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III: 14, 175-189, LNM 88 (1969)

**MEYER, Paul-André**

Processus à accroissements indépendants et positifs (Markov processes, Independent increments)

This is an exposition of the theory of subordinators (Lévy processes with increasing paths), aiming at presenting Chung's conjecture that a certain identity known to hold a.e. actually holds everywhere, also equivalent to the fact that single points are polar sets for subordinators without drift

Comment: The conjecture was proved by Kesten (see 503) who actually knew of the problem through this talk. See also 502

Keywords: Subordinators, Polar sets

Nature: Exposition

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III: 15, 190-229, LNM 88 (1969)

**MORANDO, Philippe**

Mesures aléatoires (Independent increments)

This paper consists of two talks, on the construction and structure of measures with independent values on an abstract measurable space, inspired by papers of Prekopa (*Acta Math. Acad. Sci. Hung.,* **7**, 1956 and **8**, 1957) and Kingman (*Pacific J. Math.,* **21**, 1967)

Comment: If the measurable space is not ``too'' abstract, it can be imbedded into the line, and the standard theory of Lévy processes (non-homogeneous) can be used. This simple remark reduces the interest of the general treatment: see Dellacherie-Meyer,*Probabilités et potentiel,* Chapter XIII, end of \S4

Keywords: Random measures, Independent increments

Nature: Exposition, Original additions

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V: 02, 17-20, LNM 191 (1971)

**ASSOUAD, Patrice**

Démonstration de la ``Conjecture de Chung'' par Carleson (Markov processes, Independent increments)

Chung conjectured that singletons are polar sets for driftless subordinators. This paper gives Carleson's (unpublished) analytic proof of it

Comment: See Chung,*C. R. Acad. Sci. *, **260**, 1965, p.4665. For the statement of the problem see Meyer 314. For Kesten's earlier (contrary to a statement in the paper!) probabilistic proof see Bretagnolle 503. See also *Séminaire Bourbaki * 21th year, **361**, June 1969

Keywords: Subordinators, Polar sets

Nature: Exposition

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V: 03, 21-36, LNM 191 (1971)

**BRETAGNOLLE, Jean**

Résultats de Kesten sur les processus à accroissements indépendants (Markov processes, Independent increments)

The question is to find all Lévy processes for which single points are polar. Kesten's answer (*Mem. Amer. Math. Soc.*, **93**, 1969) is almost complete and in particular proves Chung's conjecture. The proofs in this paper have been considerably reworked

Comment: See also 502 in the same volume

Keywords: Subordinators, Polar sets

Nature: Exposition, Original additions

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VI: 03, 51-71, LNM 258 (1972)

**BRETAGNOLLE, Jean**

$p$-variation de fonctions aléatoires~: 1. Séries de Rademacher 2. Processus à accoissements indépendants (Independent increments)

The main result of the paper is theorem III, which gives a necessary and sufficient condition for the sample paths of a centered Lévy process to have a.s. a finite $p$-variation on finite time intervals, for $1<p<2$: the process should have no Gaussian part, and $|x|^p$ be integrable near $0$ w.r.t. the Lévy measure $L(dx)$. The proof rests on discrete estimates on the $p$-variation of Rademacher series. Additional results on $h$-variation w.r.t. more general convex functions are given or mentioned

Comment: This paper improves on Millar,*Zeit. für W-theorie,* **17**, 1971

Keywords: $p$-variation, Rademacher functions

Nature: Original

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VIII: 07, 37-77, LNM 381 (1974)

**DUPUIS, Claire**

Mesure de Hausdorff de la trajectoire de certains processus à accroissements indépendants et stationnaires (Independent increments)

The problem is to show that, for symmetric Lévy processes with small jumps and without a Brownian part, there exists a natural Hausdorff measure for which almost every path up to time $t$ has ``length'' exactly $t$. The case of Brownian motion had been known for a long time, the case of stable processes was settled by S.J.~Taylor (*J. Math. Mech.*, **16**, 1967) whose methods are generalized here

Keywords: Hausdorff measures, Lévy processes

Nature: Original

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XIII: 10, 132-137, LNM 721 (1979)

**SIDIBÉ, Ramatoulaye**

Martingales locales à accroissements indépendants (Martingale theory, Independent increments)

It is shown here that a process with (stationary) independent increments which is a local martingale must be a true martingale

Comment: The case of non-stationary increments is considered in 1544. See also the errata sheet of vol. XV

Keywords: Local martingales, Lévy processes

Nature: Original

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XV: 43, 627-631, LNM 850 (1981)

**WANG, Jia-Gang**

Some remarks on processes with independent increments (Independent increments)

This paper contains results on non-homogeneous processes with independent increments, without fixed discontinuities, which belong to the folklore of the subject but are hard to locate in the literature. The first one is that their natural filtration, merely augmented by all sets of measure $0$, is automatically right-continuous and quasi-left-continuous. The second one concerns those processes which are multivariate point processes, i.e., have only finitely many jumps in finite intervals and are constant between jumps. It is shown how to characterize the independent increments property into a property of the process of jumps conditioned by the process of jump times. Finally, a remark is done to the order that several results extend automatically to random measures with independent increments, for which see also 1544

