20.1. Additional technical results on weak convergence Given two metric spaces S1, S2 and a measurable function f : S1 → S2, suppose S1 is equipped with some probability measure P. This induces a probability measure on S2 which is denoted by Pf−1 and is defined by Pf−1(A) = P(f−1(A) for every measurable set A ⊂ S2. Then for any random variable X : S2 → R, its expectation EPf −1 [X] is equal to ...

We introduce the notion of the mean-set (expectation) of a graph(group-) valued random element ξ and prove a generalization of the strong law of large numbers on graphs and groups. Furthermore, we prove an analogue of the classical Chebyshev’s inequality for ξ. We show that our generalized law of large numbers, as a new theoretical tool, provides a framework for practical applications; namely, ...

Let X be the branching particle diffusion corresponding to the operator Lu + β(u2 − u) on D ⊆ Rd (where β ≥ 0 and β 6≡ 0). Let λc denote the generalized principal eigenvalue for the operator L + β on D and assume that it is finite. When λc > 0 and L+β−λc satisfies certain spectral theoretical conditions, we prove that the random measure exp{−λct}Xt converges almost surely in the vague topology ...

In this paper, the issue of the law of large numbers for fuzzy variables is considered. Since in credibility theory convergence in credibility implies convergence almost sure, the strong law of large numbers is defined via convergence in credibility, while the weak law of large numbers is defined through convergence almost sure. Based on the convergence results about the unform integrability of...

Throughout this paper, let denote the set of nonnegative integer, let {X,Xn, n ∈ } be a sequence of random variables defined on probability space Ω,F, P , and put Sn ∑n k 1 Xk. The symbol C will denote a generic constant 0 < C < ∞ which is not necessarily the same one in each appearance. In 1 , Jajte studied a large class of summability method as follows: a sequence {Xn, n ≥ 1} is summable to X...

Let f(n) be a strongly additive complex-valued arithmetic function. Under mild conditions on f , we prove the following weighted strong law of large numbers: if X, X1, X2, . . . is any sequence of integrable i.i.d. random variables, then lim N→∞ ∑N n=1 f(n)Xn ∑N n=1 f(n) = EX a.s.

In the spirit of a classical result for Crump–Mode–Jagers processes, we prove a strong law of large numbers for fragmentation processes. Specifically, for self-similar fragmentation processes, including homogenous processes, we prove the almost sure convergence of an empirical measure associated with the stopping line corresponding to first fragments of size strictly smaller than η for 1 ≥ η > ...

Abstract: We prove a martingale triangular array generalization of the Chow-BirnbaumMarshall’s inequality. The result is used to derive a strong law of large numbers for martingale triangular arrays whose rows are asymptotically stable in a certain sense. To illustrate, we derive a simple proof, based on martingale arguments, of the consistency of kernel regression with dependent data. Another ...

It is observed that a wellnigh trivial application of the ergodic theorem from [3] yields a strong LLN for arbitrary concave moments. Not for publication: we found that Aaronson–Weiss essentially proved Theorem 1, see J. Aaronson, An introduction to infinite ergodic theory (AMS Math. Surv. Mon. 50, 1997), pages 65–66.