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Couplings, attractiveness and hydrodynamics for conservative particle systems

Thierry Gobron, Ellen Saada (2010)

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

Attractiveness is a fundamental tool to study interacting particle systems and the basic coupling construction is a usual route to prove this property, as for instance in simple exclusion. The derived markovian coupled process (ξt, ζt)t≥0 satisfies: (A) if ξ0≤ζ0 (coordinate-wise), then for all t≥0, ξt≤ζt a.s. In this paper, we consider generalized misanthrope models which are conservative particle systems on ℤd such that, in each transition, k particles may jump from a site x to another site y,...

Covariance structure of wide-sense Markov processes of order k ≥ 1

Arkadiusz Kasprzyk, Władysław Szczotka (2006)

Applicationes Mathematicae

A notion of a wide-sense Markov process X t of order k ≥ 1, X t ∼ W M ( k ) , is introduced as a direct generalization of Doob’s notion of wide-sense Markov process (of order k=1 in our terminology). A base for investigation of the covariance structure of X t is the k-dimensional process x t = ( X t - k + 1 , . . . , X t ) . The covariance structure of X t ∼ W M ( k ) is considered in the general case and in the periodic case. In the general case it is shown that X t ∼ W M ( k ) iff x t is a k-dimensional WM(1) process and iff the covariance function of x t has the triangular property....

Cramér type moderate deviations for Studentized U-statistics

Tze Leng Lai, Qi-Man Shao, Qiying Wang (2011)

ESAIM: Probability and Statistics

Let Tn be a Studentized U-statistic. It is proved that a Cramér type moderate deviation P(Tn ≥ x)/(1 − Φ(x)) → 1 holds uniformly in x ∈ [0, o(n1/6)) when the kernel satisfies some regular conditions.

Cramér type moderate deviations for Studentized U-statistics******

Tze Leng Lai, Qi-Man Shao, Qiying Wang (2012)

ESAIM: Probability and Statistics

Let Tn be a Studentized U-statistic. It is proved that a Cramér type moderate deviation P(Tn ≥ x)/(1 − Φ(x)) → 1 holds uniformly in x∈ [0, o(n1/6)) when the kernel satisfies some regular conditions.

Critical constants for recurrence of random walks on G -spaces

Anna Erschler (2005)

Annales de l’institut Fourier

We introduce the notion of a critical constant c r t for recurrence of random walks on G -spaces. For a subgroup H of a finitely generated group G the critical constant is an asymptotic invariant of the quotient G -space G / H . We show that for any infinite G -space c r t ≥ 1 / 2 . We say that G / H is very small if c r t < 1 . For a normal subgroup H the quotient space G / H is very small if and only if it is finite. However, we give examples of infinite very small G -spaces. We show also that critical constants for recurrence can be used...

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