On the local time of random processes in random scenery
The authors summarize the contents of this paper in the abstract as follows: ``Random walks in random scenery are processes defined by \(Z_{n}:=\sum_{k=1}^{n}\xi_{X_{1}+\dots+X_{k}}\), where basically \((X_{k},k\geq1)\) and \((\xi_{y},y\in\mathbb{Z})\) are two independent sequences of i.i.d.\ random variables. We assume here that \(X_{1}\) is \(\mathbb{Z}\)-valued, centered and with finite moments of all orders. We also assume that \(\xi_{0}\) is \(\mathbb{Z}\)-valued, centered and square integrable. In this case; \textit{H. Kesten} and \textit{F. Spitzer} [Z. Wahrscheinlichkeitstheor. Verw. Geb. 50, 5--25 (1979; Zbl 0396.60037)] proved that \((n^{-3/4}Z_{[nt]},t\geq0)\) converges in distribution as \(n\to\infty\) toward some self-similar process \((\Delta_{t},t\geq0)\) called Brownian motion in random scenery. In a previous paper, we established that \(\operatorname{P}(Z_{n}=0)\) behaves asymptotically like a constant times \(n^{-3/4}\), as \(n\to\infty\). We extend here this local limit theorem: we give a precise asymptotic result for the probability for \(Z\) to return to zero simultaneously at several times. As a byproduct of our computations, we show that \(\Delta\) admits a bi-continuous version of its local time process which is locally Hölder continuous of order \(1/4-\delta\) and \(1/6-\delta\), respectively, in the time and space variables, for any \(\delta>0\). In particular, this gives a new proof of the fact, previously obtained by \textit{D. Khoshnevisan} [Lect. Notes Math. 1832, 236--245 (2003; Zbl 1040.60087)], that the level sets of \(\Delta\) have Hausdorff dimension a.s.\ equal to \(1/4\). We also get the convergence of every moment of the normalized local time of \(Z\) toward its continuous counterpart.''
scientific article; zbMATH DE number 6369479