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XIII: 11, 138-141, LNM 721 (1979)

**REBOLLEDO, Rolando**

Décomposition des martingales locales et raréfaction des sauts (Martingale theory)

The general topic underlying this paper is that of convergence in law of a sequence of local martingales $M^n$ to a continuous Gaussian local martingale, i.e., a result analogue to the Central Limit Theorem in the Skorohod topology. This rests on three properties: tightness, convergence of the processes $<M^n,M^n>_t$ to a deterministic process, and a property of ``rarefaction of jumps''. The paper is devoted to a general discussion of the latter property

Comment: A correction is given as 1430

Keywords: Convergence in law, Tightness

Nature: Original

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XIV: 27, 227-248, LNM 784 (1980)

**JACOD, Jean**; **MÉMIN, Jean**

Sur la convergence des semimartingales vers un processus à accroissements indépendants (General theory of processes, Stochastic calculus, Martingale theory)

A method of Kabanov, Liptzer and Shiryaev is adapted to study the convergence of a sequence of semimartingales to a process with independent increments (to be completed)

Keywords: Convergence in law, Tightness

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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XV: 36, 529-546, LNM 850 (1981)

**JACOD, Jean**; **MÉMIN, Jean**

Sur un type de convergence intermédiaire entre la convergence en loi et la convergence en probabilité (Measure theory)

For simplicity we consider only real valued r.v.'s, but it is essential that the paper considers general Polish spaces instead of $**R**$. Let us define a fuzzy r.v. $X$ on $(\Omega, {\cal F},P)$ as a probability measure on $\Omega\times**R**$ whose projection on $\Omega$ is $P$. In particular, a standard r.v. $X$ defines such a measure as the image of $P$ under the map $\omega\mapsto (\omega,X(\omega))$. The space of fuzzy r.v.'s is provided with a weak topology, associated with the bounded functions $f(\omega,x)$ which are continuous in $x$ for every $\omega$, or equivalently with the functions $I_A(\omega)\,f(x)$ with $f$ bounded continuous. The main topic of this paper is the study of this topology

Comment: From this description, it is clear that this paper extends to general Polish spaces the topology of Baxter-Chacon (forgetting about the filtration), for which see 1228

Keywords: Fuzzy random variables, Convergence in law

Nature: Original

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XV: 37, 547-560, LNM 850 (1981)

**JACOD, Jean**

Convergence en loi de semimartingales et variation quadratique (General theory of processes, Stochastic calculus)

The convergence in law of cadlag processes to a cadlag process being understood in the sense of Skorohod, the problem is to find sufficient conditions under which, given semimartingales $X^n$ and $X$ such that $X^n\rightarrow X$ in law, one may deduce that $[X^n,X^n]$ converges in law to $[X,X]$. This is achieved assuming a uniform bound on the expectations of the supremum of the jumps. A version of the theorem applied to processes which are not semimartingales, but are equal to semimartingales on large sets

Keywords: Semimartingales, Skorohod topology, Convergence in law

Nature: Original

Retrieve article from Numdam

Décomposition des martingales locales et raréfaction des sauts (Martingale theory)

The general topic underlying this paper is that of convergence in law of a sequence of local martingales $M^n$ to a continuous Gaussian local martingale, i.e., a result analogue to the Central Limit Theorem in the Skorohod topology. This rests on three properties: tightness, convergence of the processes $<M^n,M^n>_t$ to a deterministic process, and a property of ``rarefaction of jumps''. The paper is devoted to a general discussion of the latter property

Comment: A correction is given as 1430

Keywords: Convergence in law, Tightness

Nature: Original

Retrieve article from Numdam

XIV: 27, 227-248, LNM 784 (1980)

Sur la convergence des semimartingales vers un processus à accroissements indépendants (General theory of processes, Stochastic calculus, Martingale theory)

A method of Kabanov, Liptzer and Shiryaev is adapted to study the convergence of a sequence of semimartingales to a process with independent increments (to be completed)

Keywords: Convergence in law, Tightness

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

XV: 36, 529-546, LNM 850 (1981)

Sur un type de convergence intermédiaire entre la convergence en loi et la convergence en probabilité (Measure theory)

For simplicity we consider only real valued r.v.'s, but it is essential that the paper considers general Polish spaces instead of $

Comment: From this description, it is clear that this paper extends to general Polish spaces the topology of Baxter-Chacon (forgetting about the filtration), for which see 1228

Keywords: Fuzzy random variables, Convergence in law

Nature: Original

Retrieve article from Numdam

XV: 37, 547-560, LNM 850 (1981)

Convergence en loi de semimartingales et variation quadratique (General theory of processes, Stochastic calculus)

The convergence in law of cadlag processes to a cadlag process being understood in the sense of Skorohod, the problem is to find sufficient conditions under which, given semimartingales $X^n$ and $X$ such that $X^n\rightarrow X$ in law, one may deduce that $[X^n,X^n]$ converges in law to $[X,X]$. This is achieved assuming a uniform bound on the expectations of the supremum of the jumps. A version of the theorem applied to processes which are not semimartingales, but are equal to semimartingales on large sets

Keywords: Semimartingales, Skorohod topology, Convergence in law

Nature: Original

Retrieve article from Numdam