random walk discretization and its associated convergence theorem in the Skorohod topology. Simulation results are also presented.From random partitions to fractional Brownian sheets.https://zbmath.org/1459.600852021-05-28T16:06:00+00:00"Durieu, Olivier"https://zbmath.org/authors/?q=ai:durieu.olivier"Wang, Yizao"https://zbmath.org/authors/?q=ai:wang.yizaoSummary: We propose discrete random-field models that are based on random partitions of \(\mathbb{N}^{2}\). The covariance structure of each random field is determined by the underlying random partition. Functional central limit theorems are established for the proposed models, and fractional Brownian sheets, with full range of Hurst indices, arise in the limit. Our models could be viewed as discrete analogues of fractional Brownian sheets, in the same spirit that the simple random walk is the discrete analogue of the Brownian motion.Quenched central limit theorem rates of convergence for one-dimensional random walks in random environments.https://zbmath.org/1459.602112021-05-28T16:06:00+00:00"Ahn, Sung Won"https://zbmath.org/authors/?q=ai:ahn.sung-won"Peterson, Jonathon"https://zbmath.org/authors/?q=ai:peterson.jonathonSummary: Unlike classical simple random walks, one-dimensional random walks in random environments (RWRE) are known to have a wide array of potential limiting distributions. Under certain assumptions, however, it is known that CLT-like limiting distributions hold for the walk under both the quenched and averaged measures. We give upper bounds on the rates of convergence for the quenched central limit theorems for both the hitting time and position of the RWRE with polynomial rates of convergence that depend on the distribution on environments.Change-point analysis using logarithmic quantile estimation.https://zbmath.org/1459.620702021-05-28T16:06:00+00:00"Tabacu, Lucia"https://zbmath.org/authors/?q=ai:tabacu.lucia"Ledbetter, Mark"https://zbmath.org/authors/?q=ai:ledbetter.markSummary: We present a new approach for estimating quantiles for change-point problems, specifically for \textit{A. N. Pettitt}'s rank test [J. R. Stat. Soc., Ser. B 41, 46--53 (1979; Zbl 0411.62025)]. We use the logarithmic quantile estimation procedure introduced by \textit{K. Thangavelu} [Quantile estimation based on the almost sure central limit theorem. Göttingen: Univ. Göttingen, Mathematisch-Naturwissenschaftliche Fakultäten (Diss.) (2005; Zbl 1095.62026)], which is based on the concept of the almost sure limit theorem. Numerical results for small data sets and simulated data are given.Some properties of the multivariate generalized hyperbolic laws.https://zbmath.org/1459.600412021-05-28T16:06:00+00:00"Fotopoulos, Stergios B."https://zbmath.org/authors/?q=ai:fotopoulos.stergios-b"Jandhyala, Venkata K."https://zbmath.org/authors/?q=ai:jandhyala.venkata-k"Paparas, Alex"https://zbmath.org/authors/?q=ai:paparas.alexSummary: The purpose of this study is to characterize multivariate generalized hyperbolic (MGH) distributions and their conditionals by considering the MGH as a subclass of the mean-variance mixing of the multivariate normal law. The essential contribution here lies in expressing MGH densities by utilizing various integral representations of the Bessel function. Moreover, in a more convenient form these modified density representations are more advantageous for deriving limiting results. The forms are also convenient for studying the transient as well as tail behavior of MGH distributions. The results include the normal distribution as a limiting form for the MGH distribution. To support the MGH model an empirical study is conducted to demonstrate the applicability of the MGH distribution for modeling not only high frequency data but also for modeling low frequency data. This is against the currently prevailing notion that the MGH model is relevant for modeling only high frequency data.Convergence to scale-invariant Poisson processes and applications in Dickman approximation.https://zbmath.org/1459.601062021-05-28T16:06:00+00:00"Bhattacharjee, Chinmoy"https://zbmath.org/authors/?q=ai:bhattacharjee.chinmoy"Molchanov, Ilya"https://zbmath.org/authors/?q=ai:molchanov.ilya-sSummary: We study weak convergence of a sequence of point processes to a scale-invariant simple point process. For a deterministic sequence \((z_n)_{n\in \mathbb{N}}\) of positive real numbers increasing to infinity as \(n \to \infty\) and a sequence \((X_k)_{k\in \mathbb{N}}\) of independent non-negative integer-valued random variables, we consider the sequence of point processes
\[
\nu_n=\sum_{k=1}^{\infty }X_k \delta_{z_k/z_n}, \quad n \in \mathbb{N},
\]
and prove that, under some general conditions, it converges vaguely in distribution to a scale-invariant Poisson process \(\eta_c\) on \((0,\infty )\) with the intensity measure having the density \(ct^{-1}, t\in (0,\infty)\). An important motivating example from probabilistic number theory relies on choosing \(X_k \sim{\text{Geom}}(1-1/p_k)\) and \(z_k=\log p_k, k \in \mathbb{N} \), where \((p_k)_{k \in \mathbb{N}}\) is an enumeration of the primes in increasing order. We derive a general result on convergence of the integrals \(\int_0^1 t \nu_n(dt)\) to the integral \(\int_0^1 t \eta_c(dt)\), the latter having a generalized Dickman distribution, thus providing a new way of proving Dickman convergence results.
We extend our results to the multivariate setting and provide sufficient conditions for vague convergence in distribution for a broad class of sequences of point processes obtained by mapping the points from \((0,\infty )\) to \(\mathbb{R}^d\) via multiplication by i.i.d. random vectors. In addition, we introduce a new class of multivariate Dickman distributions which naturally extends the univariate setting.Distribution free goodness of fit testing of grouped Bernoulli trials.https://zbmath.org/1459.620692021-05-28T16:06:00+00:00"Roberts, Leigh A."https://zbmath.org/authors/?q=ai:roberts.leigh-aSummary: Recently \textit{E. Khmaladze} [Bernoulli 22, No. 1, 563--588 (2016; Zbl 1345.60094)] has shown how to `rotate' one empirical process to another. We apply this methodology to goodness of fit tests for Bernoulli trials, generated by a single distributional family, but with covariates varying over the sample. Grouping the data, we demonstrate that goodness of fit tests after rotation to distribution free processes are easily computed, and exhibit high power to reject incorrect null hypotheses.Stability for gains from large investors' strategies in \(M_{1}/J_{1}\) topologies.https://zbmath.org/1459.601212021-05-28T16:06:00+00:00"Becherer, Dirk"https://zbmath.org/authors/?q=ai:becherer.dirk"Bilarev, Todor"https://zbmath.org/authors/?q=ai:bilarev.todor"Frentrup, Peter"https://zbmath.org/authors/?q=ai:frentrup.peterSummary: We prove continuity of a controlled SDE solution in Skorokhod's \(M_{1}\) and \(J_{1}\) topologies and also uniformly, in probability, as a nonlinear functional of the control strategy. The functional comes from a finance problem to model price impact of a large investor in an illiquid market. We show that \(M_{1}\)-continuity is the key to ensure that proceeds and wealth processes from (self-financing) càdlàg trading strategies are determined as the continuous extensions for those from continuous strategies. We demonstrate by examples how continuity properties are useful to solve different stochastic control problems on optimal liquidation and to identify asymptotically realizable proceeds.Zooming-in on a Lévy process: failure to observe threshold exceedance over a dense grid.https://zbmath.org/1459.600982021-05-28T16:06:00+00:00"Bisewski, Krzysztof"https://zbmath.org/authors/?q=ai:bisewski.krzysztof"Ivanovs, Jevgenijs"https://zbmath.org/authors/?q=ai:ivanovs.jevgenijsSummary: For a Lévy process \(X\) on a finite time interval consider the probability