The critical case of the Cramer–Lundberg theorem on the asymptotic tail behavior of the maximum of a negative drift random walk

We study the asymptotic tail behavior of the maximum $M=\max\{0,S_n,n\geqslant1\}$ of partial sums $S_n=\xi_1+\dots+\xi_n$ of independent identically distributed random variables $\xi_1,\xi_2,\dots$ with negative mean. We consider the so-called Cramer case when there exists a $\beta>0$ such that $\mathbf Ee^{\beta\xi_1}=1$. The celebrated Cramer–Lundberg approximation states the exponential decay of the large deviation probabilities of $M$ provided that $\mathbf E\xi_1e^{\beta\xi_1}$ is finite. In the present article we basically study the critical case $\mathbf E\xi_1e^{\beta\xi_1}=\infty$.