In this paper we investigate the kernel estimator of the density for a stationary reversible Markov chain. The proofs are based on a new central limit theorem for a triangular array of reversible Markov chains obtained under conditions imposed to covariances, which has interest in itself.

Chareka (2009) presented two limit theorems in which the additivity requirement of a probability measure is weakened to a one-sided version of the inclusion–exclusion formula. Counterexamples are presented which show that both his Central Limit Theorem and his Weak Law of Large Numbers are false.

In this paper we assume that X1,X2,… is a sequence of independent continuous centered random variables with finite variances σ12,σ22,…. Then we present a central limit theorem for the randomly weighted averages Sn=R1X1+⋯+RnXn, where the random weights R1,…,Rn are the cuts of (0,1) by...

A central limit theorem for a linear combination of all the maximum likelihood estimators with an increasing dimension in maximum entropy models for network data, has been established. Simulation studies illustrate the asymptotic results.

In this paper, we study the asymptotic behavior of randomly perturbed Chan–Karolyi–Longstaff–Sanders (CKLS) model with small parameter ε. When ε→0, the central limit theorem and moderate deviation principle for the solution of randomly perturbed CKLS model are obtained.

We study the parameter estimation of two-type continuous-state branching processes with immigration based on low frequency observations at equidistant time points. The ergodicity of the processes is proved. The estimators are based on the minimization of a sum of squared deviation about...

Connectivity of the support of the simple branching random walk is established in certain asymmetric cases, extending a previous result of Grill.

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...

We study subsampling estimators for the limit variance σ2=V ar(X1)+2∑k=2∞Cov(X1,Xk) of partial sums of a stationary stochastic process (Xk)k≥1. We establish L2-consistency of a non-overlapping block resampling method. Our results apply to processes that can be represented as functionals...

The Birnbaum–Saunders distribution is based on the asymptotic normality of a sum of random variables. We propose a new version of this distribution assuming that the number of terms of such a sum depends on a non-homogeneous Poisson process. The classical Birnbaum–Saunders distribution is...