that it exceeds some fixed threshold \(x>0\) while staying below \(x\) at the points of a regular grid. We establish exact asymptotic behavior of this probability as the number of grid points tends to infinity. We assume that \(X\) has a zooming-in limit, which necessarily is \(1/\alpha \)-self-similar Lévy process with \(\alpha \in (0,2]\), and restrict to \(\alpha >1\). Moreover, the moments of the difference of the supremum and the maximum over the grid points are analyzed and their asymptotic behavior is derived. It is also shown that the zooming-in assumption implies certain regularity properties of the ladder process, and the decay rate of the left tail of the supremum distribution is determined.Large deviations for interacting particle systems: joint mean-field and small-noise limit.https://zbmath.org/1459.600692021-05-28T16:06:00+00:00"Orrieri, Carlo"https://zbmath.org/authors/?q=ai:orrieri.carloSummary: We consider a system of stochastic interacting particles in \(\mathbb{R}^d\) and we describe large deviation asymptotics in a joint mean-field and small-noise limit. Precisely, a large deviation principle (LDP) is established for the empirical measure and the stochastic current, as the number of particles tends to infinity and the noise vanishes, simultaneously. We give a direct proof of the LDP using tilting and subsequently exploiting the link between entropy and large deviations. To this aim we employ consistency of suitable deterministic control problems associated to the stochastic dynamics.Interacting diffusions on sparse graphs: hydrodynamics from local weak limits.https://zbmath.org/1459.602072021-05-28T16:06:00+00:00"Oliveira, Roberto I."https://zbmath.org/authors/?q=ai:oliveira.roberto-imbuzeiro"Reis, Guilherme H."https://zbmath.org/authors/?q=ai:reis.guilherme-h"Stolerman, Lucas M."https://zbmath.org/authors/?q=ai:stolerman.lucas-mSummary: We prove limit theorems for systems of interacting diffusions on sparse graphs. For example, we deduce a hydrodynamic limit and the propagation of chaos property for the stochastic Kuramoto model with interactions determined by Erdős-Rényi graphs with constant mean degree. The limiting object is related to a potentially infinite system of SDEs defined over a Galton-Watson tree. Our theorems apply more generally, when the sequence of graphs (``decorated'' with edge and vertex parameters) converges in the local weak sense. Our main technical result is a locality estimate bounding the influence of far-away diffusions on one another. We also numerically explore the emergence of synchronization phenomena on Galton-Watson random trees, observing rich phase transitions from synchronized to desynchronized activity among nodes at different distances from the root.On the rates of convergence in weak limit theorems for normalized geometric sums.https://zbmath.org/1459.600392021-05-28T16:06:00+00:00"Hung, Tran Loc"https://zbmath.org/authors/?q=ai:hung.tran-loc"Kien, Phan Tri"https://zbmath.org/authors/?q=ai:kien.phan-triSummary: The main purpose of this paper is to establish the rates of convergence in weak limit theorems for normalized geometric sums of independent identically distributed random variables via Zolotarev's probability metric.How long is the convex minorant of a one-dimensional random walk?https://zbmath.org/1459.600562021-05-28T16:06:00+00:00"Alsmeyer, Gerold"https://zbmath.org/authors/?q=ai:alsmeyer.gerold"Kabluchko, Zakhar"https://zbmath.org/authors/?q=ai:kabluchko.zakhar-a"Marynych, Alexander"https://zbmath.org/authors/?q=ai:marynych.alexander-v"Vysotsky, Vladislav"https://zbmath.org/authors/?q=ai:vysotsky.vladislav-vSummary: We prove distributional limit theorems for the length of the largest convex minorant of a one-dimensional random walk with independent identically distributed increments. Depending on the increment law, there are several regimes with different limit distributions for this length. Among other tools, a representation of the convex minorant of a random walk in terms of uniform random permutations is utilized.Convergence properties for the partial sums of widely orthant dependent random variables under some integrable assumptions and their applications.https://zbmath.org/1459.600602021-05-28T16:06:00+00:00"He, Yongping"https://zbmath.org/authors/?q=ai:he.yongping"Wang, Xuejun"https://zbmath.org/authors/?q=ai:wang.xuejun.1|wang.xuejun"Yao, Chi"https://zbmath.org/authors/?q=ai:yao.chiSummary: Widely orthant dependence (WOD, in short) is a special dependence structure. In this paper, by using the probability inequalities and moment inequalities for WOD random variables, we study the \(L_p\) convergence and complete convergence for the partial sums respectively under the conditions of \(\mathrm{RCI}(\alpha), \mathrm{SRCI} (\alpha)\) and \(R-h\)-integrability. We also give an application to nonparametric regression models based on WOD errors by using the \(L_p\) convergence that we obtained. Finally we carry out some simulations to verify the validity of our theoretical results.Time-reversal of coalescing diffusive flows and weak convergence of localized disturbance flows.https://zbmath.org/1459.600752021-05-28T16:06:00+00:00"Bell, James"https://zbmath.org/authors/?q=ai:bell.james-h.1|bell.james-r|bell.james-f|bell.james-jSummary: We generalize the coalescing Brownian flow, also known as the Brownian web, considered as a weak flow to allow varying drift and diffusivity in the constituent diffusion processes and call these flows coalescing diffusive flows. We then identify the time-reversal of each coalescing diffusive flow and provide two distinct proofs of this identification. One of which is direct and the other proceeds by generalizing the concept of a localized disturbance flow to allow varying size and shape of disturbances, we show these new flows converge weakly under appropriate conditions to a coalescing diffusive flow and identify their time-reversals.Large deviations of radial \(SLE_{\infty}\).https://zbmath.org/1459.601742021-05-28T16:06:00+00:00"Ang, Morris"https://zbmath.org/authors/?q=ai:ang.morris"Park, Minjae"https://zbmath.org/authors/?q=ai:park.minjae"Wang, Yilin"https://zbmath.org/authors/?q=ai:wang.yilinSummary: We derive the large deviation principle for radial Schramm-Loewner evolution \((\operatorname{SLE})\) on the unit disk with parameter \(\kappa \rightarrow \infty\). Restricting to the time interval \([0,1]\), the good rate function is finite only on a certain family of Loewner chains driven by absolutely continuous probability measures \(\{\phi_t^2 (\zeta)\, d\zeta\}_{t \in [0,1]}\) on the unit circle and equals \(\int_0^1 \int_{S^1} |\phi_t'|^2/2\,d\zeta dt\). Our proof relies on the large deviation principle for the long-time average of the Brownian occupation measure by \textit{M. D. Donsker} and \textit{S. R. S. Varadhan} [Commun. Pure Appl. Math. 33, 365--393 (1980; Zbl 0504.60037)].Large deviations of a long-time average in the Ehrenfest urn model.https://zbmath.org/1459.821402021-05-28T16:06:00+00:00"Meerson, Baruch"https://zbmath.org/authors/?q=ai:meerson.baruch"Zilber, Pini"https://zbmath.org/authors/?q=ai:zilber.piniThe central limit theorem for eigenvalues.https://zbmath.org/1459.600102021-05-28T16:06:00+00:00"Aoun, Richard"https://zbmath.org/authors/?q=ai:aoun.richardLet \(V\) be a real vector space and let \(X_1,X_2,\ldots\) be independent and identically distributed \(\mathrm{GL}(V)\)-valued
random variables. Letting \(L_n\) be the random walk defined by \(L_n=X_n\cdots X_1\), one of the main results of the present
paper is a central limit theorem for \(\ln \rho(L_n)\) under second moment and irreducibility assumptions, where
\(\rho(\cdot)\) is the spectral radius. Other results include a multivariate version of this theorem for a Zariski-dense
random walk on a linear reductive group, and quantitative estimates of \(\rho(L_n)\).
