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Limit distributions for multitype branching processes of m -ary search trees

Brigitte Chauvin, Quansheng Liu, Nicolas Pouyanne (2014)

Annales de l'I.H.P. Probabilités et statistiques

Let m ≥ 3 be an integer. The so-called m -ary search treeis a discrete time Markov chain which is very popular in theoretical computer science, modelling famous algorithms used in searching and sorting. This random process satisfies a well-known phase transition: when m ≤ 26 , the asymptotic behavior of the process is Gaussian, but for m ≥ 27 it is no longer Gaussian and a limit W D T of a complex-valued martingale arises. In this paper, we consider the multitype branching process which is the continuous time version...

Limit laws of transient excited random walks on integers

Elena Kosygina, Thomas Mountford (2011)

Annales de l'I.H.P. Probabilités et statistiques

We consider excited random walks (ERWs) on ℤ with a bounded number of i.i.d. cookies per site without the non-negativity assumption on the drifts induced by the cookies. Kosygina and Zerner [15] have shown that when the total expected drift per site, δ, is larger than 1 then ERW is transient to the right and, moreover, for δ>4 under the averaged measure it obeys the Central Limit Theorem. We show that when δ∈(2, 4] the limiting behavior of an appropriately centered and scaled excited random...

Limit shapes of Gibbs distributions on the set of integer partitions : the expansive case

Michael M. Erlihson, Boris L. Granovsky (2008)

Annales de l'I.H.P. Probabilités et statistiques

We find limit shapes for a family of multiplicative measures on the set of partitions, induced by exponential generating functions with expansive parameters, ak∼Ckp−1, k→∞, p>0, where C is a positive constant. The measures considered are associated with the generalized Maxwell–Boltzmann models in statistical mechanics, reversible coagulation–fragmentation processes and combinatorial structures, known as assemblies. We prove a central limit theorem for fluctuations of a properly scaled partition...

Limit Theorems for Non-Critical Branching Processes with Continuous State Space

Kurbanov, S. (2008)

Serdica Mathematical Journal

2000 Mathematics Subject Classification: Primary 60J80, Secondary 60G99.In the paper a modification of the branching stochastic process with immigration and with continuous states introduced by Adke S. R. and Gadag V. G. (1995) is considered. Limit theorems for the non-critical processes with or without non-stationary immigration and finite variance are proved. The subcritical case is illustrated with examples.

Limit theorems for number of diffusion processes, which did not absorb by boundaries

Aniello Fedullo, Vitalii Gasanenko (2006)

Open Mathematics

We have random number of independent diffusion processes with absorption on boundaries in some region at initial time t = 0. The initial numbers and positions of processes in region is defined by the Poisson random measure. It is required to estimate the number of the unabsorbed processes for the fixed time τ > 0. The Poisson random measure depends on τ and τ → ∞.

Limit Theorems for Regenerative Excursion Processes

Mitov, Kosto (1999)

Serdica Mathematical Journal

This work is supported by Bulgarian NFSI, grant No. MM–704/97The regenerative excursion process Z(t), t = 0, 1, 2, . . . is constructed by two independent sequences X = {Xi , i ≥ 1} and Z = {Ti , (Zi (t), 0 ≤ t < Ti ), i ≥ 1}. For the embedded alternating renewal process, with interarrival times Xi – the time for the installation and Ti – the time for the work, are proved some limit theorems for the spent worktime and the residual worktime, when at least one of the means of Xi and Ti is infinite. ...

Limit theorems for some functionals with heavy tails of a discrete time Markov chain

Patrick Cattiaux, Mawaki Manou-Abi (2014)

ESAIM: Probability and Statistics

Consider an irreducible, aperiodic and positive recurrent discrete time Markov chain (Xn,n ≥ 0) with invariant distribution μ. We shall investigate the long time behaviour of some functionals of the chain, in particular the additive functional S n = ∑ i = 1 n f ( X i ) S n = ∑ i = 1 n f ( X i ) for a possibly non square integrable functionf. To this end we shall link ergodic properties of the chain to mixing properties, extending known results in the continuous time case. We will then use existing results of convergence...

Limit theorems for stationary Markov processes with L2-spectral gap

Déborah Ferré, Loïc Hervé, James Ledoux (2012)

Annales de l'I.H.P. Probabilités et statistiques

Let ( X t , Y t ) t ∈ 𝕋 be a discrete or continuous-time Markov process with state space 𝕏 × ℝ d where 𝕏 is an arbitrary measurable set. Its transition semigroup is assumed to be additive with respect to the second component, i.e. ( X t , Y t ) t ∈ 𝕋 is assumed to be a Markov additive process. In particular, this implies that the first component ( X t ) t ∈ 𝕋 is also a Markov process. Markov random walks or additive functionals of a Markov process are special instances of Markov additive processes. In this paper, the process ( Y t ) t ∈ 𝕋 is shown to satisfy the...

Limit theorems for stochastic recursions with Markov dependent coefficients

Dariusz Buraczewski, Małgorzata Letachowicz (2012)

Colloquium Mathematicae

We consider the stochastic recursion X ₙ = A ₙ X n - 1 + B ₙ for Markov dependent coefficients (Aₙ,Bₙ) ∈ ℝ⁺ × ℝ. We prove the central limit theorem, the local limit theorem and the renewal theorem for the partial sums Sₙ = X₁+ ⋯ + Xₙ.

Limite ergodique de processus de diffusion infini-dimensionnels.

Sylvie Roelly, D. Seu (1999)

Publicacions Matemàtiques

We give a temporal ergodicity criterium for the solution of a class of infinite dimensional stochastic differential equations of gradient type, where the interaction has infinite range. We illustrate our theoretical result by typical examples.

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