A limit theorem for measurable random processes and its applications

Abstract

Let the measurable random processes ξ 1 ( t ) , … , ξ n ( t ) , … {\xi _1}(t), \ldots ,{\xi _n}(t), \ldots and ξ ( t ) \xi (t) be defined on [ 0 , 1 ] [0,\;1] . There exists C C such that for all n n and t t we have E | ξ n ( t ) | p ⩽ C , p ⩾ 1 E|{\xi _n}(t){|^p} \leqslant C,\;p \geqslant 1 . The following assertion is valid: if for any finite set of points t 1 , … , t k ⊂ [ 0 , 1 ] {t_1}, \ldots ,{t_k} \subset [0,\;1] the joint distribution of ξ n ( t 1 ) , … , ξ n ( t k ) {\xi _n}({t_1}), \ldots ,{\xi _n}({t_k}) converges to the joint distribution of ξ ( t 1 ) , … , ξ ( t k ) \xi ({t_1}), \ldots ,\xi ({t_k}) , and if E | ξ n ( t ) | p → E | ξ ( t ) | p E|{\xi _n}(t){|^p} \to E|\xi (t){|^p} for all t ∈ [ 0 , 1 ] t \in [0,\;1] , then for any continuous functional f f on L p [ 0 , 1 ] {L_p}[0,\;1] the distribution of f ( ξ n ( t ) ) f({\xi _n}(t)) converges to the distribution of f ( ξ ( t ) ) f(\xi (t)) . This statement immediately implies the convergence of distributions in some limit theorems for the sums of independent random variables (for example, in one of the theorems of P. Erdös and M. Kac) and in some statistical criteria (for example, in the ω 2 {\omega ^2} -criterion of Cramér and von Mises).