Maxima of partial sums indexed by geometrical structures (Q1872311)

Let \((X_i)_{i\in \mathbb{N}^d}\) be a family of i.i.d. random variables with \(E(X_i)<0\), \(P(X_i> 0)>0\), and \(E(e^{tX_i}) <\infty\) for \(t>0\). For a rectangle or cube \(\Delta\subset \Delta_n:=\{1, \dots,n\}^d\) consider \(S_\Delta:= \sum_{i\in \Delta} X_i\) and \(W_n:=\max_{\Delta \subset\Delta_n} S_\Delta\). The author derives several limit theorems for the random variables \(W_n\) where for the dimensions \(d=1\), \(d=2\) and \(d\geq 3\) sometimes a different limit behavior appears. The results generalize the case \(d=1\), which is already treated in the classical books of W. Feller and F. Spitzer; they are motivated by possible applications in protein folding. Proofs are based on large deviation techniques, the Chen-Stein method, and inequalities for empirical processes.