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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 theorem

Nature: Original

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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 measures

Nature: Original

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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 regularity

Nature: Original

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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 regularity

Nature: Original

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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 integrals

Nature: Original

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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 law

Nature: New proof of known results

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XXV: 17, 178-195, LNM 1485 (1991)

**FERNIQUE, Xavier**

Convergence en loi de fonctions aléatoires continues ou cadlag, propriétés de compacité des lois

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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

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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

Keywords: Random distributions, Minlos theorem

Nature: Original

Retrieve article from Numdam

VIII: 08, 78-79, LNM 381 (1974)

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 measures

Nature: Original

Retrieve article from Numdam

IX: 14, 318-335, LNM 465 (1975)

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 $

Comment: For a systematic account of the theory around the time this paper was written, see Fernique's lectures in

Keywords: Gaussian processes, Sample path regularity

Nature: Original

Retrieve article from Numdam

XII: 45, 691-706, LNM 649 (1978)

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 regularity

Nature: Original

Retrieve article from Numdam

XV: 01, 1-5, LNM 850 (1981)

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 integrals

Nature: Original

Retrieve article from Numdam

XV: 02, 6-10, LNM 850 (1981)

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 law

Nature: New proof of known results

Retrieve article from Numdam

XXV: 17, 178-195, LNM 1485 (1991)

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)

Convergence en loi de variables aléatoires et de fonctions aléatoires, propriétés de compacité des lois, II

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