This paper considers the asymptotic theory of the varying environment Galton-Watson process with a countable set of types. This paper examines the convergence in Lp and almost surely of the numbers of the various types when normalised by the corresponding expected number. The harmonic functions...

Suppose a sequence of random variables {Xn} has negative drift when above a certain threshold and has increments bounded in Lp. When p2 this implies that EXn is bounded above by a constant independent of n and the particular 0sequence {Xn}. When p[less-than-or-equals, slant]2 there are...

Let (Z(t): t[greater-or-equal, slanted]0) be a supercritical age-dependent branching process and let {Yn} be the natural martingale arising in a homogeneous branching random walk. Let Z be the almost sure limit of Z(t)/EZ(t)(t--[infinity]) or that of Yn (n--[infinity]). We study the following...

This paper proves that the law of the iterated logarithm holds for a stationary negatively associated sequence of random variables with finite variance. The proof is based on a Rosenthal type maximal inequality, a Kolmogorov type exponential inequality and Stein's method.

We show that if an appropriate stopping rule is used to determine the sample size when estimating the parameters in a stationary and ergodic threshold AR(1) model, then the sequential least-squares estimator is asymptotically risk efficient. The stopping rule is also shown to be asymptotically...

In this paper we establish large increment results and moduli of continuty for a two-parameter fractional Lévy Brownian motion on rectangles in the Euclidean plane via estimating upper bounds of large deviation probabilities on suprema of the two-parameter fractional Lévy Brownian motion.

Let (H,B,[mu]) be an abstract Wiener space. Let be the set of all finite-dimensional orthogonal projections in H and for denote by [Gamma](P) the second quantization of P. It is shown that for [phi][set membership, variant][intersection operator]p1Lp(B,[mu]) and , the z-1/2-scaling...

We consider a simple bilinear process Xt=aXt-1+bXt-1Zt-1+Zt, where (Zt) is a sequence of iid N(0,1) random variables. It follows from a result by Kesten (1973, Acta Math. 131, 207-248) that Xt has a distribution with regularly varying tails of index [alpha]0 provided the equation Ea+bZ1u=1 has...

The aim of this work is to obtain sufficient conditions for stability of multidimensional jump-diffusion processes in the sense of stability in distribution and stability at the equilibrium solution. The technique employed is to construct appropriate Lyapunov functions.

In this paper we establish some new theorems on pathwise uniqueness of solutions to the stochastic differential equations of the form of for with non-Lipschitz coefficients, where is a continuous square integrable martingale and is a continuous increasing process, Z is a continuous stochastic...