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.

Consider the d-dimensional unit cube [0,1]d and portion it into n regions, A1,..., An. Select and fix a point in each one of these regions so we have x1,..., xn. Consider observable variables Yi, i = 1,..., n, satisfying the multivariate regression model Yi = g(xi) + [var epsilon]i, where g is...

By an adaptation of a method originally invented by G. Kersting [1] for the calculation of the limiting distribution of Markovian processes the central limit theorem (CLT) is proven. Only the case of equal variances is considered.

We consider Poisson’s equation for discrete-time single-birth processes, and we derive its solutions by solving a linear system of infinitely many equations. We apply the solution of Poisson’s equation to obtain the asymptotic variance. The results are further applied to birth–death...

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.

The central limit theorem is considered with respect to the transport distance W2. We discuss an alternative approach to a result of E. Rio, based on a Berry–Esseen-type bound for the entropic distance to the normal distribution.

This paper considers the asymptotic behaviors of the processes generated by the classical ergodic tent map that is defined on the unit interval. We get the uniform version of central limit theorem for the tent map by using the method of uniformly integrable entropy. An application to...

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...