Abstract Linear fractional Galton–Watson branching processes in i.i.d. random environment are, on the quenched level, intimately connected to random difference equations by the evolution of the random parameters of their linear fractional marginals. On the other hand, any random difference...

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...

In this note we give some results on the nonnegativity of odd functional moments of random variables with a decreasing density. More precisely, we prove by purely elementary arguments, Egf(X - EX) [greater-or-equal, slanted] 0 for suitable functions gf that satisfy gf(x) = -gf(-x) for all x...

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...