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 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 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≥1, describe the successive sums of the payoffs in the classical St. Petersburg game. Feller’s famous weak law, Feller (1945), states that Snnlog2n→p1 as n→∞. However, almost sure convergence fails, more precisely, lim supn→∞Snnlog2n=+∞ a.s. and lim...

Consider an i.i.d. random field {Xk:k∈Z+d}, together with a sequence of unboundedly increasing nested sets Sj=⋃k=1jHk,j≥1, where the sets Hj are disjoint. The canonical example consists of the hyperbolas Hj={k∈Z+d:|k|=j}. We are interested in the number of “hyperbolas” Hj that...

A recent paper by  Pozdnyakov and Steele (2010) is devoted to the so-called binary-plus-passive design. Two problems that the authors do not consider can be identified with the classical gambler’s ruin problem in which delays are allowed.

If Y1,Y2,… is a sequence of random variables such that Yn⟶a.s.Y as n→∞, and {τ(t),t≥0} is a family of “indices” such that τ(t)⟶a.s.∞ as t→∞, then it is pretty obvious that Yτ(t)⟶a.s.Y as t→∞. However, if one relaxes one of ⟶a.s. to ⟶p and lets the other one...

Precise asymptotics have been proved for sums like [summation operator]n=1[infinity]nr/p-2P(Sn[greater-or-equal, slanted][var epsilon]n1/p) as [var epsilon][downward right arrow]0, where {Sn, n[greater-or-equal, slanted]1} are partial sums i.i.d. random variables, and, more recently, for renewal...

Consider Z+d (d[greater-or-equal, slanted]2)--the positive d-dimensional lattice points with partial ordering [less-than-or-equals, slant], let {Xk,k[set membership, variant]Z+d} be i.i.d. random variables with mean 0, and set Sn=[summation operator]k[less-than-or-equals, slant]nXk, n[set...