On the Central Limit Theorem for Multiparameter Stochastic Processes
نویسنده
چکیده
l.INTRODUCTION AND RESULTS In recent papers Bezandry and Fernique (1990,1992), Fernique (1993) have given new convergence and tightness criteria for random processes whose sample paths are right-continuous and have leftlimits. These criteria have been applied by Bezandry and Fernique, Bloznelis and Paulauskas to prove the central limit theorem (CLT) in the Skorohod space D[0, 1]. In this paper, using recent technique of Bezandry and Fernique, we improve some results of Bickel and Wichura (1971) on weak convergence and tightness for multiparameter processes. The main results of the paper deals with stochastically continuous processes and may be viewed as an extension to multidimensional case of the weak convergence criteria due to Bezandry and Fernique (1990,1992) and of the CLT due to Bloznelis and Paulauskas (1993), Fernique (1993). Let X,X1, X2, ... be i.i.d. random processes with sample paths in Skorohod space Dk ≡ D([0, 1], R). For details about the space Dk endowed with the Skorohod topology we refer to Neuhaus (1971) and Straf (1972). Denote Sn = n−1/2(X1 + . . . + Xn − nEX). A random process X is said to satisfy the CLT in Dk (X ∈ CLT (Dk)) if the distributions of Sn converge weakly to a Gaussian distribution on Dk. For a random process X = {X(t), t ∈ [0, 1]}, k ≥ 1 define
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