## 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 $$S_n$$.

### MSC:

 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:

### References:

  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  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  Bennett, G. (1962). Probability inequalities for the sum of independent random variables. J. Amer. Statist. Assoc. 57 33–45. · Zbl 0104.11905  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  Erdős, P. (1942). On the law of the iterated logarithm. Ann. Math. 43 419–436. JSTOR: · Zbl 0063.01264  Hoeffding, W. (1963). Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc. 58 13–30. · Zbl 0127.10602  Khoshnevisan, D. (2003). Brownian sheet and quasi-sure analysis. · Zbl 1093.60053  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  Malliavin, P. (1979). Régularité de lois conditionnelles et calcul des variations stochastiques. C. R. Acad. Sci. Paris Sér. A-B 289 . · Zbl 0417.60077  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  Mountford, T. S. (1992). Quasi-everywhere upper functions. Séminaire de Probabilités XXVI. Lecture Notes in Math. 1526 95–106. Springer, Berlin. · Zbl 0765.60079  Neuhaus, G. (1971). On weak convergence of stochastic processes with multidimensional time parameter. Ann. Math. Statist. 42 1285–1295. · Zbl 0222.60013  Pickands, J. III (1967). Maxima of stationary Gaussian processes. Z. Wahrsch. Verw. Gebiete 7 190–223. · Zbl 0158.16702  Qualls, C. and Watanabe, H. (1971). An asymptotic 0–1 behavior of Gaussian processes. Ann. Math. Statist. 42 2029–2035. · Zbl 0239.60031  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  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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.