Limit Theorems for Stochastic Processes

Let us consider a sequence of processes ξ ₙ (t) such that the multivariate distribution of ξ ₙ (t₁ ),ξ ₙ (t₂ ), ⋯ ,ξ ₙ (tₖ ) tends to the multivariate distribution of ξ ₀ (t₁ ),ξ ₀ (t₂ ), ⋯ ,ξ ₀ (tₖ ) for all k and t₁ ,t₂ , ⋯ ,tₖ. Let f be the functional for which f(ξ ₙ (t)) are determined with a probability of 1, the latter being random variables (i.e, those that have probability distributions). This paper contains several sufficient conditions, for which the distributions of f(ξ ₙ (t)) tend to the distribution of f(ξ ₀ (t)) as n → ∞. Let K be the space of all functions not having discontinuities higher than simple jumps, and let us assume that ξ ₙ (t) with a probability of 1 is in K. Several topologies in K are defined. The necessary and sufficient conditions are found for all functionals f that are continuous in these topologies for which the distribution of f(ξ ₙ (t)) tends to the distribution of f(ξ ₀ (t)). The results are demonstrated in the example of topology J₁ which is defined as follows. The sequence xₙ (t) tends to x₀ (t) in topology J₁ if there exists a sequence of monotonic continuous functions λ ₙ (t) for which${gathered} λ _n (0),λ _n (1) = 1, {lim }n → ∞ { }_t | {λ _n (t) - t} | = 0, \\ {lim }n → ∞ { }_t | {x_n ( λ _n (t)) - t} | = 0. \\ {gathered} $ Theorem. The distribution of$f(ξ _n (t))$tends to the distribution of$f(ξ _0 (t))$for allfthat are continuous in topology${ J}_1 $, if and only if a) the multivariate distribution of$ξ _n (t_1 ), ⋯ ,ξ _n (t_k )$tends to the multivariate distribution of$ξ _0 (t_1 ), ⋯ ,ξ _0 (t_k )$for allk, and$t_1 ,t_2 , ⋯ ,t_k $from some setNthat is dense on$[0,1]$. b) for all$ε > 0$\[ {lim }c → 0 {lim ̄ }n → 0 P\{ { { }t - c < t_1 < t < t_2 < t + c min [ {| {ξ _n ( {t_1 } ) - ξ _n ( t )} |;| {ξ _n ( t ) - ξ _n ( {t_0 } )} |} ] > ε } \} = 0. \]