Let Ik(n) denote the kth largest intervals generated by 0, X1,..., Xn-1, 1 where X1, X2, ... are i. This note provides a complete answer to the question for which class of sequences k(n) the interval Ik(n)(n) is hit only finitely o as well as infinitely often (a.s.).

Let (Sn)n[greater-or-equal, slanted]0 be a zero-delayed nonarithmetic random walk with positive drift [mu] and ([xi]n)n[greater-or-equal, slanted]0 be a slowly varying perturbation process (see conditions (C.1)-(C.3) in Section 1). The results of this note are two weak convergence theorems for...

Consider the class of even convex functions with [phi](0)=0 and concave derivative on (0,[infinity]). Given any [phi]-integrable martingale (Mn)n[greater-or-equal, slanted]0 with increments , n[greater-or-equal, slanted]1, the Topchii-Vatutin inequality (Theory Probab. Appl. 42 (1997) 17)...

Let X1, X2,... be i.i.d. random variables with common mean [mu] [greater-or-equal, slanted] 0 and associated random walk S0 = 0, Sn = X1 + ... + Xn, n [greater-or-equal, slanted] 1. Let U(t) = [Sigma]n [greater-or-equal, slanted] 1(1/n)P(Sn [less-than-or-equals, slant] t) be the harmonic renewal...

In this paper we extend well-known results by Baum and Katz (1965) and others on the rate of convergence in the law of large numbers for sums of i.i.d. random variables to general zero-mean martingales S. For , p1/[alpha] and f(x) = x (two-sided case) OR = x+ or x- (one-sided case), it is e.g....

Let be a stochastic process adapted to the filtration and with increments X1, X2, ... Set and Ln = m1 + ... + mn for n [greater-or-equal, slanted] 1. Then we call a linear growth process (LGP) if 1. (1) [mu] [less-than-or-equals, slant] Ln/n [less-than-or-equals, slant] [nu] a.s.f.a. n...

Let be a complete separable metric space and (Fn)n[greater-or-equal, slanted]0 a sequence of i.i.d. random functions from to which are uniform Lipschitz, that is, Ln=supx[not equal to]y d(Fn(x),Fn(y))/d(x,y)[infinity] a.s. Providing the mean contraction assumption and for some , it was proved by...

Lam and Lehoczky (1991) have recently given a number of extensions of classical renewal theorems to superpositions of p independent renewal processes. In this article we want to advertise an approach that more explicitly uses a Markov renewal theoretic framework and thus leads to a simplified...

Let (S, £) be a measurable space with countably generated [sigma]-field £ and (Mn, Xn)n[greater-or-equal, slanted]0 a Markov chain with state space S x and transition kernel :S x ( [circle times operator] )--[0, 1]. Then (Mn,Sn)n[greater-or-equal, slanted]0, where Sn = X0+...+Xn for...