Convex densities and their financial applications
Convolution property and exponential bounds for symmetric monotone densities
Our first theorem states that the convolution of two symmetric densities which are k-monotone on (0,∞) is again (symmetric) k-monotone provided 0 < k ≤ 1. We then apply this result, together with an extremality approach, to derive sharp moment and exponential bounds for distributions having such shape constrained densities.
Couples de Wald indéfiniment divisibles. Exemples liés à la fonction gamma d'Euler et à la fonction zeta de Riemann
A toute mesure positive sur telle que , nous associons un couple de Wald indéfiniment divisible, i.e. un couple de variables aléatoires tel que et sont indéfiniment divisibles, , et pour tout . Plus généralement, à une mesure positive sur telle que pour tout , nous associons une “famille d’Esscher” de couples de Wald indéfiniment divisibles. Nous donnons de nombreux exemples de telles familles d’Esscher. Celles liées à la fonction gamma et à la fonction zeta de Riemann possèdent...
Cumulants de vecteurs aléatoires
Decomposability of point measures in generalized convolution algebras
Distribution of a random functional of a Ferguson-Dirichlet process over the unit sphere.
Eine Momentengleichung bei Exponentialfamilien.
Ergodic properties of locally stationary processes
Euclidean distances on signed measures and application to Berry-Esséen theorems. (Distances euclidiennes sur les mesures signées et application à des théorèmes de Berry-Esséen.)
Existence of a critical point for the infinite divisibility of squares of Gaussian vectors in with non-zero mean.
Exponential deficiency of convolutions of densities
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 ˜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∗
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 := 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...
Finite dimensional determinants as characteristic functions of quadratic Wiener functionals.
Fourier transform of the probability measures.
Generalized convolutions III
Generalized convolutions IV
Generalized convolutions V
Generalized gamma convolutions, Dirichlet means, Thorin measures, with explicit examples.
Identities in law between quadratic functionals of bivariate Gaussian processes, through Fubini theorems and symmetric projections
We present three new identities in law for quadratic functionals of conditioned bivariate Gaussian processes. In particular, our results provide a two-parameter generalization of a celebrated identity in law, involving the path variance of a Brownian bridge, due to Watson (1961). The proof is based on ideas from a recent note by J.-R. Pycke (2005) and on the stochastic Fubini theorem for general Gaussian measures proved in Deheuvels et al. (2004).
Interolation, Correlation Identities, and Inequalities for Infinitely Divisible Variables.