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Selected papers. Edited by Manfred Denker and Edward C. Waymire. (English) Zbl 1378.60010

Contemporary Mathematicians. Basel: Birkhäuser/Springer (ISBN 978-3-319-30188-4/hbk; 978-3-319-30190-7/ebook). xxi, 711 p. (2016).
The book under review is to attribute to the author in this volume of selected papers, and to represent the author’s main contributions into three parts, modes of approximation, large times for Markov processes, and stochastic foundations in applied sciences. The volume offers analytic probability theory, statistic theory, Markov processes, random dynamic systems, and applied topics in statistics, economics and geophysics with the author’s selected papers. It describes the author’s contributions, influences and related to some recent research works by his former students, colleagues and former collaborators. There are total six parts varying from theoretic results to applied works, from central limit theory, normal approximation and asymptotic expansions for stochastic processes to ergodicity for Markov processes and martingale methods for the central limit theorem, from the random dynamic systems, IID random iterations and Markov chains, stability analysis for random dynamic systems in economics to cascade representations for the Navier-Stokes equations, nonparametric statistical methods on manifolds. This volume can be very useful for researchers in probability and statistics, as well as those practitioners in economics, geophysics and statistics. Some articles provide a quick survey in certain topics, and state connections with earlier contributions of the author, and can be read by advanced graduate students in their related research. The other target audience consists of theoretic probability PhD students and researchers, stochastic processes and random dynamics researchers and applied practicers, as well as who nonetheless want to acquire the related knowledge rapidly and begin using results in their field.
In Part I, Hall first surveys the author’s PhD thesis, to have a Berry-Esseen type rate of convergence in the multidimensional central limit theorem under rather general moment conditions and to his new method of describing distributions of sums of independent random vectors in terms of the characteristic functions of those sums. The author [Ann. Math. Stat. 42, 241–259 (1971; Zbl 0224.60014)] first estimated the error of normal approximation for arbitrary continuous function for standard \(k\)-dimensional normal distribution and give the explicit rate of the weak convergence, and provided an asymptotic expansion for the weak limit with a remainder term \(o(n^{-(s-2)/2})\) uniformly over all continuous functions and \(s\geq 3\) for multidimensional central limit theorems for i.i.d random vectors. The general Edgeworth expansions of the author and J. K. Ghosh [Ann. Stat. 6, 434–451 (1978; Zbl 0396.62010)], just as importantly the methods by means of which those expansions were derived play a major role in establishing the credentials of bootstrap for distribution estimation and for constructing confidence intervals and hypothesis tests. It settles a conjecture of D. L. Wallace [Ann. Math. Stat. 29, 635–654 (1958; Zbl 0086.34004)] to have the asymptotic expansion is to be identical with a formal Edgeworth expansion of the distribution function.
The asymptotic efficiency in estimation theory is evaluated by the asymptotic variance of estimators, and in testing statistical hypotheses the critical region of a test is determined by the normal approximation. The asymptotic expansion can be realized as basis of various branches of theoretic statistics like higher-order inferential theory, prediction, model selection, resampling methods, information geometry. Yoshida’s survey emphasizes on central limit theory and asymptotic expansion applied to statistics for semimartingales and Markov chains (nonlinear time series models). The asymptotic expansion gives higher-order approximation of the distribution of the \(d\)-dimensional sum of i.i.d random vectors, the method even goes back to Chebyshev, Edgeworth and Cramer, the Cramer condition is effective to deduce the decay of the charateristic function of the sum. In [Ann. Probab. 5, 28–41 (1977; Zbl 0362.60041)], T. J. Sweeting showed a smoothing inequality for the difference of a finite measure and a finite signed measure by the integrability of their Fourier transforms and estimation of the gap between them. The author and his collaborators provided ergodicity of multidimensional diffusion processes and related limit theorems. Under assumption of mixing property, F. Goetze and C. Hipp gave asymptotic expansions for sums of weakly dependent processes that are approximated by a Markov chain [Z. Wahrscheinlichkeitstheor. Verw. Geb. 64, 211–239 (1983; Zbl 0497.60022)]. The Markovian property in practice plays an essential role in estimation of the characteristic function of an additive functional of the underlying process. Since typical statistics are expressed as a [Bhattacharya and Ghosh, 1978, loc. cit.] transform of a multidimensional additive functional that admits the Edgeworth expansion, one can obtain Edgeworth expansions for them, this enables future researchers to construct higher-order statistics for stochastic processes. There are central limit theorems for martingales, local martingals and semi-martingales. For martingales with jumps, a uniformity condition such as the conditional type Lindeberg condition is necessary to obtain central limit theorem. From aspects of limits theorem, the notion of stable convergence is fundamental since the Fisher information is random even in the limit.
