The Erdős-Rényi-Shepp law of large numbers for ballistic random walk in random environment (Q2080813)

A one-dimensional ballistic nearest-neighbor random walk in a random environment is considered. Let \[ S_n=X_1+\cdots +X_n \] be the \(n\)th partial sum of an i.i.d. sequence of random variables. The Erdős-Rényi-Shepp law describes the limiting behavior of \[ T_n=\max_{1\leq i\leq n} (S_{i+\kappa(i)}-S_i), \] \[ U_n=\max_{1\leq i\leq n} (S_{i+k}-S_i), \] \[ W_n=\max_{1\leq i\leq n-k}\max_{1\leq j\leq k} (S_{i+j}-S_i), \] and \[ V_n=\max_{1\leq i\leq n-k}\max_{1\leq j\leq k}(k/j)(S_{i+j}-S_i), \] for \[ k=\kappa(n)=[c\log n], \] and where \(c>0\) is a given constant. The authors prove an analogous statement for standard one-dimensional random walk in random environment (RWRE) and obtain among other results the full form of the Erdős-Renyi [\textit{P. Erdős} and \textit{A. Rényi}, J. Anal. Math. 23, 103--111 (1970; Zbl 0225.60015)] and \textit{L. A. Shepp} [Ann. Math. Stat. 35, 424--428 (1964; Zbl 0146.39101)] theorems.