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Expansions for Repeated Integrals of Products with Applications to the Multivariate Normal

Christopher S. Withers, Saralees Nadarajah (2012)

ESAIM: Probability and Statistics

We extend Leibniz' rule for repeated derivatives of a product to multivariate integrals of a product. As an application we obtain expansions for P(a < Y < b) for Y ~ Np(0,V) and for repeated integrals of the density of Y. When V−1y > 0 in R3 the expansion for P(Y < y) reduces to one given by [H. Ruben J. Res. Nat. Bureau Stand. B 68 (1964) 3–11]. in terms of the moments of Np(0,V−1). This is shown to be a special case of an expansion in terms of the multivariate Hermite polynomials. These...

Expansions for Repeated Integrals of Products with Applications to the Multivariate Normal

Christopher S. Withers, Saralees Nadarajah (2011)

ESAIM: Probability and Statistics

We extend Leibniz' rule for repeated derivatives of a product to multivariate integrals of a product. As an application we obtain expansions for P(a &lt; Y &lt; b) for Y ~ Np(0,V) and for repeated integrals of the density of Y. When V−1y &gt; 0 in R3 the expansion for P(Y &lt; y) reduces to one given by [H. Ruben J. Res. Nat. Bureau Stand. B 68 (1964) 3–11]. in terms of the moments of Np(0,V−1). This is shown to be a special case of an expansion in terms of the multivariate Hermite...

Exponential concentration for first passage percolation through modified Poincaré inequalities

Michel Benaïm, Raphaël Rossignol (2008)

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

We provide a new exponential concentration inequality for first passage percolation valid for a wide class of edge times distributions. This improves and extends a result by Benjamini, Kalai and Schramm (Ann. Probab.31 (2003)) which gave a variance bound for Bernoulli edge times. Our approach is based on some functional inequalities extending the work of Rossignol (Ann. Probab.35 (2006)), Falik and Samorodnitsky (Combin. Probab. Comput.16 (2007)).

Exponential deficiency of convolutions of densities

Iosif Pinelis (2012)

ESAIM: Probability and Statistics

If a probability density p(x) (x ∈ ℝk) is bounded and R(t) := ∫e〈x, tu〉p(x)dx &lt; ∞ for some linear functional u and all t ∈ (0,1), then, for each t ∈ (0,1) and all large enough n, the n-fold convolution of the t-tilted density p ˜ t ˜pt := e〈x, tu〉p(x)/R(t) is bounded. This is a corollary of a general, “non-i.i.d.” result, which is also shown to enjoy a certain optimality property. Such results and their corollaries stated in terms of the absolute integrability of the corresponding characteristic...

Exponential deficiency of convolutions of densities∗

Iosif Pinelis (2012)

ESAIM: Probability and Statistics

If a probability density p(x) (x ∈ ℝk) is bounded and R(t) := ∫e〈x, tu〉p(x)dx < ∞ for some linear functional u and all t ∈ (0,1), then, for each t ∈ (0,1) and all large enough n, the n-fold convolution of the t-tilted density p ˜ t := e〈x, tu〉p(x)/R(t) is bounded. This is a corollary of a general, “non-i.i.d.” result, which is also shown to enjoy a certain optimality property. Such results and their corollaries stated in terms of the absolute integrability of the corresponding characteristic...

Currently displaying 341 – 360 of 1158