Keywords: Poisson processes, Lévy measures

Nature: Original

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XV: 44, 632-642, LNM 850 (1981)

**SIDIBÉ, Ramatoulaye**

Mesures à accroissements indépendants et P.A.I. non homogènes (Independent increments)

This is an improved version of 1310: the classical theorem of Lévy on the structure of processes with independent increments is elegantly proved by martingale methods, in the non-homogeneous case, and it is proved that the process is a special semimartingale if and only if it is integrable

Nature: New proof of known results

Retrieve article from Numdam

Processus à accroissements indépendants et positifs (Markov processes, Independent increments)

This is an exposition of the theory of subordinators (Lévy processes with increasing paths), aiming at presenting Chung's conjecture that a certain identity known to hold a.e. actually holds everywhere, also equivalent to the fact that single points are polar sets for subordinators without drift

Comment: The conjecture was proved by Kesten (see 503) who actually knew of the problem through this talk. See also 502

Keywords: Subordinators, Polar sets

Nature: Exposition

Retrieve article from Numdam

III: 15, 190-229, LNM 88 (1969)

Mesures aléatoires (Independent increments)

This paper consists of two talks, on the construction and structure of measures with independent values on an abstract measurable space, inspired by papers of Prekopa (

Comment: If the measurable space is not ``too'' abstract, it can be imbedded into the line, and the standard theory of Lévy processes (non-homogeneous) can be used. This simple remark reduces the interest of the general treatment: see Dellacherie-Meyer,

Keywords: Random measures, Independent increments

Nature: Exposition, Original additions

Retrieve article from Numdam

V: 02, 17-20, LNM 191 (1971)

Démonstration de la ``Conjecture de Chung'' par Carleson (Markov processes, Independent increments)

Chung conjectured that singletons are polar sets for driftless subordinators. This paper gives Carleson's (unpublished) analytic proof of it

Comment: See Chung,

Keywords: Subordinators, Polar sets

Nature: Exposition

Retrieve article from Numdam

V: 03, 21-36, LNM 191 (1971)

Résultats de Kesten sur les processus à accroissements indépendants (Markov processes, Independent increments)

The question is to find all Lévy processes for which single points are polar. Kesten's answer (

Comment: See also 502 in the same volume

Keywords: Subordinators, Polar sets

Nature: Exposition, Original additions

Retrieve article from Numdam

VI: 03, 51-71, LNM 258 (1972)

$p$-variation de fonctions aléatoires~: 1. Séries de Rademacher 2. Processus à accoissements indépendants (Independent increments)

The main result of the paper is theorem III, which gives a necessary and sufficient condition for the sample paths of a centered Lévy process to have a.s. a finite $p$-variation on finite time intervals, for $1<p<2$: the process should have no Gaussian part, and $|x|^p$ be integrable near $0$ w.r.t. the Lévy measure $L(dx)$. The proof rests on discrete estimates on the $p$-variation of Rademacher series. Additional results on $h$-variation w.r.t. more general convex functions are given or mentioned

Comment: This paper improves on Millar,

Keywords: $p$-variation, Rademacher functions

Nature: Original

Retrieve article from Numdam

VIII: 07, 37-77, LNM 381 (1974)

Mesure de Hausdorff de la trajectoire de certains processus à accroissements indépendants et stationnaires (Independent increments)

The problem is to show that, for symmetric Lévy processes with small jumps and without a Brownian part, there exists a natural Hausdorff measure for which almost every path up to time $t$ has ``length'' exactly $t$. The case of Brownian motion had been known for a long time, the case of stable processes was settled by S.J.~Taylor (

Keywords: Hausdorff measures, Lévy processes

Nature: Original

Retrieve article from Numdam

XIII: 10, 132-137, LNM 721 (1979)

Martingales locales à accroissements indépendants (Martingale theory, Independent increments)

It is shown here that a process with (stationary) independent increments which is a local martingale must be a true martingale

Comment: The case of non-stationary increments is considered in 1544. See also the errata sheet of vol. XV

Keywords: Local martingales, Lévy processes

Nature: Original

Retrieve article from Numdam

XV: 43, 627-631, LNM 850 (1981)

Some remarks on processes with independent increments (Independent increments)

This paper contains results on non-homogeneous processes with independent increments, without fixed discontinuities, which belong to the folklore of the subject but are hard to locate in the literature. The first one is that their natural filtration, merely augmented by all sets of measure $0$, is automatically right-continuous and quasi-left-continuous. The second one concerns those processes which are multivariate point processes, i.e., have only finitely many jumps in finite intervals and are constant between jumps. It is shown how to characterize the independent increments property into a property of the process of jumps conditioned by the process of jump times. Finally, a remark is done to the order that several results extend automatically to random measures with independent increments, for which see also 1544

Keywords: Poisson processes, Lévy measures

Nature: Original

Retrieve article from Numdam

XV: 44, 632-642, LNM 850 (1981)

Mesures à accroissements indépendants et P.A.I. non homogènes (Independent increments)

This is an improved version of 1310: the classical theorem of Lévy on the structure of processes with independent increments is elegantly proved by martingale methods, in the non-homogeneous case, and it is proved that the process is a special semimartingale if and only if it is integrable

Nature: New proof of known results

Retrieve article from Numdam