Limit Theorems for Stochastic Processes with Independent Increments

The general results in [8] are used for the case of convergence of processes with independent increments. In particular the following results are obtained: 2.6. Theorem. Let the distributions of processes with independent increments ξ ₙ (t) converge to the distribution of a continuous probability process with independent increments ξ ₀ (t) for all t. Then, there exists an xₙ (t), such that the distribution f(ξ ₙ (t) - xₙ (t)) converges to the distribution f(ξ ₀ (t)) if the functional f is continuous in the J₁-topology (see [8]). 3.4. Theorem. Let ξ n,1 , ⋯ ,ξ n,n be independent random variables with, the same distributions, and also let η n,1 , ⋯ ,η n,n be independent random variables with the same distributions: \[ ξ _n ( t ) = ∑i t(n + 1) {ξ n,i } , η _n (t) = ∑i t(n + 1) {η n,i } . \] Further, let distributions ξ ₙ (t) and η ₙ (t) converge to the distribution ξ ₀ (t) for all t. Then, the Lévy distance between distribution functions of random variables f(ξ ₙ (t)) and f(η ₙ (t)) tends to zero as n → ∞, for all functional f, such that \[ {lim }δ → 0 { }_{ { }_t | {x(t) - y(t)} | δ } | {f(x(t)) - f(y(t))} | = 0. \]