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 Not for publication: we found that Aaronson–Weiss essentially proved Theorem 1, see J. Aaronson, An introduction to infinite ergodic theory (AMS Math.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.

Adults increase the certainty of their inductive inferences by observing more diverse instances. However, most young children fail to do so. The present study tested the hypothesis that children’s sensitivity to instance diversity is determined by three variables: ability to discriminate among instances (Discrimination); an intuition that large numbers of instances increase the strength of conc...

Abstract: We study the behaviour of stochastic processes defined as an iterated function system Xn+1 = Xn + af(Xn, Un+1) with initial value X0 = x0 and a stationary ergodic input signal (Un)n≥0 for small values of the parameter a. We obtain almost sure convergence of the path to the solution of the corresponding deterministic dynamical system defined by ẏ = F (y), where F (y) = E(f(y, U)). The ...

Heap monoids equipped with Bernoulli measures are a model of probabilistic asynchronous systems. We introduce in this framework the notion of asynchronous stopping time, which is analogous to the notion of stopping time for classical probabilistic processes. A Strong Bernoulli property is proved. A notion of cut-invariance is formulated for convergent ergodic means. Then a version of the Strong...

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 ...

We consider a reduced form model for a large pool of possibly defaultable entities. We allow the times of defaults to be correlated not only by their dependence on a common factor, but also by their dependence on the extent of past defaults, therefore with a self-exciting nature. We characterize the limiting process for the mean number of defaults and the mean loss, as the number of entities be...

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...

A classical theorem of S. Bochner states that a function f : R → C is the Fourier transform of a finite Borel measure if and only if f is positive definite. In 1938, I. Schoenberg found a beautiful converse to Bochner’s theorem. We present a non-technical derivation of of Schoenberg’s theorem that relies chiefly on the law of large numbers of classical probability theory.

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...