Reviewer: Fraser Daly (Edinburgh)Spectral potential, Kullback action, and large deviation principle for finitely-additive measures.https://zbmath.org/1459.600662021-05-28T16:06:00+00:00"Bakhtin, V. I."https://zbmath.org/authors/?q=ai:bakhtin.victor-iSummary: The known large deviation principle for empirical measures, generated by a sequence if i.i.d. random variables, is extended to the case of finitely-additive and nonnormalized distributions. For the Kullback-Leibler information function we prove a least action principle and gauge identities, linking the Kullback-Leibler information function with its Legendre dual functional.Central limit theorems for gaps of generalized Zeckendorf decompositions.https://zbmath.org/1459.600622021-05-28T16:06:00+00:00"Li, Ray"https://zbmath.org/authors/?q=ai:li.ray"Miller, Steven J."https://zbmath.org/authors/?q=ai:miller.steven-jIt is well known as Zeckendorf's theorem that every positive integer can be decomposed uniquely as the sum of non-adjacent (shifted) Fibonacci numbers $F_1=1$, $F_2=2$, $F_n=F_{n-1}+F_{n-2}$, $n\geq 3$. See [\textit{C. G. Lekkerkerker}, Simon Stevin 29, 190--195 (1952; Zbl 0049.03101); \textit{D. E. Daykin}, J. Lond. Math. Soc. 35, 143--160 (1960; Zbl 0215.06801); \textit{E. Zeckendorf}, Bull. Soc. R. Sci. Liège 41, 179--182 (1972; Zbl 0252.10011); Fibonacci Q. 10, 365--372 (1972; Zbl 0252.10053)]. Lekkerkerker also proved, using a continued fraction approach, that the average number of such summands for integers in $[F_n,F_{n+1})$ is $n/(\alpha^2+1)+O(1)$, where $\alpha=(1+\sqrt 5)/2$ is the golden mean. These results have been generalized to certain classes of linear recurrence relations with appropriate notions of decompositions. It has also been proved, that the distribution of the number of summands in the decomposition of an $M\in[G_n,G_{n+1})$ converges to a normal distribution as $n\to\infty$. See [\textit{M. Koloğlu} et al., Fibonacci Q. 49, No. 2, 116--130 (2011; Zbl 1225.11021); \textit{S. J. Miller} and \textit{Y. Wang}, J. Comb. Theory, Ser. A 119, No. 7, 1398--1413 (2012; Zbl 1245.05006)] and references therein.
The authors of the present very rich paper state and prove Theorem 1.9 (generalized Lekkerkerker's theorem for gaps of decompositions), Theorem 1.10 (variance is linear for gaps of decompositions), and Theorem 1.11 (Gaussian behavior for gaps of decompositions). Loosely speaking, for any nonnegative integer $g$, the average number of gaps of size $g$ in many generalized Zeckendorf decompositions is $nC_\mu+d_\mu+o(1)$ for constants $C_\mu>0$ and $d_\mu$ depending on $g$ and the recurrence, the variance of the number of gaps of size $g$ is similarly $nC_\sigma+d_\sigma+o(1)$ for constants $C_\sigma>0$ and $d_\sigma$, and the number of gaps of size $g$ of an $M\in[G_n,G_{n+1})$, properly normalized, converges to the standard normal $N(0, 1)$ as $n\to\infty$. They do this by means of their Theorem 3.1, which they prove using the method of moments. The theorem is a general result regarding the convergence to a normal distribution of an associated two-dimensional recurrence.
Reviewer: Andreas N. Philippou (Patras)Multi-point nonequilibrium umbrella sampling and associated fluctuation relations.https://zbmath.org/1459.822472021-05-28T16:06:00+00:00"Whitelam, Stephen"https://zbmath.org/authors/?q=ai:whitelam.stephenLimit theorem for the Robin Hood game.https://zbmath.org/1459.601152021-05-28T16:06:00+00:00"Angel, Omer"https://zbmath.org/authors/?q=ai:angel.omer"Matzavinos, Anastasios"https://zbmath.org/authors/?q=ai:matzavinos.anastasios"Roitershtein, Alexander"https://zbmath.org/authors/?q=ai:roitershtein.alexanderSummary: In its simplest form, the Robin Hood game is described by the following urn scheme: every day the Sheriff of Nottingham puts \(s\) balls in an urn. Then Robin chooses \(r\) \((r < s)\) balls to remove from the urn. Robin's goal is to remove balls in such a way that none of them are left in the urn indefinitely. Let \(T_n\) be the random time that is required for Robin to take out all \(s \cdot n\) balls put in the urn during the first \(n\) days. Our main result is a limit theorem for \(T_n\) if Robin selects the balls uniformly at random. Namely, we show that the random variable \(T_n \cdot n^{- s \slash r}\) converges in law to a Fréchet distribution as \(n\) goes to infinity.The law of logarithm for arrays of random variables under sub-linear expectations.https://zbmath.org/1459.600732021-05-28T16:06:00+00:00"Xu, Jia-pan"https://zbmath.org/authors/?q=ai:xu.jia-pan"Zhang, Li-xin"https://zbmath.org/authors/?q=ai:zhang.lixin|zhang.li-xinSummary: Under the framework of sub-linear expectation initiated by Peng, motivated by the concept of extended negative dependence, we establish a law of logarithm for arrays of row-wise extended negatively dependent random variables under weak conditions. Besides, the law of logarithm for independent and identically distributed arrays is derived more precisely and the sufficient and necessary conditions for the law of logarithm are obtained.On the law of large numbers for compositions of independent random semigroups.https://zbmath.org/1459.470162021-05-28T16:06:00+00:00"Sakbaev, V. Zh."https://zbmath.org/authors/?q=ai:sakbaev.vsevolod-zhSummary: We study random linear operators in Banach spaces and random one-parameter semigroups of such operators. For compositions of independent random semigroups of linear operators in the Hilbert space, we obtain sufficient conditions for fulfilment of the law of large numbers and give examples of its violation.Asymptotic entropy of the ranges of random walks on discrete groups.https://zbmath.org/1459.600932021-05-28T16:06:00+00:00"Chen, Xinxing"https://zbmath.org/authors/?q=ai:chen.xinxing"Xie, Jiansheng"https://zbmath.org/authors/?q=ai:xie.jiansheng"Zhao, Minzhi"https://zbmath.org/authors/?q=ai:zhao.minzhiSummary: Inspired by \textit{I. Benjamini} et al. [Ann. Inst. Henri Poincaré, Probab. Stat. 46, No. 4, 1080--1092 (2010; Zbl 1208.82046)] and \textit{D. Windisch} [Electron. J. Probab. 15, Paper No. 36, 1143--1160 (2010; Zbl 1226.60070)], we consider the entropy of the random walk ranges \(R_n\) formed by the first \(n\) steps of a random walk \(S\) on a discrete group. In this setting, we show the existence of \(h_R :={\lim}_{n\rightarrow\infty}\frac{H(R_n)}{n}\) called the asymptotic entropy of the ranges. A sample version of the above statement in the sense of \textit{C. E. Shannon} [Bell Syst. Tech. J. 27, 379--423, 623--656 (1948; Zbl 1154.94303)] is also proved. This answers a question raised by Windisch [loc. cit.]. We also present a systematic characterization of the vanishing asymptotic entropy of the ranges. Particularly, we show that \(h_R = 0\) if and only if the random walk either is recurrent or escapes to negative infinity without left jump. By introducing the weighted digraphs \(\Gamma_n\) formed by the underlying random walk, we can characterize the recurrence property of \(S\) as the vanishing property of the quantity \({\lim}_{n\rightarrow\infty}\frac{H(\Gamma_n)}{n}\) which is an analogue of \(h_R \).Multivariate central limit theorems for random simplicial complexes.https://zbmath.org/1459.600172021-05-28T16:06:00+00:00"Akinwande, Grace"https://zbmath.org/authors/?q=ai:akinwande.grace-itunuoluwa"Reitzner, Matthias"https://zbmath.org/authors/?q=ai:reitzner.matthiasSummary: Consider a Poisson point process within a convex set in a Euclidean space. The Vietoris-Rips complex is the clique complex over the graph connecting all pairs of points with distance at most \(\delta\). Summing powers of the volume of all \(k\)-dimensional faces defines the volume-power functionals of these random simplicial complexes. The asymptotic behavior of the volume-power functionals of the Vietoris-Rips complex is investigated as the intensity of the underlying Poisson point process tends to infinity and the distance parameter goes to zero. Univariate and multivariate central limit