Weak convergence of mutually independent $X_n^B$ and $X_n^A$ under weak convergence of $ X_n\equiv X_n^B-X_n^A $
Volume 33 / 2006
Abstract
For each $n\geq 1,$ let $\{ v_{n,k}, k\geq 1\} $ and $\{ u_{n,k}, k\geq 1\} $ be mutually independent sequences of nonnegative random variables and let each of them consist of mutually independent and identically distributed random variables with means $\overline {v}_n$ and $\overline {u}_n,$ respectively. Let $X_n^B(t)={(1/ c_n)}\sum _{j=1}^{[nt]}(v_{n,j} -\overline {v}_n),$ $X_n^A(t)={(1/ c_n)}\sum _{j=1}^{[nt]}(u_{n,j}-\overline {u}_n),\ t\geq 0,$ and $ X_n=X_n^B-X_n^A.$ The main result gives conditions under which the weak convergence $X_n\mathrel {\mathop {\rightarrow }\limits ^ { D}}X,$ where $X$ is a Lévy process, implies $X_n^B\mathrel {\mathop {\rightarrow }\limits ^{ D}}X^B$ and $X_n^A\mathrel {\mathop {\rightarrow }\limits ^{ D}}X^A,$ where $X^B$ and $ X^A$ are mutually independent Lévy processes and $X=X^B-X^A$.
