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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 $\\mathbf{P}(M^{(a)}=y\\Delta)$ as $a\\to 0$ in the case, when $y\\to \\infty$ and $ay=O(1)$. 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