theorems are proven. Analogous results for the Čech complex are given.Oriented distance point of view on random sets.https://zbmath.org/1459.600202021-05-28T16:06:00+00:00"Dambrine, Marc"https://zbmath.org/authors/?q=ai:dambrine.marc"Puig, Bénédicte"https://zbmath.org/authors/?q=ai:puig.benedicteSummary: Motivated by free boundary problems under uncertainties, we consider the oriented distance function as a way to define the expectation for a random compact or open set. In order to provide a law of large numbers and a central limit theorem for this notion of expectation, we also address the question of the convergence of the level sets of \(f_n\) to the level sets of \(f\) when \((f_n)\) is a sequence of functions uniformly converging to \(f\). We provide error estimates in term of Hausdorff convergence. We illustrate our results on a free boundary problem.Current statistics and depinning transition for a one-dimensional Langevin process in the weak-noise limit.https://zbmath.org/1459.601782021-05-28T16:06:00+00:00"Tizón-Escamilla, Nicolás"https://zbmath.org/authors/?q=ai:tizon-escamilla.nicolas"Lecomte, Vivien"https://zbmath.org/authors/?q=ai:lecomte.vivien"Bertin, Eric"https://zbmath.org/authors/?q=ai:bertin.eric-mOn the law of the iterated logarithm without assumptions about the existence of moments.https://zbmath.org/1459.600722021-05-28T16:06:00+00:00"Petrov, V. V."https://zbmath.org/authors/?q=ai:petrov.valentin-vladimirovichSummary: Sufficient conditions are found for the applicability of the generalized law of the iterated logarithm for sums of random variables without conditions of independence and existence of moments.A Bernstein-type inequality for suprema of random processes with applications to model selection in non-Gaussian regression.https://zbmath.org/1459.600442021-05-28T16:06:00+00:00"Baraud, Yannick"https://zbmath.org/authors/?q=ai:baraud.yannickSummary: Let $(X_{t})_{t\in T}$ be a family of real-valued centered random variables indexed by a countable set $T$. In the first part of this paper, we establish exponential bounds for the deviation probabilities of the supremum $Z=\sup t_{t\in T} X_t$ by using the generic chaining device introduced in [\textit{M. Talagrand}, The generic chaining. Upper and lower bounds of stochastic processes. Berlin: Springer (2005; Zbl 1075.60001)]. Compared to concentration-type inequalities, these bounds offer the advantage to hold under weaker conditions on the family $(X_{t})_{t\in T}$. The second part of the paper is oriented towards statistics. We consider the regression setting $Y=f+\xi$ where $f$ is an unknown vector of $\mathbb R^{n}$ and $\xi$ is a random vector the components of which are independent, centered and admit finite Laplace transforms in a neighborhood of 0. Our aim is to estimate $f$ from the observation of $Y$ by mean of a model selection approach among a collection of linear subspaces of $\mathbb R^{n}$. The selection procedure we propose is based on the minimization of a penalized criterion the penalty of which is calibrated by using the deviation bounds established in the first part of this paper. More precisely, we study suprema of random variables of the form $X_{t}=\sum_{i=1}^{n}t_{i}\xi_{i}$ when $t$ varies among the unit ball of a linear subspace of $\mathbb R^{n}$. We finally show that our estimator satisfies some oracle-type inequality under suitable assumptions on the metric structures of the linear spaces of the collection.On the strong law of large numbers for sequences of pairwise independent random variables.https://zbmath.org/1459.600712021-05-28T16:06:00+00:00"Korchevsky, V. M."https://zbmath.org/authors/?q=ai:korchevsky.valery-mSummary: We establish new sufficient conditions for the applicability of the strong law of large numbers (SLLN) for sequences of pairwise independent, nonidentically distributed random variables. These results generalize \textit{N. Etemadi}'s extension of Kolmogorov's SLLN for identically distributed random variables [Z. Wahrscheinlichkeitstheor. Verw. Geb. 55, 119--122 (1981; Zbl 0438.60027)]. Some of the obtained results hold with an arbitrary norming sequence in place of the classical normalization.On the order of approximation in limit theorems for negative-binomial sums of strictly stationary \(m\)-dependent random variables.https://zbmath.org/1459.600612021-05-28T16:06:00+00:00"Hung, Tran Loc"https://zbmath.org/authors/?q=ai:hung.tran-loc"Kien, Phan Tri"https://zbmath.org/authors/?q=ai:kien.phan-triSummary: During the last several decades, the results related to geometric random sums of independent identically distributed random variables have become interesting results in probability theory and in insurance, risk theory, queuing theory and stochastic finance, etc. The negative-binomial random sums are extensions of geometric random sums. Up to the present, the negative-binomial random sums of independent identically distributed random variables have attracted much attention since they appear in many fields. However, for the strictly stationary \(m\)-dependent summands, such negative-binomial random sums are rarely studied. It seems that the complexity of the dependent structure has limited the use of classical tools based on independent random variables such as characteristic function. Up to now, a number of methods have been used to overcome the dependency such as the truncate method, Stein method, and Heinrich's method. The paper deals with the order of approximation in weak limit theorems for normalized negative-binomial sums of strictly stationary \(m\)-dependent random variables generated by a sequence of independent, identically distributed random variables, using the moving average techniques. The orders of approximation in desired theorems are established in terms of the so-called Zolotarev \(\zeta\)-metric. The obtained results are extensions and generalizations of several known results related to negative-binomial random sums and geometric random sums of independent, identically distributed random variables.On the convergence of a multidimensional workload in a service system to a stable process.https://zbmath.org/1459.600592021-05-28T16:06:00+00:00"Garai, E. S."https://zbmath.org/authors/?q=ai:garai.elena-sSummary: A service system model introduced by \textit{I. Kaj} and \textit{M. S. Taqqu} [Prog. Probab. 60, 383--427 (2008; Zbl 1154.60020)] is considered. We prove a limit theorem on the convergence of finite-dimensional distributions of the integral workload process with a multidimensional resource to the corresponding distributions of a multidimensional stable process.Functional central limit theorems for single-stage sampling designs.https://zbmath.org/1459.620132021-05-28T16:06:00+00:00"Boistard, Hélène"https://zbmath.org/authors/?q=ai:boistard.helene"Lopuhaä, Hendrik P."https://zbmath.org/authors/?q=ai:lopuhaa.hendrik-p"Ruiz-Gazen, Anne"https://zbmath.org/authors/?q=ai:ruiz-gazen.anneSummary: For a joint model-based and design-based inference, we establish functional central limit theorems for the Horvitz-Thompson empirical process and the Hájek empirical process centered by their finite population mean as well as by their super-population mean in a survey sampling framework. The results apply to single-stage unequal probability sampling designs and essentially only require conditions on higher order correlations. We apply our main results to a Hadamard differentiable statistical functional and illustrate its limit behavior by means of a computer simulation.Limit behavior of a compound Poisson process with switching.https://zbmath.org/1459.600762021-05-28T16:06:00+00:00"Borodin, A. N."https://zbmath.org/authors/?q=ai:borodin.andrei-nikollaevichSummary: The paper deals with the limit behavior of a compound Poisson process with switching. The switching is provided by Bernoulli's random variables. Under a suitable normalization, the limit process is a Brownian motion with switching variance.Estimation of the limit variance for sums under a new weak dependence condition.https://zbmath.org/1459.600632021-05-28T16:06:00+00:00"Manel, Kacem"https://zbmath.org/authors/?q=ai:manel.kacem"Véronique, Maume-Deschamps"https://zbmath.org/authors/?q=ai:veronique.maume-deschampsSummary: We prove a self-normalized central limit theorem for a mixing class of processes introduced in [the first author et al., Commun. Stat., Theory Methods 45, No. 5, 1241--1259 (2016; Zbl 1338.60070)]. This class is larger than more classical strongly mixing processes and thus our result is more general than [\textit{M. Peligrad} and \textit{Q.