Limit theorems for random sums of dependent d-dimensional random vectors

Beśka, Marek; Kłopotowski, Andrzej; Słomiński, Leszek

doi:10.1007/bf00537224

Cited by 19 publications

(13 citation statements)

References 2 publications

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“…The original proof can be found in [2] and [7], and further extensions to Hilbert spaces in [10]. Theorem 3.2.…”

Section: Tightness By Decouplingmentioning

confidence: 99%

Stochastic integration with respect to cylindrical Lévy processes

Jakubowski¹,

Riedle²

2017

Ann. Probab.

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A cylindrical Lévy process does not enjoy a cylindrical version of the semimartingale decomposition which results in the need to develop a completely novel approach to stochastic integration. In this work, we introduce a stochastic integral for random integrands with respect to cylindrical Lévy processes in Hilbert spaces. The space of admissible integrands consists of adapted stochastic processes with values in the space of Hilbert-Schmidt operators. Neither the integrands nor the integrator is required to satisfy any moment or boundedness condition. The integral process is characterised as an adapted, Hilbert space valued semi-martingale with càdlàg trajectories.

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“…The original proof can be found in [2] and [7], and further extensions to Hilbert spaces in [10]. Theorem 3.2.…”

Section: Tightness By Decouplingmentioning

confidence: 99%

Stochastic integration with respect to cylindrical Lévy processes

Jakubowski¹,

Riedle²

2017

Ann. Probab.

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“…The first one is found in Beśka et al (1982) and the second one is a version of the martingale CLT (see, e.g., Hall and Heyde, 1980). …”

Section: Theorem A1 (Poissonian Conditional Limit Theorem)mentioning

confidence: 99%

Double asymptotics for the chi-square statistic

Rempała

Wesołowski

2016

Statistics & Probability Letters

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“…The proof will be based on the following version of the Poisson central limit theorem for row-wise dependent triangular arrays obtained by Beśka et al (1982). Theorem 2.2.…”

Section: Past Conditioningmentioning

confidence: 99%

Poisson process via martingale and related characteristics

Wesołowski

1999

Journal of Applied Probability

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The classical martingale characterizations of the Poisson process were obtained for point process or purely discontinuous martingale i.e. under additional assumptions on properties of trajectories. Here our aim is to search for related characterizations without relying on properties of trajectories. Except for a new martingale characterization, results based on conditional moments jointly involving the past and the nearest future are presented.

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Limit theorems for random sums of dependent d-dimensional random vectors

Cited by 19 publications

References 2 publications

Stochastic integration with respect to cylindrical Lévy processes

Stochastic integration with respect to cylindrical Lévy processes

Double asymptotics for the chi-square statistic

Poisson process via martingale and related characteristics

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