Shao presents recent developments on normal approximation by Stein’s method and strong Gaussian approximation, and introduced the normal approximation by first stating the author’s (1977) results on approximated density and the author’s (1975) Berry-Essen type inequality for the multidimensional central limit theory. The results proved by Stein’s method are mainly the first-order approximation. It is still unclear whether Stein’s method can be used to prove an Edgeworth expansion. Chapter 4 consists of five papers by the author and his coauthors.
In Part II, Large time asymptotics for Markov processes, Varadhan discusses the central limit theorem for Markov chains via martingale methods. Markov proved the central limit theorem for sums of stationary Markov chains consisting of leaving large gaps to create enough independence but not large enough to make a difference in the sum, Paul Lévy observed that if the summands form a stationary sequence and the partial sums is a martingale relative to the natural filtration then the central limit theorem is valid under virtually no additional conditions. In [Sov. Math., Dokl. 19, 392–394 (1978; Zbl 0395.60057); translation from Dokl. Akad. Nauk SSSR 239, 766–767 (1978)], M. I. Gordin and B. A. Lifshits provide a central limit theorem for functions of Markov chains using explicitly the martingale approximation, and the author extended this to continuous time Markov processess [Z. Wahrscheinlichkeitstheor. Verw. Geb. 60, 185–201 (1982; Zbl 0468.60034)].
The finite dimensional distributions of a Markov process are determined by its initial distribution and its transition function. Kurtz reviews ideas of ergodicity of Markov processes by proving the existence of stationary distributions for the process, and gives the central study on the uniqueness of stationary distributions in terms of Harris recurrence. It is necessary to study ergodicity for processes without Harris recurrent. The author provided a splitting, and G. K. Basak and the author provided asymptotic flatness [Ann. Probab. 20, No. 1, 312–321 (1992; Zbl 0749.60073)]. In [Ann. Probab. 13, 385–396 (1985; Zbl 0575.60075)], the author considered diffusions under the assumption that diffusion terms is the square root of a positive definite matrix, the diffusion (mod 1) is a Markov process which has a unique ergodic stationary distribution under additional regularity assumption. Chapter 7 collects 6 papers of the author on criteria for recurrence and existence of invariant measures, functional central limit theorem and the law of the iterative logarithm for Markov processes, central limit theorem for diffusions with periodic coefficients and almost periodic coefficients, stability in distribution and the speed of convergence to equilibrium and to normality.