-M. Shao}, J. Multivariate Anal. 52, No. 1, 140--157 (1995; Zbl 0816.62027); \textit{S. Strongway}, Appl. Math., Ser. B (Engl. Ed.) 15, No. 1, 45--54 (2000; Zbl 0969.62057)] ones. The fact that some conditionally independent processes satisfy this kind of mixing properties motivated our study. We investigate the weak consistency as well as the asymptotic normality of the estimator of the variance that we propose.A hypothesis-testing perspective on the \(G\)-normal distribution theory.https://zbmath.org/1459.620272021-05-28T16:06:00+00:00"Peng, Shige"https://zbmath.org/authors/?q=ai:peng.shige"Zhou, Quan"https://zbmath.org/authors/?q=ai:zhou.quanSummary: The \(G\)-normal distribution was introduced by \textit{S. Peng} [Abel Symp. 2, 541--567 (2007; Zbl 1131.60057)] as the limiting distribution in the central limit theorem for sublinear expectation spaces. Equivalently, it can be interpreted as the solution to a stochastic control problem where we have a sequence of random variables, whose variances can be chosen based on all past information. In this note we study the tail behavior of the \(G\)-normal distribution through analyzing a nonlinear heat equation. Asymptotic results are provided so that the tail ``probabilities'' can be easily evaluated with high accuracy. This study also has a significant impact on the hypothesis testing theory for heteroscedastic data; we show that even if the data are generated under the null hypothesis, it is possible to cheat and attain statistical significance by sequentially manipulating the error variances of the observations.Area fluctuations on a subinterval of Brownian excursion.https://zbmath.org/1459.601722021-05-28T16:06:00+00:00"Meerson, Baruch"https://zbmath.org/authors/?q=ai:meerson.baruchPhase uniqueness for the Mallows measure on permutations.https://zbmath.org/1459.820992021-05-28T16:06:00+00:00"Starr, Shannon"https://zbmath.org/authors/?q=ai:starr.shannon-l"Walters, Meg"https://zbmath.org/authors/?q=ai:walters.megSummary: For a positive number \(q\), the Mallows measure on the symmetric group is the probability measure on \(\operatorname{S}_{n}\) such that \(P_{n,q}(\pi)\) is proportional to \(q\)-to-the-power-\(\mathrm{inv}(\pi)\) where \(\mathrm{inv}(\pi)\) equals the number of inversions: \(\mathrm{inv}(\pi)\) equals the number of pairs \(i < j\) such that \(\pi_{i} > \pi_{j}\). One may consider this as a mean-field model from statistical mechanics. The weak large deviation principle may replace the Gibbs variational principle for characterizing equilibrium measures. In this sense, we prove the absence of phase transition, i.e., phase uniqueness.{
\copyright 2018 American Institute of Physics}Random conductance models with stable-like jumps: heat kernel estimates and Harnack inequalities.https://zbmath.org/1459.602132021-05-28T16:06:00+00:00"Chen, Xin"https://zbmath.org/authors/?q=ai:chen.xin.1"Kumagai, Takashi"https://zbmath.org/authors/?q=ai:kumagai.takashi"Wang, Jian"https://zbmath.org/authors/?q=ai:wang.jian.2Summary: We establish two-sided heat kernel estimates for random conductance models with non-uniformly elliptic (possibly degenerate) stable-like jumps on graphs. These are long range counterparts of the well known two-sided Gaussian heat kernel estimates by \textit{M. T. Barlow} [Ann. Probab. 32, No. 4, 3024--3084 (2004; Zbl 1067.60101)] for nearest neighbor (short range) random walks on the supercritical percolation cluster. Unlike the cases of nearest neighbor conductance models, we cannot use parabolic Harnack inequalities since even elliptic Harnack inequalities do not hold in the present setting. As an application, we establish the local limit theorem for the models.Fluctuation scaling limits for positive recurrent jumping-in diffusions with small jumps.https://zbmath.org/1459.601662021-05-28T16:06:00+00:00"Yamato, Kosuke"https://zbmath.org/authors/?q=ai:yamato.kosuke"Yano, Kouji"https://zbmath.org/authors/?q=ai:yano.koujiSummary: For positive recurrent jumping-in diffusions with small jumps, we establish distributional limits of the fluctuations of inverse local times and occupation times on the half line. For this purpose, we introduce and utilize eigenfunctions with modified Neumann boundary condition and apply the Krein-Kotani correspondence.Exact and asymptotic properties of \(\delta \)-records in the linear drift model.https://zbmath.org/1459.601172021-05-28T16:06:00+00:00"Gouet, R."https://zbmath.org/authors/?q=ai:gouet.raul"Lafuente, M."https://zbmath.org/authors/?q=ai:lafuente.marianela"López, F. J."https://zbmath.org/authors/?q=ai:lopez-fernandez.francisco-jose|ariza-lopez.francisco-javier"Sanz, G."https://zbmath.org/authors/?q=ai:sanz.gerardoHigher variations for free Lévy processes.https://zbmath.org/1459.460622021-05-28T16:06:00+00:00"Anshelevich, Michael"https://zbmath.org/authors/?q=ai:anshelevich.michael"Wang, Zhichao"https://zbmath.org/authors/?q=ai:wang.zhichaoSummary: For a general free Lévy process, we prove the existence of its higher variation processes as limits in distribution, and identify the limits in terms of the Lévy-Itô representation of the original process. For a general free compound Poisson process, this convergence holds almost uniformly. This implies joint convergence in distribution to a \(k\)-tuple of higher variation processes, and so the existence of \(k\)-fold stochastic integrals as almost uniform limits. If the existence of moments of all orders is assumed, the result holds for free additive (not necessarily stationary) processes and more general approximants. In the appendix we note relevant properties of symmetric polynomials in non-commuting variables.Oscillating Gaussian processes.https://zbmath.org/1459.600822021-05-28T16:06:00+00:00"Ilmonen, Pauliina"https://zbmath.org/authors/?q=ai:ilmonen.pauliina"Torres, Soledad"https://zbmath.org/authors/?q=ai:torres.soledad"Viitasaari, Lauri"https://zbmath.org/authors/?q=ai:viitasaari.lauriSummary: In this article we introduce and study oscillating Gaussian processes defined by \(X_t=\alpha_+Y_t\mathbf{1}_{Y_t >0}+\alpha_-Y_t\mathbf{1}_{Y_t <0}\), where \(\alpha_+,\alpha_->0\) are free parameters and \(Y\) is either stationary or self-similar Gaussian process. We study the basic properties of \(X\) and we consider estimation of the model parameters. In particular, we show that the moment estimators converge in \(L^p\) and are, when suitably normalised, asymptotically normal.Graphons, permutons and the Thoma simplex: three mod-Gaussian moduli spaces.https://zbmath.org/1459.600072021-05-28T16:06:00+00:00"Féray, Valentin"https://zbmath.org/authors/?q=ai:feray.valentin"Méliot, Pierre-Loïc"https://zbmath.org/authors/?q=ai:meliot.pierre-loic"Nikeghbali, Ashkan"https://zbmath.org/authors/?q=ai:nikeghbali.ashkanThe paper under review is devoted to the theory of graphons, which are limit objects for sequences of graphs \((G_{n})\) on increasing numbers of vertices. Given a graphon \(\gamma\), one can construct a model of random graphs \((G_{n}(\gamma))\) which converges to \(\gamma\). Convergence of a sequence of graphs to a graphon is known, through, e.g., work of Lovász and co-authors, to be closely linked to convergence of (various definitions of) subgraph densities \(t(F,G_{n})\), informally the ratio of the number of occurrences of the fixed graph \(F\) in \(G_{n}\) to the maximum possible such number.
