Large deviations of Shepp statistics (Q2882307)

Let \(X_1,X_2,\dots,\) be i.i.d. zero mean and nondegenerate random variables on some probability space \(\Omega\) satisfying \(R(h)= Ee^{hX_1}<\infty\) \((0< h< h^+)\). Consider a random walk \(S_n= \sum^n_{i=1} X_i\), \(S_0= 0\) with steps \(X_1,X_2\dots\). Let \(S_{n,m}= S_{n+m}- S_m\). Asymptotics of large deviations of \(M_n=\max_{1\leq i\leq n}S_i\) are known. Define the Shepp statistic NEWLINE\[NEWLINE\rho_{n,m}= \max_{k\leq m}\,M_{n,k},NEWLINE\]NEWLINE where \(M_{n,k}= \max_{l\leq n}S_{l,k}\) \((M_{n,k}=^dM_n)\).NEWLINENEWLINE In this paper, among others, the author proves that NEWLINE\[NEWLINEP(M_{n,m}\geq \theta n- u|M_{n,i}< \theta n- u, i= 1,\dots, m-1)\sim p(\theta)P(M_n\geq \theta n- u)NEWLINE\]NEWLINE holds uniformly on \(\theta\in[\theta_1, \theta_2]\subset (0,m^+)\) and \(|u|\leq\gamma_n= o(\sqrt{n})\), where \(p(\theta)\) is a complicated but clear form and using this result obtains the probability \(P(\rho_{n,m}\geq \theta n- u)\) under some restriction on \(m\).