We consider the first descending ladder epoch $\tau =min\{n\ge 1:S_n\le 0\}$ of a random walk $S_n={\sum }_1^n{\xi }_i,n\ge 1$ with i.d.d. summands having a negative drift $\mathbb{E}\xi =-a<0$. Let ${\xi }^{+}=max(0,{\xi }_1)$. It is well-known that, for any $\alpha >1$, the finiteness of $\mathbb{E}{\left({\xi }^{+}\right)}^{\alpha }$ implies the finiteness of $\mathbb{E}{\tau }^{\alpha }$ and, for any $\lambda >0$, the finiteness of $\mathbb{E}exp\left(\lambda {\xi }^{+}\right)$ implies that of $\mathbb{E}exp\left(c\tau \right)$ where $c>0$ is, in general, another constant that depends on the distribution of ${\xi }_1$. We consider the intermediate case, assuming that $\mathbb{E}exp\left(g\left({\xi }^{+}\right)\right)<\infty$ for a positive increasing function $g$ such that ${lim\; inf}_{x\to \infty }g\left(x\right)/logx=\infty$ and ${lim\; sup}_{x\to \infty }g\left(x\right)/x=0$, and that $\mathbb{E}exp\left(\lambda {\xi }^{+}\right)=\infty$, for all $\lambda >0$. Assuming a few further technical assumptions, we show that then $\mathbb{E}exp\left((1-\varepsilon )g\left((1-\delta )a\tau \right)\right)<\infty$, for any $\varepsilon ,\delta \in (0,1)$.

我们考虑第一个下降阶梯时期$\tau =分钟\{n\ge 1：{\mathrm{小号}}_n\le 0\}$随机游走${\mathrm{小号}}_n={\sum }_1^n{\xi }_{\mathrm{一世}},n\ge 1$具有负漂移的 idd 和数$\mathbb{乙}\xi =-\mathrm{一个}<0$. 让${\xi }^{+}=最大限度(0,{\xi }_1)$. 众所周知，对于任何$\alpha >1$, 的有限性$\mathbb{乙}{\left({\xi }^{+}\right)}^{\alpha }$意味着有限性$\mathbb{乙}{\tau }^{\alpha }$并且，对于任何$\lambda >0$, 的有限性$\mathbb{乙}经验\left(\lambda {\xi }^{+}\right)$意味着$\mathbb{乙}经验\left(C\tau \right)$在哪里$C>0$是，一般来说，另一个常数，取决于分布${\xi }_1$. 我们考虑中间情况，假设$\mathbb{乙}经验\left(G\left({\xi }^{+}\right)\right)<\infty$为正增函数$G$这样${林inf}_{X\to \infty }G\left(X\right)/日志X=\infty$和${林燮}_{X\to \infty }G\left(X\right)/X=0$， 然后$\mathbb{乙}经验\left(\lambda {\xi }^{+}\right)=\infty$， 对所有人$\lambda >0$. 假设一些进一步的技术假设，我们证明了$\mathbb{乙}经验\left((1-\epsilon )G\left((1-\delta )\mathrm{一个}\tau \right)\right)<\infty$, 对于任何$\epsilon ,\delta \in (0,1)$.