We study the exact asymptotics for the distribution of the first time$\tau_x$ a L\'evy process $X_t$ crosses a negative level $-x$. We prove that$\mathbf P(\tau_x>t)\sim V(x)\mathbf P(X_t\ge 0)/t$ as $t\to\infty$ for acertain function $V(x)$. Using known results for the large deviations of randomwalks we obtain asymptotics for $\mathbf P(\tau_x>t)$ explicitly in both lightand heavy tailed cases. We also apply our results to find asymptotics for thedistribution of the busy period in an M/G/1 queue.
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机译：我们研究了第一次过程$ \ tau_x $的精确渐近性L'evy过程$ X_t $越过负值$ -x $。我们证明$ \ mathbf P（\ tau_x> t）\ sim V（x）\ mathbf P（X_t \ ge 0）/ t $作为$ t \ to \ infty $的确定函数$ V（x）$使用已知的随机游走偏差的结果，我们可以得出在轻尾和重尾情况下$ \ mathbf P（\ tau_x> t）$的渐近性。我们还应用我们的结果找到M / G / 1队列中忙碌时段分布的渐近性。
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