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 $

W. Szczotka¹

¹ Institute of Mathematics University of Wroc/law Pl. Grunwaldzki 2/4 50-384 Wroc/law, Poland

Applicationes Mathematicae, Tome 33 (2006) no. 1, pp. 41-49

Résumé

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 $X=X^B-X^A$.

Détail

DOI : 10.4064/am33-1-3

Keywords: each geq geq geq mutually independent sequences nonnegative random variables each consist mutually independent identically distributed random variables means overline overline respectively t sum overline t sum overline geq n b x main result gives conditions under which weak convergence mathrel mathop rightarrow limits where process implies mathrel mathop rightarrow limits mathrel mathop rightarrow limits where mutually independent processes b x