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Recently, a number of limit theorems related to $\\tau_u$ were proved including the law of large numbers, the central limit theorem and large deviations theorems. We obtain a precise asymptotics of the sequence ${\\mathbb P}[\\tau_u = \\log u/\\rho ]$, $\\rho >0$, $u\\to \\infty $ which considerably improves the previous results. 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Recently, a number of limit theorems related to $\\tau_u$ were proved including the law of large numbers, the central limit theorem and large deviations theorems. We obtain a precise asymptotics of the sequence ${\\mathbb P}[\\tau_u = \\log u/\\rho ]$, $\\rho >0$, $u\\to \\infty $ which considerably improves the previous results. 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