An algorithm is developed for exact simulation from distributions that are defined as fixed points of maps between spaces of probability measures. The fixed points of the class of maps under consideration include examples of limit distributions of random variables studied in the probabilistic analysis of algorithms. Approximating sequences for the densities of the fixed points with explicit error bounds are constructed. The sampling algorithm relies on a modified rejection method.

Empirical studies of data traffic in high-speed networks suggest that network traffic exhibits self-similarity and long-range dependence. Cumulative network traffic has been modeled using the so-called ON/OFF model. It was shown that cumulative network traffic can be approximated by either fractional Brownian motion or stable Lévy motion, depending on how many sources are active in the model. In this paper we consider exceedances of a high threshold by the sequence of lengths of ON-periods. If the cumulative network traffic converges to stable Lévy motion, the number of exceedances converges to a Poisson limit. The same holds in the fractional Brownian motion case, provided a very high threshold is used. Finally, we show that the number of exceedances obeys the central limit theorem.

Motivated by various applications in queueing systems, this work is devoted to continuous-time Markov chains with countable state spaces that involve both fast-time scale and slow-time scale with the aim of approximating the time-varying queueing systems by their quasistationary counterparts. Under smoothness conditions on the generators, asymptotic expansions of probability vectors and transition probability matrices are constructed. Uniform error bounds are obtained, and then sequences of occupation measures and their functionals are examined. Mean square error estimates of a sequence of occupation measures are obtained; a scaled sequence of functionals of occupation measures is shown to converge to a Gaussian process with zero mean. The representation of the variance of the limit process is also explicitly given. The results obtained are then applied to treat Mt/Mt/1 queues and Markov-modulated fluid buffer models.

In various fields, such as teletraffic and economics, measured time series have been reported to adhere to multifractal scaling. Classical cascading measures possess multifractal scaling, but their increments form a nonstationary process. To overcome this problem, we introduce a construction of random multifractal measures based on iterative multiplication of stationary stochastic processes, a special form of T-martingales. We study the ℒ2-convergence, nondegeneracy, and continuity of the limit process. Establishing a power law for its moments, we obtain a formula for the multifractal spectrum and hint at how to prove the full formalism.

Galton-Watson forests consisting of N roots (or trees) and n nonroot vertices are studied. The limit distributions of the number of leaves in such a forest are obtained. By a leaf we mean a vertex from which there are no arcs emanating.

We give (i) the Cramér power spectral measure of the general shot noise process with random excitation and non-Poisson stationary driving point processes and (ii) the Bartlett power spectral measure of the self-exciting Hawkes point process with random excitation, also called the Hawkes branching point process with random fertility rate. The latter is obtained via the isometry formula for integrals with respect to the canonical martingale measure associated with a marked point process.

Empirical point processes of exceedances play an important role in extreme value theory, and their limiting behaviour has been extensively studied. Here, we provide explicit bounds on the accuracy of approximating an exceedance process by a compound Poisson or Poisson cluster process, in terms of a Wasserstein metric that is generally more suitable for the purpose than the total variation metric. The bounds only involve properties of the finite, empirical sequence that is under consideration, and not of any limiting process. The argument uses Bernstein blocks and Lindeberg's method of compositions.

The purpose of this paper is to study ageing properties of first-passage times of increasing Markov chains. We extend the literature to some new ageing classes, such as the IFR(2), NBU(2), DRLLt and NBULt classes. We also give sufficient conditions in the finite case, that are more efficient computationally, just in terms of the transition matrix K, in the discrete case, or the generator matrix Q, in the continuous case. For the uniformizable, continuous-time Markov processes, we derive conditions in terms of the discrete uniformized Markov chain for the NBU(2) and the NBULt classes. In the last section, a review of the main results in this direction in the literature is given, and we compare some of the conditions stated in this paper with others given in the literature about some other ageing classes. Some examples where these results are applied are given.