Limit Theorems for Stochastic Processes

Let us consider a sequence of processes ₙ (t) such that the multivariate distribution of ₙ (t₁), ₙ (t₂), , ₙ (tₖ) tends to the multivariate distribution of ₀ (t₁), ₀ (t₂), , ₀ (tₖ) for all k and t₁, t₂, , tₖ. Let f be the functional for which f (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 (ₙ (t) ) tend to the distribution of f (₀ (t) ) as n. Let K be the space of all functions not having discontinuities higher than simple jumps, and let us assume that 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 (ₙ (t) ) tends to the distribution of f (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 ₙ (t) for whichgathered ₙ (0), ₙ (1) = 1, ₍ ₜ | ₙ (t) - t | = 0, \\ ₍ ₜ | xₙ (ₙ (t) ) - t | = 0. \\ gathered Theorem. The distribution off (ₙ (t) ) tends to the distribution off (₀ (t) ) for allfthat are continuous in topology J₁, if and only if a) the multivariate distribution of ₙ (t₁), , ₙ (tₖ) tends to the multivariate distribution of ₀ (t₁), , ₀ (tₖ) for allk, andt₁, t₂, , tₖ from some setNthat is dense on0, 1. b) for all > 0\ ₂ ₀ { }₍ ₀ P\ { { ₓ - ₂ \} = 0. \