In Part III, Dynamic systems and iterated maps, Athreya first introduces dynamic systems, IID random iterations, Markov chains, to get criteria for the random i.i.d dynamic sequence to be Harris irreducible and a law of large numbers, a central limit theorem for null recurrent Markov chains and a Brownian motion, and an application to Monte Carlo methods for estimating integrals with respect to improper measures. Waymire further explains the random dynamic systems and selected works of the author, emphasizes on existence and uniqueness of invariant measures under conditions in which the Markov process may not be irreducible. In the absence of irreducibility, even ergodic Markov chains can have a nontrivial tail sigmafield, and fail to have good mixing properties. For existence, uniqueness and stability, the author considered a representation by iterations of i.i.d. random maps. In [J. Theor. Probab. 12, No. 4, 1067–1087 (1999; Zbl 0961.60064)], the author and M. Majumdar developed an important splitting method for the unique invariant measure. The role of splitting is best understood in terms of H. Furstenberg’s backward iteration [Trans. Am. Math. Soc. 108, 377–428 (1963; Zbl 0203.19102)], a powerful idea introduced for i.i.d products of random matrices. The essential feature of nondecreasing maps on an interval is that the backward iteration are respectively nondecreasing and nonincreasing sequences, therefore having limits and squeezing all other such iterates. The author and Majumdar [1999, loc. cit.] showed that splitting was not necessary for the existence of unique invariant probability when the maps are assumed nonincreasing. In [Ann. Probab. 16, No. 3, 1333–1347 (1988; Zbl 0652.60028)], the author and O. Lee proved a functional central limit theorem with weak convergence to Brownian motion. The reciprocity between the bounds on the “random Lipschitz coefficient” of P. Diaconis and D. Freedman [SIAM Rev. 41, No. 1, 45–76 (1999; Zbl 0926.60056)] and the “random Lyapounov growth rate” of the homeomorphic inverse map, implicit in A. Brandt’s theorems [Adv. Appl. Probab. 18, 211–220 (1986; Zbl 0588.60056)] is not an accident but occurs for a more general class of non-irreducible Markov processes. Chapter 10 contains 4 papers of the author from asymptotics of a class of Markov processes, random iterations of two quadratic maps, existence of unique invariant measures for Markov processes and Dubins and Freedman theorem. In Part IV, Stochastic foundations in applied sciences. I: Economics, Kamihigashi and Stachurski review two standard economic models in optimal growth model and the models of production and growth with overlapping generations. Seminal work by the author and his coauthors showed that for existence, uniqueness, and stability in models such as this one sector stochastic optimal growth model, continuity of the optimal policy is unnecessary. In [in: Limit theorems in probability and statistics. Fourth Hungarian colloquium on limit theorems in probability and statistics, Balatonlelle, Hungary, June 28–July 2, 1999. Vol I. Budapest: János Bolyai Mathematical Society. 181–200 (2002; Zbl 1033.60078)], the author and E. C. Waymire introduced weaker mixing conditions maintaining an order theoretic flavor to study local splitting conditions in conjunction with a recurrence condition ensuring drift back to the set where splitting occurs. In [Sankhyā, Ser. A 72, No. 1, 170–190 (2010; Zbl 1209.60016)], the author et al. showed that when the overlapping generations model satisfies the splitting conditions, the equilibrium capital stock process satisfies central limit theorem. Roy further explains the dynamic resource allocation problems, and stated that the important problem for the economist is to understand the asymptotic behavior of this dynamic system and how it depends on initial conditions, the nature of the transition function and the distribution of shocks. In models of optimal economic growth under uncertainty, the optimal policy function of a stochastic stationary dynamic optimization problem of maximizing the expected discounted sum of returns from consumption or resource harvests over time. The author and his coauthors have developed various sufficient conditions for the existence of a unique and stable invariant distribution for a Markov process defined by i.i.d. random monotone maps on some suitably defined subset. Chapter 13 has two papers of the author and his coauthors. In Part V, Stochastic foundations in applied science. II: Geophysics, Thomann and Waymire provide some overview and context for the salient features of the author and V. K. Gupta [SIAM J. Appl. Math. 37, 485–498 (1979; Zbl 0417.60080); ibid. 44, 33–39 (1984; Zbl 0537.76078)] which stands out for the important theoretical insights provided to contemporary understanding of the advection-dispersion in fluid media over a range of space-time scales. Deeper understanding of distinguished particle systems has been a subject of considerable interest from the point of view of hydrodynamic limits. The author, Gupta and Sposito (1981) expressed a vision with regard to multi-scaling phenomena being reported, and considered two spatial scales of heterogeneity embodied in the flow