Let $(\xi_k,\eta_k)_{k\in\mathbb{N}}$ be independent identically distributed random vectors with arbitrarily dependent positive components. We call a (globally) perturbed random walk a random sequence $T\,{:\!=}\, (T_k)_{k\in\mathbb{N}}$ defined by $T_k\,{:\!=}\, \xi_1+\cdots+\xi_{k-1}+\eta_k$ for $k\in\mathbb{N}$ . Consider a general branching process generated by T and let $N_j(t)$ denote the number of the jth generation individuals with birth times $\leq t$ . We treat early generations, that is, fixed generations j which do not depend on t. In this setting we prove counterparts for $\mathbb{E}N_j$ of the Blackwell theorem and the key renewal theorem, prove a strong law of large numbers for $N_j$ , and find the first-order asymptotics for the variance of $N_j$ . Also, we prove a functional limit theorem for the vector-valued process $(N_1(ut),\ldots, N_j(ut))_{u\geq0}$ , properly normalized and centered, as $t\to\infty$ . The limit is a vector-valued Gaussian process whose components are integrated Brownian motions.

We discuss weak convergence of the number of busy servers in a G/G/∞ queue in the J1-topology on the Skorokhod space. We prove two functional limit theorems with random and nonrandom centering, thereby solving two open problems stated in Mikosch and Resnick (2006). A new integral representation for the limit Gaussian process is given.

A critical branching process {Zk, k = 0, 1, 2, ...} in a random environment is considered. A conditional functional limit theorem for the properly scaled process {log Zpu, 0 ≤ u < ∞} is established under the assumptions that Zn > 0 and p ≪ n. It is shown that the limiting process is a Lévy process conditioned to stay nonnegative. The proof of this result is based on a limit theorem describing the distribution of the initial part of the trajectories of a driftless random walk conditioned to stay nonnegative.

We show that simple diffusion processes are weak limits of piecewise continuous processesconstructed within a totally deterministic framework. The proofs are based on thecontinuous mapping theorem and the functional central limit theorem.

Consider an irreducible, aperiodic and positive recurrent discrete time Markov chain(Xn,n ≥0) with invariant distribution μ. We shall investigate thelong time behaviour of some functionals of the chain, in particular the additivefunctional \hbox{$S_{n}=\sum_{i=1}^{n}f(X_{i})$}${\mathit{S}}_{\mathit{n}}\mathrm{=}{\sum }_{\mathit{i}\mathrm{=}\mathrm{1}}^{\mathit{n}}\mathit{f}\mathrm{\left(}{\mathit{X}}_{\mathit{i}}\mathrm{\right)}$ for a possibly non square integrable functionf. To thisend we shall link ergodic properties of the chain to mixing properties, extending knownresults in the continuous time case. We will then use existing results of convergence tostable distributions, obtained in [M. Denker and A. Jakubowski, Stat. Probab.Lett. 8 (1989) 477–483; M. Tyran-Kaminska, StochasticProcess. Appl. 120 (2010) 1629–1650; D. Krizmanic, Ph.D. thesis(2010); B. Basrak, D. Krizmanic and J. Segers, Ann. Probab. 40(2012) 2008–2033] for stationary mixing sequences. Contrary to the usualL^2$\mathrm{L}{\mathrm{2}}_{}^{}$ frameworkstudied in [P. Cattiaux, D. Chafai and A. Guillin, ALEA, Lat. Am. J. Probab. Math.Stat. 9 (2012) 337–382], where weak forms of ergodicity aresufficient to ensure the validity of the Central Limit Theorem, we will need here strongergodic properties: the existence of a spectral gap, hyperboundedness (orhypercontractivity). These properties are also discussed. Finally we give explicitexamples.

We study the (two-sided) exit time and position of a random walk outside boundaries which are regularly varying functions of smaller order at infinity than the square root. A natural domain of interest is those random walks which are attracted without centring to a normal law, or are relatively stable. These are shown to have ‘stable’ exit positions, in that the overshoot of the curved boundary is of smaller order of magnitude (in probability) than the boundary, as the boundary expands. Surprisingly, this remains true regardless of the shape of the boundary. Furthermore, within the same natural domain of interest, norming of the exit position by, for example, the square root of the exit time (in the finite-variance case), produces limiting distributions which are computable from corresponding functionals of Brownian motion. We give a functional limit theorem for attraction of normed sums to general infinitely divisible random variables, as a means of making this, and more general, computations. These kinds of theorems have applications in sequential analysis, for example.

By imbedding the multitype version of the standard epidemic model in a multiparameter process, we derive a functional limit theorem for the total cost of the epidemics.

A security of supply model for energy is described. The energy demand is a non-random periodic function of time and the supply model is a continuous-time Markov chain. Asymptotic normality is proved for the two important reliability quantities: the amount of energy not supplied and the time the demand is not met within a time interval.