A limit result for \(U\)-statistics of binary variables (Q1266779)

Let \(X_1,X_2,\dots\) be a sequence of independent and identically distributed binary random variables taking the values 1 and \(-1\). This paper considers symmetric \(U\)-type statistics of the form \(U_{k,n}=\sum_{1\leq i_1\neq i_2\neq \dots\neq i_k\leq n}X_{i_1}\cdots X_{i_k}\) based on these binary variables. An analogue of the almost sure decomposition for \(U_{k,n}\) given by \textit{H. Teicher} and \textit{C.-H. Zhang} [ibid. 9, No. 1, 161-170 (1996; Zbl 0837.60033)] is established. This result is then applied to obtain a result akin to the law of the iterated logarithm for this class of statistics.