The main concern of the paper under review is to show that for a graphon \(\gamma\) and a fixed graph \(F\) the sequence \(t(F,G_{n}(\gamma))\) has the property that generically it is mod-Gaussian after a suitable renormalisation. Being mod-Gaussian is a property of a sequence \((X_{n})\) of random variables which can be defined in terms of the Laplace transforms of the random variables. If a sequence is mod-Gaussian, desirable consequences include a central limit theorem, moderate deviation principle, estimate of speed of convergence, local limit theorem and concentration inequality. One of the methods for proving that a sequence is mod-Gaussian is through considering estimates of cumulants. One of the estimates needed to get this approach to work can be approached through dependency graphs. The word ``generically'' above is a technical limitation of the method: for example, it transpires that the standard Erdős-Rényi random graphs are singular so this theory gives less insight there, the theory is best for models where there is less symmetry.
In addition to the results on graphons proved in the paper, there is a theory of permutons (limit objects for permutations) and limit objects for random integer partitions. The authors also prove several results based on the mod-Gaussian property for these random combinatorial objects. An application to the Ising model in statistical physics is also given.
Reviewer: David B. Penman (Colchester)An elementary renormalization-group approach to the generalized central limit theorem and extreme value distributions.https://zbmath.org/1459.600572021-05-28T16:06:00+00:00"Amir, Ariel"https://zbmath.org/authors/?q=ai:amir.arielScaling limit of a biased random walk on a critical branching random walk.https://zbmath.org/1459.602122021-05-28T16:06:00+00:00"Andriopoulos, George"https://zbmath.org/authors/?q=ai:andriopoulos.georgeSummary: Relying on powerful resistance techniques developed in \textit{D. A. Croydon} [Ann. Inst. Henri Poincaré, Probab. Stat. 54, No. 4, 1939--1968 (2018; Zbl 1417.60067)], the recent paper [the author, ``Invariance principles for random walks in random environment on trees'' \url{arXiv:1812.10197}] investigates random walks in random environment on tree-like spaces and their scaling limits in a certain regime, that is when the potential of the random walk in random environment converges. We introduce and summarise a result from the author [loc. cit.]. We choose to review the example of a novel scaling continuum limit of a biased random walk on large critical branching random walk. In this case the diffusion that is not on natural scale is identified as a Brownian motion on a continuum random fractal tree with its canonical metric replaced by a distorted resistance metric. This example allows the least technical presentation (compared to the others covered in the main article). Moreover, it is nevertheless of current interest, given its relation to critical percolation.Quenched lower large deviation for the first passage time of the frog model in random initial configurations.https://zbmath.org/1459.602032021-05-28T16:06:00+00:00"Kubota, Naoki"https://zbmath.org/authors/?q=ai:kubota.naokiSummary: We consider the so-called frog model with random initial configurations. The dynamics of this model are described as follows: Some particles are randomly assigned to any site of the multidimensional cubic lattice. Initially, only particles at the origin are active and these independently perform simple random walks. The other particles are sleeping and do not move at first. When sleeping particles are hit by an active particle, they become active and start moving in a similar fashion. In this paper, we study the behavior of the first passage time at which an active particle reaches a target site, and provide an upper bound on a quenched lower large deviation for the first passage time.Convergence rates for a class of estimators based on Stein's method.https://zbmath.org/1459.600642021-05-28T16:06:00+00:00"Oates, Chris J."https://zbmath.org/authors/?q=ai:oates.chris-j"Cockayne, Jon"https://zbmath.org/authors/?q=ai:cockayne.jon"Briol, François-Xavier"https://zbmath.org/authors/?q=ai:briol.francois-xavier"Girolami, Mark"https://zbmath.org/authors/?q=ai:girolami.mark-aSummary: Gradient information on the sampling distribution can be used to reduce the variance of Monte Carlo estimators via Stein's method. An important application is that of estimating an expectation of a test function along the sample path of a Markov chain, where gradient information enables convergence rate improvement at the cost of a linear system which must be solved. The contribution of this paper is to establish theoretical bounds on convergence rates for a class of estimators based on Stein's method. Our analysis accounts for (i) the degree of smoothness of the sampling distribution and test function, (ii) the dimension of the state space, and (iii) the case of non-independent samples arising from a Markov chain. These results provide insight into the rapid convergence of gradient-based estimators observed for low-dimensional problems, as well as clarifying a curse-of-dimension that appears inherent to such methods.Non-asymptotic bounds for percentiles of independent non-identical random variables.https://zbmath.org/1459.600552021-05-28T16:06:00+00:00"Xia, Dong"https://zbmath.org/authors/?q=ai:xia.dongSummary: This note displays an interesting phenomenon for the percentiles of independent but non-identical random variables. Let \(X_1, \ldots, X_n\) be independent random variables obeying non-identical continuous distributions and \(X^{(1)} \geq \cdots \geq X^{(n)}\) be the corresponding order statistics. For \(p \in(0, 1)\), we investigate the \(100(1 - p)\%\)th percentile \(X^{(\lfloor p n \rfloor)}\) and prove the non-asymptotic bounds for \(X^{(\lfloor p n \rfloor)}\). In particular, for a wide class of distributions, we discover an intriguing connection between their median and the harmonic mean of the associated standard deviations. For example, if \(X_k \sim \mathcal{N}(0, \sigma_k^2)\) for \(k = 1, \ldots, n\) and \(p = \frac{1}{2}\), we show that its median \(|\mathrm{Med}(X_1, \ldots, X_n) | = O_P(n^{1 \slash 2} \cdot(\sum_{k = 1}^n \sigma_k^{- 1})^{- 1})\) as long as \(\{\sigma_k \}_{k = 1}^n\) satisfy certain mild non-dispersion property.From statistical polymer physics to nonlinear elasticity.https://zbmath.org/1459.823472021-05-28T16:06:00+00:00"Cicalese, Marco"https://zbmath.org/authors/?q=ai:cicalese.marco"Gloria, Antoine"https://zbmath.org/authors/?q=ai:gloria.antoine"Ruf, Matthias"https://zbmath.org/authors/?q=ai:ruf.matthiasSummary: A polymer-chain network is a collection of interconnected polymer-chains, made themselves of the repetition of a single pattern called a monomer. Our first main result establishes that, for a class of models for polymer-chain networks, the thermodynamic limit in the canonical ensemble yields a hyperelastic model in continuum mechanics. In particular, the discrete Helmholtz free energy of the network converges to the infimum of a continuum integral functional (of an energy density depending