Favourite sites, favourite values and jump sizes for random walk and Brownian motion (Q5933564)

Let \(L(t,x)\) be the local time at \(x\) up to time \(t\) of a simple symmetric random walk \(\{S_s\}_{s\in\mathbb{N}}\) or a one-dimensional standard Brownian motion \(\{B_s\}_{s\geq 0}\) and consider the largest favourite (= most visited) site up to time \(t\) NEWLINE\[NEWLINEU(t)= \sup_x \Biggl\{x: L(t, x)= \sup_y L(t,y)\Biggr\}.NEWLINE\]NEWLINE The study of the asymptotic behaviour of \(U\) was initiated by \textit{P. Erdős} and \textit{P. Révész} [in: Mathematical structures, computational mathematics, mathematical modelling 2, 152-157 (1984; Zbl 0593.60072)] and \textit{R. F. Bass} and \textit{P. S. Griffin} [Z. Wahrscheinlichkeitstheorie Verw. Gebiete 70, 417-436 (1985; Zbl 0554.60076)] where (among other results) a law of the iterated logarithm-type theorem for \(U\) and the maximum local time \(L^*(t)= \sup_x L(t,x)\) was proved. In the present paper the authors extend these results as follows: Let \(\varphi(t)= (2t\log\log t)^{1/2}\).NEWLINENEWLINENEWLINE(1) The limit set of \(\{(U(t)/\varphi(t); L^*(t)/\varphi(t)): t\geq 3\}\) as \(t\to\infty\) is almost surely the simplex \(\{(x; y): y\geq 0,|x|+ y\leq 1\}\).NEWLINENEWLINENEWLINE(2) One has almost surely \(\limsup_{t\to\infty} {U(t)- U(t-)\over \varphi(t)}= 1\) where ``\(t-\)'' stands for the left limit at (for \(t\in\mathbb{N}\): predecessor of) \(t\).NEWLINENEWLINENEWLINENote that the upper bound in (2) follows readily from the usual law of the iterated logarithm for random walks. The proofs of (1), (2) are based around a (known) approximation theorem that says that one can approximate Brownian motion itself and its local time simultaneously through the corresponding discrete objects for a simple random walk. This allows to reduce all calculations to local times of a one-dimensional Brownian motion. Crucial ingredients are highly technical, sharp estimates for \(d\)-dimensional Brownian motion \(B_t\), its local time \(L(t,x)\), \(d\)-dimensional Bessel processes \(R_t\sim\|B_t\|\) and various functionals thereof. These estimates are only used for \(d=2\) but the theorems are stated and proved in any dimension \(d\).