Moments of the superdiffusive elephant random walk with general step distribution (Q2081106)

The elephant random walk is the following simple discrete-time random walk with memory. Here, the authors consider a general i.i.d.\ sequence \((\xi_n)_{n\in\mathbb N}\) of step sizes. Depending on a memory parameter \(\alpha\in[0,1]\), the elephant random walk \((X_n)_{n\in\mathbb N}\) is recursively given by \(X_1=\xi_1\) and \[ X_{n+1}=\begin{cases}X_{U(n)} & \text{with probability } \alpha,\\ \xi_{n+1} & \text{with probability } 1-\alpha,\end{cases} \] where \(U(n)\) is uniformly distributed on \(\{1,\ldots,n\}\) and independent of \(X_1,\ldots,X_n\). Generalizing earlier results for special step size distributions, the authors prove that in the superdiffusive regime \(\alpha\in(\frac12,1]\) the centered random walk \(\sum_{k=1}^nX_k-n\mathbb E[\xi_1]\) normalized by \(n^\alpha\) converges almost surely (provided \(E[\xi_1^2]<\infty\)) and in \(L^p\) (provided \(E[|\xi_1|^p]<\infty\)) to a non-degenerate random variable \(Q\) for general step size distributions. The proof relies on \(L^p\)-boundedness of an underlying martingale difference. The authors further calculate the first four moments of \(Q\) in terms of the centered first four moments of \(\xi_1\) by solving recursive equations of mixed centered moments of the step sizes.