Local limit theorem for the maximum of a random walk

Consider a family of $Δ$-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 $\sup_{a\le a_0}\mathbf{E}[\max\{0,X^{(a)}\}^{2+\varepsilon}]<\infty$ for some $\varepsilon>0$. Assume that $X^{(a)}\xrightarrow[]{w} X^{(0)}$ 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 $\mathbf{P}(M^{(a)}=yΔ)$ as $a\to 0$ in the case, when $y\to \infty$ and $ay=O(1)$. This asymptotics follows from a representation of $\mathbf{P}(M^{(a)}=yΔ)$ via a geometric sum and a uniform renewal theorem, which is also proved in this paper.

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