# Poisson perturbations

ESAIM: Probability and Statistics (2010)

- Volume: 3, page 131-150
- ISSN: 1292-8100

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topBarbour, Andrew D., and Xia, Aihua. "Poisson perturbations." ESAIM: Probability and Statistics 3 (2010): 131-150. <http://eudml.org/doc/197774>.

@article{Barbour2010,

abstract = {
Stein's method is used to prove approximations in total variation to the
distributions of integer valued random variables by (possibly signed)
compound Poisson measures. For sums of independent random variables,
the results obtained are very explicit, and improve upon earlier
work of Kruopis (1983) and Čekanavičius (1997);
coupling methods are used to derive concrete expressions for the error
bounds. An example is given to illustrate the potential for application
to sums of dependent random variables.
},

author = {Barbour, Andrew D., Xia, Aihua},

journal = {ESAIM: Probability and Statistics},

keywords = {Stein's method; signed compound Poisson measure; total variation;
coupling.; coupling; sums of random variables; compound measures},

language = {eng},

month = {3},

pages = {131-150},

publisher = {EDP Sciences},

title = {Poisson perturbations},

url = {http://eudml.org/doc/197774},

volume = {3},

year = {2010},

}

TY - JOUR

AU - Barbour, Andrew D.

AU - Xia, Aihua

TI - Poisson perturbations

JO - ESAIM: Probability and Statistics

DA - 2010/3//

PB - EDP Sciences

VL - 3

SP - 131

EP - 150

AB -
Stein's method is used to prove approximations in total variation to the
distributions of integer valued random variables by (possibly signed)
compound Poisson measures. For sums of independent random variables,
the results obtained are very explicit, and improve upon earlier
work of Kruopis (1983) and Čekanavičius (1997);
coupling methods are used to derive concrete expressions for the error
bounds. An example is given to illustrate the potential for application
to sums of dependent random variables.

LA - eng

KW - Stein's method; signed compound Poisson measure; total variation;
coupling.; coupling; sums of random variables; compound measures

UR - http://eudml.org/doc/197774

ER -

## References

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- A.D. Barbour and S. Utev, Solving the Stein equation in compound Poisson approximation. Adv. in Appl. Probab.30 (1998) 449-475.
- A.D. Barbour and S. Utev, Compound Poisson approximation in total variation. Stochastic Process. Appl., to appear.
- V. Cekanavicius, Asymptotic expansions in the exponent: A compound Poisson approach. Adv. in Appl. Probab.29 (1997) 374-387.
- P. Eichelsbacher and M. Roos, Compound Poisson approximation for dissociated random variables via Stein's method (1998) preprint.
- J. Kruopis, Precision of approximations of the generalized Binomial distribution by convolutions of Poisson measures. Lithuanian Math. J.26 (1986) 37-49.
- T. Lindvall, Lectures on the coupling method. Wiley, New York (1992).
- E.L. Presman, Approximation of binomial distributions by infinitely divisible ones. Theory. Probab. Appl.28 (1983) 393-403.
- D.A. Raikov, On the decomposition of Gauss and Poisson laws. Izv. Akad. Nauk Armyan. SSR Ser. Mat.2 (1938) 91-124.
- M. Roos, Stein-Chen method for compound Poisson approximation. Ph.D. Dissertation, University of Zürich (1993).

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