I: 05, 54-71, LNM 39 (1967)
FERNIQUE, Xavier
Séries de distributions aléatoires indépendantes (2 talks) (
Miscellanea)
This is part of X.~Fernique's research on random distributions (probability measures on ${\cal D}'$, and more generally on the dual space $E'$ of a nuclear LF space $E$) and their characteristic functions, which are exactly, according to Minlos' theorem, the continuous positive definite functions on $E$ assuming the value $1$ at $0$. Here it is proved that a series of independent random distributions converges a.s. if and only if the product of their characteristic functions converges pointwise to a continuous limit, and converges a.s. after centering if and only if the product of absolute values converges
Comment: See for further results
Ann. Inst. Fourier, 17-1, 1967;
Invent. Math.,
3, 1967, and
C.R. Acad. Sc.,
266, 1968 for the extension of Lévy's continuity theorem (also presented at Séminaire Bourbaki, June 1966,
311)
Keywords: Random distributions,
Minlos theoremNature: Original
Retrieve article from Numdam
VIII: 08, 78-79, LNM 381 (1974)
FERNIQUE, Xavier
Une démonstration simple du théorème de R.M.~Dudley et M.~Kanter sur les lois 0-1 pour les mesures stables (
Miscellanea)
The theorem concerns stable laws on a linear space, and asserts that every measurable linear subspace has probability 0 or 1. The title describes accurately the paper
Keywords: Stable measuresNature: Original Retrieve article from Numdam
IX: 14, 318-335, LNM 465 (1975)
FERNIQUE, Xavier
Des résultats nouveaux sur les processus gaussiens (
Gaussian processes)
Given a centered Gaussian process indexed by an arbitrary set~$T$, a major problem has been to find conditions implying that the sample functions are a.s. bounded, or a.s. continuous in the natural metric associated with the covariance. Here new necessary conditions for boundedness are given, which turn out to be sufficient in the case of stationary processes on $
R^n$. The conditions given here involve the existence of a majorizing measure, an idea which became crucial in the theory
Comment: For a systematic account of the theory around the time this paper was written, see Fernique's lectures in
École d'Été de Saint-Four~IV, LNM
480, 1974. For the definitive solution, see chapter 11 of Ledoux-Talagrand
Probability in Banach spaces, Springer 1991
Keywords: Gaussian processes,
Sample path regularityNature: Original Retrieve article from Numdam
XII: 45, 691-706, LNM 649 (1978)
FERNIQUE, Xavier
Caractérisation de processus à trajectoires majorées ou continues (
Miscellanea,
Gaussian processes)
The methods which lead the author to necessary and sufficient conditions for boundedness or continuity of stationary Gaussian processes are extended and applied to non-stationary Gaussian processes and non-Gaussian processes
Keywords: Sample path regularityNature: Original Retrieve article from Numdam
XV: 01, 1-5, LNM 850 (1981)
FERNIQUE, Xavier
Sur les lois de certaines intégrales associées à des mouvements browniens (
Brownian motion)
Let $(Z_n)$ be a sequence of independent standard Brownian motions. Define by induction a sequence of processes $U_k$ by $U_0=Z_0$, $U_k(t)=\int_0^tU_{k-1}(s)dZ_k(s)$. Let $g_k(x)$ be the density of the random variable $U_k(1)$. Then the decrease at infinity of $g_k(x)$ is of the order $\exp(-C|x|^{\alpha})$ with $\alpha=2/(k+1)$ (slightly incorrect statement, see the paper for details)
Keywords: Iterated stochastic integralsNature: Original Retrieve article from Numdam
XV: 02, 6-10, LNM 850 (1981)
FERNIQUE, Xavier
Sur le théorème de Kantorovitch-Rubinstein dans les espaces polonais (
Measure theory)
The theorem asserts the existence, given two probability measures $\mu,\nu$ on a complete separable metric space $(S,d)$, of a measure $\pi$ on $S\times S$ with marginals $\mu$ and $\nu$ such that $\int d(x,y)\,\pi(dx,dy)$ realizes a suitable distance between $\mu$ and $\nu$. An elementary proof is given here by reduction to the compact case
Keywords: Convergence in lawNature: New proof of known results Retrieve article from Numdam
XXV: 17, 178-195, LNM 1485 (1991)
FERNIQUE, Xavier
Convergence en loi de fonctions aléatoires continues ou cadlag, propriétés de compacité des lois Retrieve article from Numdam
XXVII: 22, 216-232, LNM 1557 (1993)
FERNIQUE, Xavier
Convergence en loi de variables aléatoires et de fonctions aléatoires, propriétés de compacité des lois, II Retrieve article from Numdam
