Convergence of the \(p\)-series for stationary sequences
Let \((X_n)_n\) be a stationary sequence and consider for \(1< p<\infty\) the non-integrable function, known as the \(p\)-series, \[ G_p(x)= (p-1)^{1/p} \Biggl(\sum^\infty_{i= 1} \Biggl|{X_i(x)\over i}\Biggr|^p\Biggr)^{1/p}. \] The author studies the limit \(\lim_{p\to 1^+} G_p(x)\). This \(p\)-series is associated with the maximal function \[ N^*(x)= \sup_n {1\over n}\# \Biggl\{k; {| X_k(x)|\over k}\geq {1\over n}\Biggr\}. \] In the first part of the paper, the author proves that if \((X_n)_n\) is an i.i.d. sequence of nonnegative random variables with \(E(X_1)< \infty\), then \hskip 17mm (a) \(\lim_{n\to\infty} {N_n(x)\over n}= E(X_1)\) (a.e.), where \(N_n(x)= \#\{k; X_k(x)/k\geq 1/n\}\). \hskip 17mm (b) \(\lim_{p\to 1^+} G_p(x)= E(X_1)\) (a.e.). In the second part of the paper, the author extends this result to the case \(X_k= f(T^kx)\), where \((X,{\mathcal F},\mu, T)\) is an ergodic dynamical system, and \(f\geq 0\), \(f\in L(\log L)\). He shows that in this case one has that \(\sup_{1< p<\infty} G_p(x)\) is an integrable function.
scientific article; zbMATH DE number 1144086