On the distribution of the maximum cumulative sum of independent random variables

T. V. Arak

Teoriâ veroâtnostej i ee primeneniâ, Tome 19 (1974) no. 2, pp. 257-277

Résumé

Let $X_1,\dots,X_n$ be independent random variables with $\mathbf EX_k=0$ and $\mathbf E|x_k|^3=\gamma_{3k}\infty$. Let $$ S_0=0,\quad S_k=\sum_{i=1}^kX_i\quad(k=1,\dots,n),\quad\overline{S_n}=\max\limits_{0\le k\le n}S_k,\quad B_k^2=\sum_{i=1}^k\mathbf DX_i. $$ In the paper, some bounds for $$ \Delta_n(x)=\mathbf P\{\overline{S_n}\}-\sqrt{\frac2\pi}\int_0^{x/B_n}e^{-y^2/2}\,dy\quad(x\ge0) $$ are obtained. The main result is the following Theorem. {\em Let $x\ge0$. Then $$ |\Delta_n(x)|\le C\sum_{k=1}^n\frac{x+\rho_k}{x+\rho_k+B_k}\cdot\frac{B_n\gamma_{3k}}{(B_k^2+x^2)(B_n+x)B_{k-1,n}} $$ where $\rho_k=\max\limits_{i\le k}\gamma_{3i}/\mathbf DX_i$ and} $B_{k-1,n}=(\sum_{i=k}^n\mathbf DX_i)^{1/2}$.