On the properties for increments of a local time -- a look through the set of limit points (Q5937056)

Let \(l(x,t)\) be a local time for the standard Brownian motion. The article contains the analogue of the functional law of iterated logarithm for the increments of \(l(x,t)\). Denote by \(K\) the classical Strassen limit set and define \(L_t(s)=l(0,t+s)-l(0,t).\) For any \(h\in (0,1)\) and \(t\in [0,1-h]\) denote NEWLINE\[NEWLINEI_{t,h}(x)=L_t(hx)/(h\log h^{-1})^{0.5},\quad 0\leq x\leq 1.NEWLINE\]NEWLINE One of the main results is the following Theorem 1.1: NEWLINE\[NEWLINE\lim_{h\to 0}\sup_{0\leq t\leq {1-h}}\inf_{k\in K}\|I_{t,h}-k\|_\infty =0\quad\text{a.s.}NEWLINE\]NEWLINE and, for each \(k\in K,\) NEWLINE\[NEWLINE\lim_{h\to 0}\inf_{0\leq t\leq{1-h}}\|I_{t,h}-k\|_{\infty}=0\quad\text{a.s.}NEWLINE\]NEWLINE The comparison with known results is presented.