In this paper we introduce discrete-time semi-Markov random evolutions (DTSMREs) and study asymptotic properties, namely, averaging, diffusion approximation, and diffusion approximation with equilibrium by the martingale weak convergence method. The controlled DTSMREs are introduced and Hamilton–Jacobi–Bellman equations are derived for them. The applications here concern the additive functionals (AFs), geometric Markov renewal chains (GMRCs), and dynamical systems (DSs) in discrete time. The rates of convergence in the limit theorems for DTSMREs and AFs, GMRCs, and DSs are also presented.

A simple model for a randomly oscillating variable is suggested, which is a variant of the two-state random velocity model. As in the latter model, the variable keeps rising or falling with constant velocity for some time before randomly reversing its direction. In contrast however, its propensity to reverse depends on its current value and it is for this desirable feature that the model is proposed here. This feature has two implications: (a) neither the changing variable nor its velocity is Markovian, although the joint process is, and (b) the linear differential equations arising in the case of our model do not have constant coefficients. The results given in this paper are meant to illustrate the straightforward nature of some of the calculations involved and to highlight the relationship with one-dimensional diffusions.

A simple derivation of the explicit form of the transition density of a planar random motion at finite speed, based on some specific properties of the wave propagation on the plane R2, is given.

We consider the random motion of a particle that moves with constant finite speed in the space ℝ4 and, at Poisson-distributed times, changes its direction with uniform law on the unit four-sphere. For the particle's position, X(t) = (X1(t), X2(t), X3(t), X4(t)), t > 0, we obtain the explicit forms of the conditional characteristic functions and conditional distributions when the number of changes of directions is fixed. From this we derive the explicit probability law, f(x, t), x ∈ ℝ4, t ≥ 0, of X(t). We also show that, under the Kac condition on the speed of the motion and the intensity of the switching Poisson process, the density, p(x,t), of the absolutely continuous component of f(x,t) tends to the transition density of the four-dimensional Brownian motion with zero drift and infinitesimal variance σ2 = ½.