The main goal of this paper is to study the accuracy of approximation for the distributions of negative-binomial random sums of independent, identically distributed random variables by the gamma distribution.
Let be the collection of all -optimal solutions for a stochastic process with locally bounded trajectories defined on a topological space. For sequences of such stochastic processes and of nonnegative random variables we give sufficient conditions for the (closed) random sets to converge in distribution with respect to the Fell-topology and to the coarser Missing-topology.
We consider a centered Gaussian random field X = X t : t ∈ T with values in a Banach space defined on a parametric set T equal to ℝm or ℤm. It is supposed that the distribution of X t is independent of t. We consider the asymptotic behavior of closed convex hulls W n = convX t : t ∈ T n, where (T n) is an increasing sequence of subsets of T. We show that under some conditions of weak dependence for the random field under consideration and some sequence (b n)n≥1 with probability 1, (in the sense...
Particle filter algorithms approximate a sequence of distributions by a sequence of empirical measures generated by a population of simulated particles. In the context of Hidden Markov Models (HMM), they provide approximations of the distribution of optimal filters associated to these models. For a given set of observations, the behaviour of particle filters, as the number of particles tends to infinity, is asymptotically Gaussian, and the asymptotic variance in the central limit theorem depends...
Particle filter algorithms approximate a sequence of distributions by a sequence of empirical measures generated by a population of simulated particles. In the context of Hidden Markov Models (HMM), they provide approximations of the distribution of optimal filters associated to these models. For a given set of observations, the behaviour of particle filters, as the number of particles tends to infinity, is asymptotically Gaussian, and the asymptotic variance in the central limit theorem depends...
We present some extensions of the Borel-Cantelli Lemma in terms of moments. Our result can be viewed as a new improvement to the Borel-Cantelli Lemma. Our proofs are based on the expansion of moments of some partial sums by using Stirling numbers. We also give a comment concerning the results of Petrov V.V., A generalization of the Borel-Cantelli Lemma, Statist. Probab. Lett. 67 (2004), no. 3, 233–239.
In the present paper, by using the inequality due to Talagrand's isoperimetric method, several versions of the bounded law of iterated logarithm for a sequence of independent Banach space valued random variables are developed and the upper limits for the non-random constant are given.
In the present paper, by using the inequality due to Talagrand’s isoperimetric method, several versions of the bounded law of iterated logarithm for a sequence of independent Banach space valued random variables are developed and the upper limits for the non-random constant are given.
For a sequence of blockwise m-dependent random variables {Xn,n ≥ 1}, conditions are provided under which almost surely where {bn,n ≥ 1} is a sequence of positive constants. The results are new even when bn ≡ nr,r > 0. As special case, the Brunk-Chung strong law of large numbers is obtained for sequences of independent random variables. The current work also extends results of Móricz [Proc. Amer. Math. Soc.101 (1987) 709–715], and Gaposhkin [Teor. Veroyatnost. i Primenen. 39 (1994) 804–812]....
Suppose that {Xn, n ≥ 0} is a stationary Markov chain and V is a certain function on a phase space of the chain, called an observable. We say that the observable satisfies the central limit theorem (CLT) if [...] [...] converge in law to a normal random variable, as N → +∞. For a stationary Markov chain with the L2 spectral gap the theorem holds for all V such that V (X0) is centered and square integrable, see Gordin [7]. The purpose of this article is to characterize a family of observables V for...
A probability theory on IFS-events has been constructed in [3], and axiomatically characterized in [4]. Here using a general system of axioms it is shown that any probability on IFS-events can be decomposed onto two probabilities on a Lukasiewicz tribe, hence some known results from [5], [6] can be used also for IFS-sets. As an application of the approach a variant of Central limit theorem is presented.