only on the local deformation gradient) and the discrete Gibbs measure converges (in the sense of a large deviation principle) to a measure supported on minimizers of the integral functional. Our second main result establishes the small temperature limit of the obtained continuum model (provided the discrete Hamiltonian is itself independent of the temperature), and shows that it coincides with the \(\Gamma\)-limit of the discrete Hamiltonian, thus showing that thermodynamic and small temperature limits commute. We eventually apply these general results to a standard model of polymer physics from which we derive nonlinear elasticity. We moreover show that taking the \(\Gamma\)-limit of the Hamiltonian is a good approximation of the thermodynamic limit at finite temperature in the regime of large number of monomers per polymer-chain (which turns out to play the role of an effective inverse temperature in the analysis).A note on limit theory for mildly stationary autoregression with a heavy-tailed GARCH error process.https://zbmath.org/1459.621712021-05-28T16:06:00+00:00"Hwang, Eunju"https://zbmath.org/authors/?q=ai:hwang.eunjuSummary: A first-order mildly stationary autoregression with a heavy-tailed GARCH error process is considered to study the limit theory for the least squared estimator of the autoregression coefficient \(\rho = \rho_n \in [0, 1)\). A Gaussian limit theory is established as \(\rho_n\) converges to the unity as \(n \rightarrow \infty\), with rate condition \((1 - \rho_n) n \rightarrow \infty\), as in [\textit{L. Giraitis} and \textit{P. C. B. Phillips}, J. Time Ser. Anal. 27, No. 1, 51--60 (2006; Zbl 1114.62087)], who have discussed the limit theory in case that errors are martingale difference sequences. This work addresses asymptotic results in a case of heavy-tailed GARCH errors, and extends the existing one by allowing errors to follow heavy-tailed process as well as conditional heteroscedasticity.Theory of generalized discrepancies on a ball of arbitrary finite dimensions and algorithms for finding low-discrepancy point sets.https://zbmath.org/1459.460402021-05-28T16:06:00+00:00"Ishtiaq, Amna"https://zbmath.org/authors/?q=ai:ishtiaq.amna"Michel, Volker"https://zbmath.org/authors/?q=ai:michel.volker"Scheffler, Hans Peter"https://zbmath.org/authors/?q=ai:scheffler.hans-peter|scheffler.hans-peter.1Summary: One of the objectives of this paper is to extend the idea of the generalized discrepancy to the ball of arbitrary finite dimensions and to study its properties. We first construct orthonormal systems in higher dimensions, Sobolev spaces as well as particular differential operators. The discrepancy is then derived from an error estimate for numerical integration on a ball. We also present some new statistical and numerical properties of the generalized discrepancy which have also been unknown in three dimensions before. In addition, this paper focuses on constructing and implementing novel algorithms in order to obtain low-discrepancy point grids on the ball. Such low-discrepancy grids are uniformly distributed and are appropriate for quadrature rules (sometimes also called cubature rules) and as centres of localized trial functions in tomographic inverse problems on the ball. For this purpose, we also consider the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm and examine different BFGS updates and line search methods.The semi-infinite asymmetric exclusion process: large deviations via matrix products.https://zbmath.org/1459.600682021-05-28T16:06:00+00:00"González Duhart, Horacio"https://zbmath.org/authors/?q=ai:gonzalez-duhart.horacio"Mörters, Peter"https://zbmath.org/authors/?q=ai:morters.peter"Zimmer, Johannes"https://zbmath.org/authors/?q=ai:zimmer.johannesSummary: We study the totally asymmetric exclusion process on the positive integers with a single particle source at the origin. \textit{T. M. Liggett} [Trans. Am. Math. Soc. 213, 237--261 (1975; Zbl 0322.60086)] has shown that the long term behaviour of this process has a phase transition: If the particle production rate at the source and the original density are below a critical value, the stationary measure is a product measure, otherwise the stationary measure is spatially correlated. Following the approach of \textit{B. Derrida} et al. [J. Phys. A, Math. Gen. 26, No. 7, 1493--1517 (1993; Zbl 0772.60096)] it was shown by \textit{S. Grosskinsky} [Phase transitions in nonequilibrium stochastic particle systems with local conservation laws. Munich: TU Munich (PhD Thesis) (2004)] that these correlations can be described by means of a matrix product representation. In this paper we derive a large deviation principle with explicit rate function for the particle density in a macroscopic box based on this representation. The novel and rigorous technique we develop for this problem combines spectral theoretical and combinatorial ideas and is potentially applicable to other models described by matrix products.On asymptotic expansions in the ``interval'' CLT for sums of independent random vectors.https://zbmath.org/1459.600652021-05-28T16:06:00+00:00"Rozovsky, L. V."https://zbmath.org/authors/?q=ai:rozovskii.leonid-viktorovichSummary: We study the remainder term taking into account the asymptotic expansions in the multidimensional central limit theorem for sums of independent random vectors. The dependence of the remainder term on the measure of hitting set is studied.Quantum algorithmic randomness.https://zbmath.org/1459.810292021-05-28T16:06:00+00:00"Bhojraj, Tejas"https://zbmath.org/authors/?q=ai:bhojraj.tejasSummary: Quantum Martin-Löf randomness (q-MLR) for infinite qubit sequences was introduced by \textit{A. Nies} and \textit{V. B. Scholz} [J. Math. Phys. 60, No. 9, 092201, 11 p. (2019; Zbl 07116343)]. We define a notion of quantum Solovay randomness, which is equivalent to q-MLR. The proof of this goes through a purely linear algebraic result about approximating density matrices by subspaces. We then show that random states form a convex set. Martin-Löf absolute continuity is shown to be a special case of q-MLR. Quantum Schnorr randomness is introduced. A quantum analog of the law of large numbers is shown to hold for quantum Schnorr random states.
{\copyright 2021 American Institute of Physics}The inverse gamma distribution and Benford's law.https://zbmath.org/1459.600582021-05-28T16:06:00+00:00"Durst, Rebecca F."https://zbmath.org/authors/?q=ai:durst.rebecca-f"Huynh, Chi"https://zbmath.org/authors/?q=ai:huynh.chi"Lott, Adam"https://zbmath.org/authors/?q=ai:lott.adam"Miller, Steven J."https://zbmath.org/authors/?q=ai:miller.steven-j"Palsson, Eyvindur A."https://zbmath.org/authors/?q=ai:palsson.eyvindur-ari"Touw, Wouter"https://zbmath.org/authors/?q=ai:touw.wouter"Vriend, Gert"https://zbmath.org/authors/?q=ai:vriend.gertSummary: According to Benford's Law, many data sets have a bias towards lower leading digits (about 30\% are 1's). The applications of Benford's Law vary: from detecting tax, voter and image fraud to determining the possibility of match-fixing in competitive sports. There are many common distributions that exhibit such bias, i.e. they are almost Benford.
