On dynamical Gaussian random walks. (English) Zbl 1090.60066

Let \((\omega_j)\) be iid random variables and associate to each \(\omega_j\) independently Poisson processes with intensity \(1\); the jump times of the \(j\)th processes are denoted by \(\tau_j(k)\), \(k\in\mathbb N\). Finally, write \((\omega_j^k)_{j,k\in\mathbb N}\) for an array of independent copies of the \(\omega_j\)’s. Define a process \(X=\{X_j(t), t\geq 0\}_{j\in\mathbb N}\) by \[ X_j(0) := \omega_j, \qquad X_j(t) := \omega_j^k\quad\forall\,t\in [\tau_j(k),\tau_j(k+1)). \] A dynamical random walk is then given as the partial sum \(S_n(t) := X_1(t)+\cdots+X_n(t)\). Throughout this paper it is assumed that the distribution of \(\omega_1\) is standard normal. Denote by \(U_t^n(s) := S_{[ns]}(t) / \sqrt n\) the rescaled dynamical walk. The main theorems of this paper concern results on the convergence behaviour of the two-parameter field \(U_t^n(s)\) as \(n\to\infty\).
Firstly, as \(n\to\infty\) the random fields \(U^n_t(s)\) converge weakly in \(D([0,1]^2)\) to the continuous centered Gaussian random field \(U_t(s)\) with covariance function \[ \mathbb E(U_t(s)U_{t'}(s')) = \exp(-|t-t'|)\min(s,s'),\quad s,t,s',t'\in [0,1]. \] Here \(D([0,1]^2)\) is the two-parameter Skorokhod-type space consisting of all càdlàg-functions on \([0,1]^2\) where we have a partial ordering on \([0,1]^2\) induced by \((s,t)\prec (s',t')\) if \(s\leq s'\) and \(t\leq t'\); this construction is due to G. Neuhaus [Ann. Math. Stat. 42, 1285–1295 (1971; Zbl 0222.60013)]. The limiting random field can be interpreted as \(U_t(s) = e^{-t} B(s,e^{2t})\) where \(B\) is a standard Brownian sheet. Another way to see \(\{U_t(\cdot)\}_{t}\) is to interpret it as Ornstein-Uhlenbeck process in classical Wiener space. The above theorem, therefore, gives a construction of this process.
The authors prove then a two-sided maximal inequality (large-deviation-type result) estimating \(\mathbb P(\sup_t S_n(t) \geq z_n\sqrt n)\) below and above by (a constant times) \(z_n^2\bar\Phi(z_n)\) where \(z_n\) is a fixed sequence converging to infinity and satisfying \(z_n = o(\sqrt{n/\log n})\) and \(\bar\Phi = 1-\Phi\), \(\Phi\) being the standard normal cdf. This result has a path-by-path consequence which furnishes an analogue of Erdős’ integral test. Set for any positive measurable function \(H\) \[ J(H) = \int_1^\infty H^4(t)\,\bar\Phi(H(t))\frac{dt}{t}. \] Then, with probability one, \(J(H)<\infty\) implies that \( \sup_t S_n(t)< H(n)\sqrt n\) for finally all \(n\), while \(J(H)=\infty\) entails that there is some \(t_0\) such that \(S_n(t_0) \geq H(n)\sqrt n\) for infinitely many \(n\). This immediately yields a \(\log\log\log\)-type result for \(S_n\).


60J25 Continuous-time Markov processes on general state spaces
60J05 Discrete-time Markov processes on general state spaces
60F10 Large deviations
60G17 Sample path properties
28C20 Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.)


Zbl 0222.60013
Full Text: DOI arXiv


[1] Bass, R. F. and Pyke, R. (1987). A central limit theorem for \(D(A)\)-valued processes. Stochastic Process. Appl. 24 109–131. · Zbl 0617.60020
[2] Benjamini, I., Häggström, O., Peres, Y. and Steif, J. (2003). Which properties of a random sequence are dynamically sensitive? Ann. Probab. 31 1–34. · Zbl 1021.60055
[3] Bennett, G. (1962). Probability inequalities for the sum of independent random variables. J. Amer. Statist. Assoc. 57 33–45. · Zbl 0104.11905
[4] Bickel, P. J. and Wichura, M. J. (1971). Convergence criteria for multiparamater stochastic processes and some applications. Ann. Math. Statist. 42 1656–1670. · Zbl 0265.60011
[5] Erdős, P. (1942). On the law of the iterated logarithm. Ann. Math. 43 419–436. JSTOR: · Zbl 0063.01264
[6] Hoeffding, W. (1963). Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc. 58 13–30. · Zbl 0127.10602
[7] Khoshnevisan, D. (2003). Brownian sheet and quasi-sure analysis. · Zbl 1093.60053
[8] Kuelbs, J. (1973). Some results for probability measures on linear topological vector spaces with an application to Strassen’s \(\log\log\) law. J. Funct. Anal. 14 28–43. · Zbl 0292.60007
[9] Malliavin, P. (1979). Régularité de lois conditionnelles et calcul des variations stochastiques. C. R. Acad. Sci. Paris Sér. A-B 289 . · Zbl 0417.60077
[10] Meyer, P.-A. (1982). Note sur les processus d’Ornstein–Uhlenbeck (Appendice: Un resultat de D. Williams). Séminaire de Probabilités XVI. Lecture Notes in Math. 920 95–133. Springer, Berlin. · Zbl 0481.60041
[11] Mountford, T. S. (1992). Quasi-everywhere upper functions. Séminaire de Probabilités XXVI. Lecture Notes in Math. 1526 95–106. Springer, Berlin. · Zbl 0765.60079
[12] Neuhaus, G. (1971). On weak convergence of stochastic processes with multidimensional time parameter. Ann. Math. Statist. 42 1285–1295. · Zbl 0222.60013
[13] Pickands, J. III (1967). Maxima of stationary Gaussian processes. Z. Wahrsch. Verw. Gebiete 7 190–223. · Zbl 0158.16702
[14] Qualls, C. and Watanabe, H. (1971). An asymptotic 0–1 behavior of Gaussian processes. Ann. Math. Statist. 42 2029–2035. · Zbl 0239.60031
[15] Straf, M. L. (1972). Weak convergence of stochastic processes with several parameters. Proc. Sixth Berkeley Symp. Math. Statist. Probab. 2 187–221. Univ. California Press, Berkeley. · Zbl 0255.60019
[16] Walsh, J. B. (1986). An introduction to stochastic partial differential equations. École d’Été de Probabilités de Saint-Flour XIV–1984 . Lecture Notes in Math. 1180 . Springer, Berlin. · Zbl 0608.60060
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