Local limit theorem for the maximum of a random walk

Abstract

Consider a family of $\Delta$-latticed aperiodic random walks $\{S^{(a)},0\le a\le a_0\}$ with increments $X_i^{(a)}$ and non-positive drift $-a$. Suppose that ${\sup }_{a\le a_0}\mathbf{E}[(X^{(a)}{)}^2]<\infty$ and ${\sup }_{a\le a_0}\mathbf{E}[\max \{0,X^{(a)}{\}}^{2+\epsilon }]<\infty$ for some $\epsilon >0$. Assume that $X^{(a)}\stackrel{w}{\to }{X}^{(0)}$ as $a\to 0$ and denote by $M^{(a)}={\max }_{k\ge 0}{S}_k^{(a)}$ the maximum of the random walk $S^{(a)}$. In this paper we provide the asymptotics of $\mathbf{P}(M^{(a)}=y\Delta )$ as $a\to 0$ in the case, when $y\to \infty$ and $ay=O(1)$. This asymptotics follows from a representation of $\mathbf{P}(M^{(a)}=y\Delta )$ via a geometric sum and a uniform renewal theorem, which is also proved in this paper.

19 pages