Compound Poisson Approximations in $$\ell _p$$-norm for Sums of Weakly Dependent Vectors

Čekanavičius, Vydas; Vellaisamy, P.

doi:10.1007/s10959-020-01042-9

Cited by 4 publications

(4 citation statements)

References 28 publications

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“…In this paper we use the Stein method to estimate the Wasserstein distance between a nonnegative integer-valued random vector and a Poisson random vector. This problem has been studied by several authors, mostly in terms of the total variation distance; among others we mention [1,3,4,6,13,28,29]. Furthermore, we use our abstract result on multivariate Poisson approximation to derive a limit theorem for the Poisson process approximation.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…, Y (n) are m-dependent. Note that the case of sums of 1-dependent random vectors has recently been treated in [13] using metrics that are weaker than the total variation distance. To the best of our knowledge, this is the first paper where the Poisson approximation of the sum of m-dependent Bernoulli random vectors is investigated in terms of the Wasserstein distance.…”

Section: Sum Of M-dependent Bernoulli Random Vectorsmentioning

confidence: 99%

See 1 more Smart Citation

Multivariate Poisson and Poisson process approximations with applications to Bernoulli sums and -statistics

Pianoforte

Turin

2022

J. Appl. Probab.

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This article derives quantitative limit theorems for multivariate Poisson and Poisson process approximations. Employing the solution of the Stein equation for Poisson random variables, we obtain an explicit bound for the multivariate Poisson approximation of random vectors in the Wasserstein distance. The bound is then utilized in the context of point processes to provide a Poisson process approximation result in terms of a new metric called $d_\pi$ , stronger than the total variation distance, defined as the supremum over all Wasserstein distances between random vectors obtained by evaluating the point processes on arbitrary collections of disjoint sets. As applications, the multivariate Poisson approximation of the sum of m-dependent Bernoulli random vectors, the Poisson process approximation of point processes of U-statistic structure, and the Poisson process approximation of point processes with Papangelou intensity are considered. Our bounds in $d_\pi$ are as good as those already available in the literature.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Sum Of M-dependent Bernoulli Random Vectorsmentioning

confidence: 99%

Multivariate Poisson and Poisson process approximations with applications to Bernoulli sums and -statistics

Pianoforte

Turin

2022

J. Appl. Probab.

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“…, X n become independent random variables. See Kumar [14,15], Röllin [19] and Čekanavičius and Vellaisamy [7,8] for a similar locally dependent setup.…”

Section: Bounds For Binomial Approximationmentioning

confidence: 99%

Binomial Approximation to Locally Dependent CDO

Kumar¹,

Vellaisamy²

2022

Preprint

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In this paper, we develop Stein's method for binomial approximation using the stop-loss metric that allows one to obtain a bound on the error term between the expectation of call functions. We obtain the results for a locally dependent collateralized debt obligation (CDO), under certain conditions on moments. The results are also exemplified for an independent CDO. Finally, it is shown that our bounds are sharper than the existing bounds.

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“…, x d ) ∈ R d . The accuracy of the multivariate Poisson approximation has mostly been studied in terms of the total variation distance; among others we mention [1,3,4,6,13,26,27]. In contrast, we consider the Wasserstein distance.…”

Section: E[g(x)] − E[g(p)]mentioning

confidence: 99%

Multivariate Poisson and Poisson process approximations with applications to Bernoulli sums and $U$-statistics

Pianoforte,

Turin

2021

Preprint

View full text Add to dashboard Cite

This article derives quantitative limit theorems for multivariate Poisson and Poisson process approximations. Employing the solution of Stein's equation for Poisson random variables, we obtain an explicit bound for the multivariate Poisson approximation of random vectors in the Wasserstein distance. The bound is then utilized in the context of point processes to provide a Poisson process approximation result in terms of a new metric called d π , stronger than the total variation distance, defined as the supremum over all Wasserstein distances between random vectors obtained by evaluating the point processes on arbitrary collections of disjoint sets. As applications, the multivariate Poisson approximation of the sum of m-dependent Bernoulli random vectors, the Poisson process approximation of point processes of U -statistic structure and the Poisson process approximation of point processes with Papangelou intensity are considered. Our bounds in d π are as good as those already available in the literature.

show abstract

Compound Poisson Approximations in $$\ell _p$$-norm for Sums of Weakly Dependent Vectors

Cited by 4 publications

References 28 publications

Multivariate Poisson and Poisson process approximations with applications to Bernoulli sums and -statistics

Multivariate Poisson and Poisson process approximations with applications to Bernoulli sums and -statistics

Binomial Approximation to Locally Dependent CDO

Multivariate Poisson and Poisson process approximations with applications to Bernoulli sums and $U$-statistics

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