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The exit path of a Markov chain with rare transitions

Olivier Catoni, Raphaël Cerf (2010)

ESAIM: Probability and Statistics

We study the exit path from a general domain after the last visit to a set of a Markov chain with rare transitions. We prove several large deviation principles for the law of the succession of the cycles visited by the process (the cycle path), the succession of the saddle points gone through to jump from cycle to cycle on the cycle path (the saddle path) and the succession of all the points gone through (the exit path). We estimate the time the process spends in each cycle of the cycle path...

The first exit of almost strongly recurrent semi-Markov processes

Joachim Domsta, Franciszek Grabski (1995)

Applicationes Mathematicae

Let ( · ) , n ∈ N, be a sequence of homogeneous semi-Markov processes (HSMP) on a countable set K, all with the same initial p.d. concentrated on a non-empty proper subset J. The subrenewal kernels which are restrictions of the corresponding renewal kernels on K×K to J×J are assumed to be suitably convergent to a renewal kernel P (on J×J). The HSMP on J corresponding to P is assumed to be strongly recurrent. Let [ π j ; j ∈ J] be the stationary p.d. of the embedded Markov chain. In terms of the averaged...

The fractional mixed fractional brownian motion and fractional brownian sheet

Charles El-Nouty (2007)

ESAIM: Probability and Statistics


We introduce the fractional mixed fractional Brownian motion and fractional Brownian sheet, and investigate the small ball behavior of its sup-norm statistic. Then, we state general conditions and characterize the sufficiency part of the lower classes of some statistics of the above process by an integral test. Finally, when we consider the sup-norm statistic, the necessity part is given by a second integral test.

The generalized weighted probability measure on the symmetric group and the asymptotic behavior of the cycles

Ashkan Nikeghbali, Dirk Zeindler (2013)

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

The goal of this paper is to analyse the asymptotic behaviour of the cycle process and the total number of cycles of weighted and generalized weighted random permutations which are relevant models in physics and which extend the Ewens measure. We combine tools from combinatorics and complex analysis (e.g. singularity analysis of generating functions) to prove that under some analytic conditions (on relevant generating functions) the cycle process converges to a vector of independent Poisson variables...

The infinite valley for a recurrent random walk in random environment

Nina Gantert, Yuval Peres, Zhan Shi (2010)

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

We consider a one-dimensional recurrent random walk in random environment (RWRE). We show that the – suitably centered – empirical distributions of the RWRE converge weakly to a certain limit law which describes the stationary distribution of a random walk in an infinite valley. The construction of the infinite valley goes back to Golosov, see Comm. Math. Phys.92 (1984) 491–506. As a consequence, we show weak convergence for both the maximal local time and the self-intersection local time of the...

The large deviation principle for certain series

Miguel A. Arcones (2010)

ESAIM: Probability and Statistics

We study the large deviation principle for stochastic processes of the form { ∑ k = 1 ∞ x k ( t ) ξ k : t ∈ T } , where { ξ k } k = 1 ∞ is a sequence of i.i.d.r.v.'s with mean zero and x k ( t ) ∈ ℝ . We present necessary and sufficient conditions for the large deviation principle for these stochastic processes in several situations. Our approach is based in showing the large deviation principle of the finite dimensional distributions and an exponential asymptotic equicontinuity condition. In order to get the exponential asymptotic equicontinuity condition,...

The large deviation principle for certain series

Miguel A. Arcones (2004)

ESAIM: Probability and Statistics

We study the large deviation principle for stochastic processes of the form { ∑ k = 1 ∞ x k ( t ) ξ k : t ∈ T } , where { ξ k } k = 1 ∞ is a sequence of i.i.d.r.v.’s with mean zero and x k ( t ) ∈ ℝ . We present necessary and sufficient conditions for the large deviation principle for these stochastic processes in several situations. Our approach is based in showing the large deviation principle of the finite dimensional distributions and an exponential asymptotic equicontinuity condition. In order to get the exponential asymptotic equicontinuity condition,...

The law of large numbers and a functional equation

Maciej Sablik (1998)

Annales Polonici Mathematici

We deal with the linear functional equation (E) g ( x ) = ∑ i = 1 r p i g ( c i x ) , where g:(0,∞) → (0,∞) is unknown, ( p ₁ , . . . , p r ) is a probability distribution, and c i ’s are positive numbers. The equation (or some equivalent forms) was considered earlier under different assumptions (cf. [1], [2], [4], [5] and [6]). Using Bernoulli’s Law of Large Numbers we prove that g has to be constant provided it has a limit at one end of the domain and is bounded at the other end.

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