Rate of convergence in a~boundary problem

K. A. Borovkov

Teoriâ veroâtnostej i ee primeneniâ, Tome 27 (1982) no. 1, pp. 141-142

Résumé

Let $\{\xi_i\}_{i\ge 1}$and $\{\tau_i\}_{i\ge 1}$ be the sequences of i. i . d. r. v. 's such that $\tau_1\ge 0$ a. s., $\mathbf E\xi_1=0$, $\mathbf E\xi_1^2=\mathbf E\tau_1=1$ and $$ S_n=\sum_{i=1}^n\xi_i,\quad T_n=\sum_{i=1}^n\tau_i,\quad\nu(t)=max\{k\ge 0:\,T_k\le t\}. $$ We investigate the rate of convergence \begin{gather*} \mathbf P\{g^-(n^{-1}T_k)^{-1/2}S_k^+(n^{-1}T_k),\ k\le \nu(n)\}\to\\ \to\mathbf P\{g^-(t)(t)^+(t),\ 0\le t\le 1\},\qquad n\to\infty \end{gather*} where $w(t)$ is a standard Wiener process and $g^\pm(t)$ are Lipschitz functions.