Expansions for Gaussian processes and Parseval frames.
Expansions for Repeated Integrals of Products with Applications to the Multivariate Normal
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
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...
Expectation and variance of the volume covered by a large number of independent random sets
Expectation, conditional expectation and martingales in local fields.
Expectation of Random Polytopes.
Expected lengths of minimum spanning trees for non-identical edge distributions.
Expected number of real zeros of Lévy and harmonizable stable random polynomials.
Expected values in the alternative set theory and their applications to some limit theorems
This article presents an alternative approach to statistical moments within non-standard models and by the help of these moments some limit theorems are reformulated and proved.
Explicit bounds for the approximation error in Benford's law.
Explicit error bounds for Markov chain Monte Carlo [Book]
Explicit forms of Wick tensor powers in general white noise spaces.
Explicit formulae of distributions and densities of characteristics of a dynamic advertising and pricing model
We analyze the optimal sales process of a stochastic advertising and pricing model with constant demand elasticities. We derive explicit formulae of the densities of the (optimal) sales times and (optimal) prices when a fixed finite number of units of a product are to be sold during a finite sales period or an infinite one. Furthermore, for any time t the exact distribution of the inventory, i.e. the number of unsold items, at t is determined and will be expressed in terms of elementary functions....
Explicit formulas for correlation functions of ground states of the 1 dimensional XY model
Explicit formulas for transition intensities in the queueing system
Explicit Karhunen-Loève expansions related to the Green function of the Laplacian
Karhunen-Loève expansions of Gaussian processes have numerous applications in Probability and Statistics. Unfortunately the set of Gaussian processes with explicitly known spectrum and eigenfunctions is narrow. An interpretation of three historical examples enables us to understand the key role of the Laplacian. This allows us to extend the set of Gaussian processes for which a very explicit Karhunen-Loève expansion can be derived.
Explicit parametrix and local limit theorems for some degenerate diffusion processes
For a class of degenerate diffusion processes of rank 2, i.e. when only Poisson brackets of order one are needed to span the whole space, we obtain a parametrix representation of McKean–Singer [J. Differential Geom.1 (1967) 43–69] type for the density. We therefrom derive an explicit gaussian upper bound and a partial lower bound that characterize the additional singularity induced by the degeneracy. This particular representation then allows to give a local limit theorem with the usual convergence...
Explicit solutions of some fractional partial differential equations via stable subordinators.
Explosion en temps fini pour l’équation de Schrödinger non linéaire stochastique surcritique
Explosion time in stochastic differential equations with small diffusion.