Based on an R2-valued random sample {(yi,xi),1≤i≤n} on the simple linear regression model yi=xiβ+α+εi with unknown error variables εi, least squares processes (LSPs) are introduced in D[0,1] for the unknown slope β and intercept α, as well as for the unknown β when α=0. These LSPs...

For a suitable definition of the local time of a random walk strong invariance principles are proved, saying that this local time is like that of a Wiener process. Consequences of these results are LIL statements for the local time of a general enough class of random walks. One of the tools for...

We establish moduli of continuity and large increment properties for stationary increment Gaussian processes in order to study the path behavior of infinite series of independent Ornstein-Uhlenbeck processes. The existence and continuity of the latter infinite series type Gaussian processes are...

Let X1, X2... be a sequence of positive, independent, identically distributed (i.i.d.) random variables with S0 = 0, Sn = X1 + ... + Xn, n [greater-or-equal, slanted] 1. Denote by [tau]i = sup{nSn [less-than-or-equals, slant] t }. In this paper we establish almost sure lower and upper bounds for...

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.