Asymptotic independence of three statistics of the maximal increments of random walks and Levy processes

Abstract: Let $H(x) = \inf\{n:\, \exists\, k x\}$ be the first epoch that an increment of the size larger than $x>0$ of a random walk $S$ occurs and consider the path functionals: $$ R_n = \max_{m\in\{0, \ldots, n\}}\{S_{n} - S_m\}, R_n^* = \max_{m,k\in\{0, \ldots, n\}, m\leq k} \{S_{k}-S_m\} \text{and} O_x=R_{H(x)}-x.$$ The main result states that, under Cram\'{e}r's condition on the step-size distribution of $S$, the statistics $R_n$, $R_n^* -y$ and $O_{x+y}$ are asymptotically independent as $\min\{n,y,x\}\uparrow\infty$. Furthermore, we establish a novel Spitzer-type identity characterising the limit law $O_\infty$ in terms of the one-dimensional marginals of $S$. If $y=\gamma^{-1}\log n$, where $\gamma$ is the Cram\'er coefficient, our results together with the classical theorem of Iglehart (1972) imply the existence of a joint weak limit of the three statistics and identify its law. As corollary we obtain a new factorization of the exponential distribution as a convolution of the asymptotic overshoot $O_\infty$ and the stationary distribution of the reflected random walk $R$. We prove analogous results for the corresponding statistics of a L\'{e}vy process. This is joint work with M. Pistorius.