163 research outputs found
On exceedance times for some processes with dependent increments
Let be a random walk with a negative drift and i.i.d.
increments with heavy-tailed distribution and let be its
supremum. Asmussen & Kl{\"u}ppelberg (1996) considered the behavior of the
random walk given that , for large, and obtained a limit theorem, as
, for the distribution of the quadruple that includes the time
\rtreg=\rtreg(x) to exceed level , position Z_{\rtreg} at this time,
position Z_{\rtreg-1} at the prior time, and the trajectory up to it (similar
results were obtained for the Cram\'er-Lundberg insurance risk process). We
obtain here several extensions of this result to various regenerative-type
models and, in particular, to the case of a random walk with dependent
increments. Particular attention is given to describing the limiting
conditional behavior of . The class of models include Markov-modulated
models as particular cases. We also study fluid models, the Bj{\"o}rk-Grandell
risk process, give examples where the order of is genuinely different
from the random walk case, and discuss which growth rates are possible. Our
proofs are purely probabilistic and are based on results and ideas from
Asmussen, Schmidli & Schmidt (1999), Foss & Zachary (2002), and Foss,
Konstantopoulos & Zachary (2007).Comment: 17 page
Regular Variation in a Fixed-Point Problem for Single- and Multiclass Branching Processes and Queues
Tail asymptotics of the solution to a fixpoint problem of type is derived under heavy-tailed conditions allowing both
dependence between and and the tails to be of the same order of
magnitude. Similar results are derived for a -class version with
applications to multitype branching processes and busy periods in multiclass
queues.Comment: 19 pages, 1 figur
Point processes with finite-dimensional conditional probabilities
AbstractWe study the structure of point processes N with the property that the P(θtN∈·|Ft) vary in a finite-dimensional space where θt is the shift and Ft the σ-field generated by the counting process up to time t. This class of point processes is strictly larger than Neuts’ class of Markovian arrival processes. On the one hand, it allows for more general features like interarrival distributions which are matrix-exponential rather than phase type, on the other the probabilistic interpretation is a priori less clear. Nevertheless, the properties are very similar. In particular, finite-dimensional distributions of interarrival times, moments, Laplace transforms, Palm distributions, etc., are shown to be given by two fundamental matrices C,D just as for the Markovian arrival process. We also give a probabilistic interpretation in terms of a piecewise deterministic Markov process on a compact convex subset of Rp, whose jump times are identical to the epochs of N
Heavy Tails, Importance Sampling and Cross-Entropy
We consider the problem of estimating P (Y1+ ... +Yn > x) by importance sampling when the Yi are i.i.d. and heavy-tailed. The idea is to exploit the cross-entropy method as a tool for choosing good parameters in the importance sampling distribution; in doing so, we use the asymptotic description that given P(Y1+ ... +Yn > x,) n-1 of the Yi have distribution F and one the conditional distribution of Y given Y > x. We show in some parametric examples (Pareto and Weibull) how this leads to precise answers, which as demonstrated numerically, are close to being variance minimal within the parametric class under consideration. Related problems for M/G/1 and GI/G/1 queues are also discussed
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