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Limit theorems for one and two-dimensional random walks in random scenery

Fabienne Castell, Nadine Guillotin-Plantard, Françoise Pène (2013)

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

Random walks in random scenery are processes defined by Z n : = ∑ k = 1 n ξ X 1 + ⋯ + X k , where ( X k , k ≥ 1 ) and ( ξ y , y ∈ ℤ d ) are two independent sequences of i.i.d. random variables with values in ℤ d and ℝ respectively. We suppose that the distributions of X 1 and ξ 0 belong to the normal basin of attraction of stable distribution of index α ∈ ( 0 , 2 ] and β ∈ ( 0 , 2 ] . When d = 1 and α ≠ 1 , a functional limit theorem has been established in (Z. Wahrsch. Verw. Gebiete50 (1979) 5–25) and a local limit theorem in (Ann. Probab.To appear). In this paper, we establish the convergence in...

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 the painting of graphs by clusters

Olivier Garet (2001)

ESAIM: Probability and Statistics

We consider a generalization of the so-called divide and color model recently introduced by Häggström. We investigate the behavior of the magnetization in large boxes of the lattice ℤ d and its fluctuations. Thus, Laws of Large Numbers and Central Limit Theorems are proved, both quenched and annealed. We show that the properties of the underlying percolation process deeply influence the behavior of the coloring model. In the subcritical case, the limit magnetization is deterministic and the Central...

Limit Theorems for the painting of graphs by clusters

Olivier Garet (2010)

ESAIM: Probability and Statistics

We consider a generalization of the so-called divide and color model recently introduced by Häggström. We investigate the behavior of the magnetization in large boxes of the lattice ℤ d and its fluctuations. Thus, Laws of Large Numbers and Central Limit Theorems are proved, both quenched and annealed. We show that the properties of the underlying percolation process deeply influence the behavior of the coloring model. In the subcritical case, the limit magnetization is deterministic and the Central Limit...

Limited space double channel Markovian queue with heterogeneus servers.

O.P. Sharma, J. Dass (1990)

Trabajos de Investigación Operativa

For a double channel Markovian queue with finite waiting space and unequal service rates at the two counters, the difference equations satisfied by the Laplace transforms of the state probabilities at finite time are solved and the state probabilities have been obtained. The closed form of the state probabilities can be used to obtain the important parameters of the system.

Limiting curlicue measures for theta sums

Francesco Cellarosi (2011)

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

We consider the ensemble of curves {γα, N: α∈(0, 1], N∈ℕ} obtained by linearly interpolating the values of the normalized theta sum N−1/2∑n=0N'−1exp(πin2α), 0≤N'&lt;N. We prove the existence of limiting finite-dimensional distributions for such curves as N→∞, when α is distributed according to any probability measure λ, absolutely continuous w.r.t. the Lebesgue measure on [0, 1]. Our Main Theorem generalizes a result by Marklof [Duke Math. J.97 (1999) 127–153] and Jurkat and van Horne [Duke...

Limits of determinantal processes near a tacnode

Alexei Borodin, Maurice Duits (2011)

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

We study a Markov process on a system of interlacing particles. At large times the particles fill a domain that depends on a parameter ε &gt; 0. The domain has two cusps, one pointing up and one pointing down. In the limit ε ↓ 0 the cusps touch, thus forming a tacnode. The main result of the paper is a derivation of the local correlation kernel around the tacnode in the transition regime ε ↓ 0. We also prove that the local process interpolates between the Pearcey process and the GUE minor process....

Lipschitz percolation.

Dirr, Nicolas, Dondl, Patrick W., Grimmett, Geoffrey R., Holroyd, Alexander E., Scheutzow, Michael (2010)

Electronic Communications in Probability [electronic only]

Local percolative properties of the vacant set of random interlacements with small intensity

Alexander Drewitz, Balázs Ráth, Artëm Sapozhnikov (2014)

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

Random interlacements at level u is a one parameter family of connected random subsets of ℤ d , d ≥ 3 (Ann. Math.171(2010) 2039–2087). Its complement, the vacant set at level u , exhibits a non-trivial percolation phase transition in u (Comm. Pure Appl. Math.62 (2009) 831–858; Ann. Math.171 (2010) 2039–2087), and the infinite connected component, when it exists, is almost surely unique (Ann. Appl. Probab.19(2009) 454–466). In this paper we study local percolative properties of the vacant set of random interlacements...

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