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Betti numbers of random real hypersurfaces and determinants of random symmetric matrices

Damien Gayet, Jean-Yves Welschinger (2016)

Journal of the European Mathematical Society

We asymptotically estimate from above the expected Betti numbers of random real hypersurfaces in smooth real projective manifolds. Our upper bounds grow as the square root of the degree of the hypersurfaces as the latter grows to infinity, with a coefficient involving the Kählerian volume of the real locus of the manifold as well as the expected determinant of random real symmetric matrices of given index. In particular, for large dimensions, these coefficients get exponentially small away from...

Block distribution in random strings

Peter J. Grabner (1993)

Annales de l'institut Fourier

For almost all infinite binary sequences of Bernoulli trials ( p , q ) the frequency of blocks of length k ( N ) in the first N terms tends asymptotically to the probability of the blocks, if k ( N ) increases like log 1 p N - log 1 p N - ψ ( N ) (for p q ) where ψ ( N ) tends to + . This generalizes a result due to P. Flajolet, P. Kirschenhofer and R.F. Tichy concerning the case p = q = 1 2 .

Bounds on regeneration times and limit theorems for subgeometric Markov chains

Randal Douc, Arnaud Guillin, Eric Moulines (2008)

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

This paper studies limit theorems for Markov chains with general state space under conditions which imply subgeometric ergodicity. We obtain a central limit theorem and moderate deviation principles for additive not necessarily bounded functional of the Markov chains under drift and minorization conditions which are weaker than the Foster–Lyapunov conditions. The regeneration-split chain method and a precise control of the modulated moment of the hitting time to small sets are employed in the proof....

Bounds on tail probabilities for negative binomial distributions

Peter Harremoës (2016)

Kybernetika

In this paper we derive various bounds on tail probabilities of distributions for which the generated exponential family has a linear or quadratic variance function. The main result is an inequality relating the signed log-likelihood of a negative binomial distribution with the signed log-likelihood of a Gamma distribution. This bound leads to a new bound on the signed log-likelihood of a binomial distribution compared with a Poisson distribution that can be used to prove an intersection property...

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