Convergence and regularity of multiparameter strong martingales

Walsh, John B.

doi:10.1007/bf00533258

Cited by 33 publications

(11 citation statements)

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“…Let { ^( z ) } be an increasing set of complete sub-a-fields of J^and let X be an adapted process which vanishes on the axes. We will use Walsh's ((1979) pages 179-180) definition of a stopping domain D in the discrete case. Note that his definition extends D to be continuous rather than discrete.…”

Section: Appendix Amentioning

confidence: 99%

Poisson convergence for point processes on the plane

Ivanoff

1985

J Aust Math Soc A

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A compensator is defined for a point process in two dimensions. It is shown that a Poisson process is characterized by a continuous deterministic compensator. Sufficient conditions are given for convergence in distribution of a sequence of two-dimensional point processes in the Skorokhod topology to a Poisson process when the corresponding sequence of compensators converges pointwise in probability to a continuous deterministic function.1980 Mathematics subject classification (Amer. Math. Soc): primary 60 G 55; secondary 60 F 17, 60 G 48.

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Section: Appendix Amentioning

confidence: 99%

Poisson convergence for point processes on the plane

Ivanoff

1985

J Aust Math Soc A

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“…qui ont été introduites par Cairoli et Walsh (1975), possèdent des propriétés qui ressemblent beaucoup à celles des martingales à un indice. Par exemple, l'inégalité maximale démontrée par Walsh (1979) pour les martingales fortes, permet de prouver la convergence de toute martingale forte bornée dans L1 et l'existence de versions continues à droite de telles martingales.…”

Section: Remarquesunclassified

Differents types de martingales a deux indices

Nualart

1983

Lecture Notes in Mathematics

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“…The study of multiparameter processes goes back to the 70' and the theory developed for years covers multiple properties of random fields (we refer to the recent books [22] and [2] for a modern review). For instance, Cairoli and Walsh [10,34,35] have deeply investigated the extension of the martingale and stochastic integral theories to the two-parameter framework. A vast literature also concerns the Markovian aspects of random fields.…”

Section: Introductionmentioning

confidence: 99%

A set-indexed Ornstein-Uhlenbeck process

Balança

Herbin

2012

Electron. Commun. Probab.

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The purpose of this article is a set-indexed extension of the well-known Ornstein-Uhlenbeck process. The first part is devoted to a stationary definition of the random field and ends up with the proof of a complete characterization by its $L^2$-continuity, stationarity and set-indexed Markov properties. This specific Markov transition system allows to define a general \emph{set-indexed Ornstein-Uhlenbeck (SIOU) process} with any initial probability measure. Finally, in the multiparameter case, the SIOU process is proved to admit a natural integral representation.Comment: 13 page

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Convergence and regularity of multiparameter strong martingales

Cited by 33 publications

References 3 publications

Poisson convergence for point processes on the plane

Poisson convergence for point processes on the plane

Differents types de martingales a deux indices

A set-indexed Ornstein-Uhlenbeck process

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