Let , 0<aT[less-than-or-equals, slant]T<[infinity], and {W(t);0[less-than-or-equals, slant]t<[infinity]} be a standard Wiener process. This exposition studies the almost sure behaviour of inf0[less-than-or-equals, slant]t[less-than-or-equals, slant]T-aTsup0[less-than-or-equals, slant]s[less-than-or-equals, slant]aT [gamma]TW(t+s)-W(t) as T -->[infinity], under varying conditions on aT and T/aT. The following analogue of Lévy's modulus of continuity of a Wiener Process is also given: and this may be viewed as the exact "modulus of non-differentiability" of a Wiener Process.

Let U1, U2,... be a sequence of independent, uniform (0, 1) r.v.'s and let R1, R2,... be the lengths of increasing runs of {Ui}, i.e., X1=R1=inf{i:Ui+1<Ui},..., Xn=R1+R2+...+Rn=inf{i:i>Xn-1,Ui+1Ui}. The first theorem states that the sequence can be approximated by a Wiener process in strong sense. Let [tau](n) be the largest...</ui},...,>

A class of iterated processes is studied by proving a joint functional limit theorem for a pair of independent Brownian motions. This Strassen method is applied to prove global (t -- [infinity]), as well as local (t -- 0), LIL type results for various iterated processes. Similar results are also...

We prove a strong approximation for the spatial Kesten-Spitzer random walk in random scenery by a Wiener process.

We prove that the number Z(N) of level crossings of a two-parameter simple random walk in its first NxN steps is almost surely N3/2+o(1) as N--[infinity]. The main ingredient is a strong approximation of Z(N) by the crossing local time of a Brownian sheet. Our result provides a useful algorithm...

In this paper we study ratios of local times of a random walk in random environment. Strong and weak limit theorems are obtained.

We study the asymptotic behaviour of the occupation time process [integral operator]t0 IA(W1(L2(s)))ds, t [greater-or-equal, slanted] 0, where W1 is a standard Wiener process and L2 is a Wiener local time process at zero that is independent from W1. We prove limit laws, as well as almost sure...

For a simple symmetric random walk in dimension d[greater-or-equal, slanted]3, a uniform strong law of large numbers is proved for the number of sites with given local time up to time n.

A limit theorem is given for the quadratic variation of the local time of a random walk and that of a Wiener process.

This paper presents a lower bound for the distance between local time and mesure du voisinage of Brownian motion.