These include the exponential and the Weibull distributions. Motivated by these examples and the fact that the underlying distribution of factors in protein structure follows an inverse gamma distribution, we determine the closeness of this distribution to a Benford distribution as its parameters change.Complementary densities of Lévy walks: typical and rare fluctuations.https://zbmath.org/1459.822752021-05-28T16:06:00+00:00"Rebenshtok, A."https://zbmath.org/authors/?q=ai:rebenshtok.a"Denisov, S."https://zbmath.org/authors/?q=ai:denisov.s-v|denisov.s-i"Hänggi, P."https://zbmath.org/authors/?q=ai:hanggi.peter"Barkai, E."https://zbmath.org/authors/?q=ai:barkai.edi|barkai.eliSummary: Strong anomalous diffusion is a recurring phenomenon in many fields, ranging from the spreading of cold atoms in optical lattices to transport processes in living cells. For such processes the scaling of the moments follows \(\langle x| (t)|^q\rangle\sim t^{q\nu(q)}\) and is characterized by a bi-linear spectrum of the scaling exponents, \(q\nu(q)\). Here we analyze Lévy walks, with power law distributed times of flight \(\psi(\tau)\sim\tau^{-(1+\alpha)}\), demonstrating sharp bi-linear scaling. Previously we showed that for \(\alpha >1\) the asymptotic behavior is characterized by two complementary densities corresponding to the bi-scaling of the moments of \(x(t)\). The first density is the expected generalized central limit theorem which is responsible for the low-order moments \(0<q<\alpha\). The second one, a non-normalizable density (also called infinite density) is formed by rare fluctuations and determines the time evolution of higher-order moments. Here we use the Faà di Bruno formalism to derive the moments of sub-ballistic super-diffusive Lévy walks and then apply the Mellin transform technique to derive exact expressions for their infinite densities. We find a uniform approximation for the density of particles using Lévy distribution for typical fluctuations and the infinite density for the rare ones. For ballistic Lévy walks \(0<\alpha <1\) we obtain mono-scaling behavior which is quantified.Phase transition in random contingency tables with non-uniform margins.https://zbmath.org/1459.620902021-05-28T16:06:00+00:00"Dittmer, Samuel"https://zbmath.org/authors/?q=ai:dittmer.samuel-j"Lyu, Hanbaek"https://zbmath.org/authors/?q=ai:lyu.hanbaek"Pak, Igor"https://zbmath.org/authors/?q=ai:pak.igorSummary: For parameters \(n,\delta ,B\), and \(C\), let \(X=(X_{k\ell })\) be the random uniform contingency table whose first \(\lfloor n^{\delta } \rfloor\) rows and columns have margin \(\lfloor BCn \rfloor\) and the last \(n\) rows and columns have margin \(\lfloor Cn \rfloor \). For every \(0<\delta <1\), we establish a sharp phase transition of the limiting distribution of each entry of \(X\) at the critical value \(B_c=1+\sqrt{1+1/C} \). In particular, for \(1/2<\delta <1\), we show that the distribution of each entry converges to a geometric distribution in total variation distance whose mean depends sensitively on whether \(B<B_c\) or \(B>B_c\). Our main result shows that \(\mathbb{E}[X_{11}]\) is uniformly bounded for \(B<B_c\) but has sharp asymptotic \(C(B-B_c) n^{1-\delta }\) for \(B>B_c\). We also establish a strong law of large numbers for the row sums in top right and top left blocks.A Berry-Esseen theorem for Pitman's \(\alpha\)-diversity.https://zbmath.org/1459.600702021-05-28T16:06:00+00:00"Dolera, Emanuele"https://zbmath.org/authors/?q=ai:dolera.emanuele"Favaro, Stefano"https://zbmath.org/authors/?q=ai:favaro.stefanoSummary: This paper contributes to the study of the random number \(K_n\) of blocks in the random partition of \(\{1,\ldots,n\}\) induced by random sampling from the celebrated two parameter Poisson-Dirichlet process. For any \(\alpha \in (0,1)\) and \(\theta > -\alpha\) \textit{J. Pitman} [Combinatorial stochastic processes. LNM 1875. Berlin: Springer (2006; Zbl 1103.60004)] showed that \(n^{-\alpha} K_n \stackrel{\text{a.s.}}{\longrightarrow} S_{\alpha,\theta}\) as \(n \rightarrow +\infty\), where the limiting random variable, referred to as Pitman's \(\alpha\)-diversity, is distributed according to a polynomially scaled Mittag-Leffler distribution function. Our main result is a Berry-Esseen theorem for Pitman's \(\alpha\)-diversity \(S_{\alpha,\theta}\), namely we show that
\[
\text{sup}_{x \geq 0}\biggl\vert \mathsf{P}\biggl[\frac{K_n}{n^{\alpha}} \leq x \biggr]-\mathsf{P}[S_{\alpha,\theta}\leq x]\biggr\vert \leq\frac{C(\alpha,\theta)}{n^{\alpha}}
\]
holds for every \(n \in \mathbb{N}\) with an explicit constant term \(C(\alpha,\theta)\), for \(\alpha \in (0,1)\) and \(\theta >0\). The proof relies on three intermediate novel results which are of independent interest: (i) a probabilistic representation of the distribution of \(K_n\) in terms of a compound distribution; (ii) a quantitative version of the Laplace's approximation method for integrals; (iii) a refined quantitative bound for Poisson approximation. An application of our Berry-Esseen theorem is presented in the context of Bayesian nonparametric inference for species sampling problems, quantifying the error of a posterior approximation that has been extensively applied to infer the number of unseen species in a population.Eigenvectors of the 1-dimensional critical random Schrödinger operator.https://zbmath.org/1459.600132021-05-28T16:06:00+00:00"Rifkind, Ben"https://zbmath.org/authors/?q=ai:rifkind.ben"Virág, Bálint"https://zbmath.org/authors/?q=ai:virag.balintSummary: The purpose of this paper is to understand in more detail the shape of the eigenvectors of the random Schrödinger operator \(H = \Delta + V\) on \(\ell^2(\mathbb{Z})\). Here \(\Delta\) is the discrete Laplacian and \(V\) is a random potential. It is well known that under certain assumptions on \(V\) the spectrum of this operator is pure point and its eigenvectors are exponentially localized; a phenomenon known as Anderson Localization. We restrict the operator to \(\mathbb{Z}_n\) and consider the critical model,
\[
(H_n \psi)_\ell =\psi_{\ell-1,n} +\psi_{\ell+1,n} + v_{\ell,n} \psi_\ell,\quad \psi_0 = \psi_{n+1}=0,
\]
with \(v_k\) are independent random variables with mean 0 and variance \(\sigma^2/n\).
We show that the scaling limit of the shape of a uniformly chosen eigenvector of \(H_n\) is
\[
\exp \left(- \frac{|t-U|}{4} + \frac{\mathcal{Z}_{t-U}}{\sqrt{2}} \right),
\]
where \(U\) is uniform on \([0,1]\) and \(\mathcal{Z}\) is an independent two sided Brownian motion started from 0.Large deviations for random projections of \(\ell^{p}\) balls.https://zbmath.org/1459.600672021-05-28T16:06:00+00:00"Gantert, Nina"https://zbmath.org/authors/?q=ai:gantert.nina"Kim, Steven Soojin"https://zbmath.org/authors/?q=ai:kim.steven-soojin"Ramanan, Kavita"https://zbmath.org/authors/?q=ai:ramanan.kavitaSummary: Let \(p\in [1,\infty]\). Consider the projection of a uniform random vector from a suitably normalized \(\ell^{p}\) ball in \(\mathbb{R}^{n}\) onto an independent random vector from the unit sphere. We show that sequences of such random projections, when suitably normalized, satisfy a large deviation principle (LDP) as the dimension \(n\) goes to \(\infty\), which can be viewed as an annealed LDP. We also establish a quenched LDP (conditioned on a fixed sequence of projection directions) and show that for \(p\in (1,\infty]\) (but not for \(p=1\)), the corresponding rate function is ``universal,'' in the sense that it coincides for ``almost every'' sequence of projection directions. We also analyze some exceptional sequences of directions in the ``measure zero'' set, including the sequence of directions corresponding to the classical Cramér's theorem, and show that those sequences of directions yield LDPs with rate functions that are distinct from the universal rate function of the quenched LDP. Lastly, we identify a variational formula that relates the annealed and quenched LDPs, and analyze the minimizer of this variational formula. These large deviation results complement the central limit theorem for convex sets, specialized to the case of sequences of \(\ell^{p}\) balls.