velocity. The essential parameters that determine the model of advection-dispersion of solute concentrations are a drift rate and a dispersion rate. These rates can be used to described the phenomena in different ways (i) to define local fluxes of solute concentration or (ii) to define local mean and variance-covariances of stochastic displacements of individual particles. Flandoli and Romito presents the simple fluid motions, the classical Navier-Stokes equations, with direct cascade. In [C. R. (Dokl.) Acad. Sci. URSS, n. Ser. 30, 301–305 (1941; JFM 67.0850.06)], A. Kolmogoroff introduced the K41 theory (the very innovative ideas) on the averaged flow are superposed the pulsation of the first order consisting in disorderly displacements of separate fluid volumes and turbulence. The Fourier formulation of the Navier-Stokes equations is still obscure. In [in: Probability and partial differential equations in modern applied mathematics. Selected papers presented at the 2003 IMA summer program. New York, NY: Springer. 27–40 (2005; Zbl 1104.35027)], the author et al. made some advanced results on Picard iteration and deterministic cascade representation, and Y. Le Jan and A. S. Sznitman provided a stochastic cascade representation [Probab. Theory Relat. Fields 109, No. 3, 343–366 (1997; Zbl 0888.60072)]. The solution of the system is represented by the expectation of a recursive functional over a tree of branching particles. A branching event triggers the multiplication by two functionals corresponding to the two branches rooted at integer lattice and a death event the evaluation of the external force. The Navier-Stokes equation in dimension three fits into the general scheme in the stochastic cascade representation. The author and his coauthors contributed to the analysis with a generalization of the stochastic cascade with a degree of freedom of conceptual importance (majorizing kernels). The comparison equation for the infinite dimensional ODE system is obtained essentially by neglecting any geometric information about the directions of the data in the system. Chapter 16 organizes 5 papers of the author and his group.
In Part VI, Stochastic foundations in applied sciences. III: Statistics, Dryden, Le, Preston and Wood first introduce the construction of confidence regions and multi-sample tests via bootstrap methods, and gave the algorithms for bootstrapping confidence region and bootstrapping test for equality of means as well as nonpivotal bootstrapping confidence region. They illustrated the idea of using geodesics to model the variability of the data leading to the concept of principal geodesic component analysis, various Riemannian geometry concepts are developed from parallel transport to transfer data to tangent spaces to the method of solving a nonparametric smoothing problem on the sphere, from the the Jacobi field to the method of principal flows.
Huckemann and Hotz further review intrinsic and extrinsic means on manifolds and problems involved, and recall the statistics of shape from Aristotle’s student Theophrast of Eresos discussing tree shape and speed of growth. Statistical analysis of shape can be approached by parameters describing either shape or size, and it can be approached by taking an underlying data space into a Euclidean space modulo a group action. The naive quotient gives a non-Hausdorff space, all configurations can be rescaled to arbitrary size, in any neighborhood of the shape of configurations with all landmarks coinciding, a dead end to statistical ambition. In generalize procrustes analysis, a procrustes means and a corresponding (co)variance are defined. In [Ann. Stat. 31, No. 1, 1–29 (2003; Zbl 1020.62026); ibid. 33, No. 3, 1225–1259 (2005; Zbl 1072.62033)], the author and V. Patrangenaru introduced the Bhattacharya and Patrangenaru strong law and Bhattacharya-Patrangearu asymptotic central limit theorem which covers many important cases, cases with non-manifold data-spaces are not covered, and introduced the intrinsic means and moments of a probability measure on a Riemannian manifold. This work develops nonparametric statistical inference procedures for measures of location of distributions on general manifolds, nonparametric inference procedure for estimation and testing problems for means on manifolds. They later analyze the asymptotic theory for extrinsic mean and intrinsic mean on Riemannian manifolds. For readers who are interested in the related topics, we recommend to read those original articles to see the insights and developments of concepts and theorems. There are many classical results in asymptotic expansions, central limit theorem, random dynamic system and stochastic foundation of the author and his coauthors’ work. This volume is a very important contribution to the asymptotic expansions. This book will be a valuable source for anyone who is interested in the asymptotic expansion theory, and will be a reference book for years to come.

MSC:

60-03 History of probability theory
01A75 Collected or selected works; reprintings or translations of classics
01A70 Biographies, obituaries, personalia, bibliographies
01A60 History of mathematics in the 20th century
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