## Bernoulli

• Bernoulli
• Volume 12, Number 3 (2006), 501-514.

### Compound Poisson process approximation for locally dependent real-valued random variables via a new coupling inequality

Michael V. Boutsikas

#### Abstract

We present a general and quite simple upper bound for the total variation distance $d TV$ between any stochastic process $( X i ) i ∈Γ$ defined over a countable space $Γ$, and a compound Poisson process on $Γ .$ This result is sufficient for proving weak convergence for any functional of the process $( X i ) i ∈Γ$ when the real-valued $X i$ are rarely non-zero and locally dependent. Our result is established after introducing and employing a generalization of the basic coupling inequality. Finally, two simple examples of application are presented in order to illustrate the applicability of our results.

#### Article information

Source
Bernoulli, Volume 12, Number 3 (2006), 501-514.

Dates
First available in Project Euclid: 28 June 2006

https://projecteuclid.org/euclid.bj/1151525133

Digital Object Identifier
doi:10.3150/bj/1151525133

Mathematical Reviews number (MathSciNet)
MR2232729

Zentralblatt MATH identifier
1114.60022

#### Citation

Boutsikas, Michael V. Compound Poisson process approximation for locally dependent real-valued random variables via a new coupling inequality. Bernoulli 12 (2006), no. 3, 501--514. doi:10.3150/bj/1151525133. https://projecteuclid.org/euclid.bj/1151525133

#### References

• [1] Arratia, R., Goldstein, L. and Gordon, L. (1989) Two moments suffice for Poisson approximations: the Chen-Stein method. Ann. Probab., 17, 9-25.
• [2] Arratia, R., Goldstein, L. and Gordon, L. (1990) Poisson approximation and the Chen-Stein method. Statist. Sci., 5, 403-434.
• [3] Barbour A.D. and Chryssaphinou O. (2001) Compound Poisson approximation: a user´s guide. Ann. Appl. Probab., 11, 964-1002.
• [4] Barbour, A.D. and Mansson, M. (2002) Compound Poisson process approximation. Ann. Probab., 30, 1492-1537.
• [5] Barbour, A.D., Holst, L. and Janson, S. (1992) Poisson Approximation. Oxford: Clarendon Press.
• [6] Barbour, A.D., Chryssaphinou, O. and Vaggelatou, E. (2001) Applications of compound Poisson approximation. In Ch. A. Charalambides, M.V. Koutras and N. Balakrishnan (eds), Probability and Statistical Models with Applications, pp. 41-62. Boca Raton, FL: Chapman & Hall.
• [7] Boutsikas, M.V. and Koutras, M.V. (2000) A bound for the distribution of the sum of discrete associated or negatively associated random variables. Ann. Appl. Probab., 10, 1137-1150.
• [8] Boutsikas, M.V. and Koutras, M.V. (2001) Compound Poisson approximation for sums of dependent random variables. In Ch.A. Charalambides, M.V. Koutras, N. Balakrishnan (eds), Probability and Statistical Models with Applications, pp. 63-86. Boca Raton, FL: Chapman & Hall.
• [9] Boutsikas, M.V. and Koutras, M.V. (2002) Modelling claim exceedances over thresholds. Insurance Math. Econom., 30, 67-83.
• [10] Boutsikas, M.V. and Vaggelatou, E. (2002) On the distance between convex-ordered random variables. Adv. Appl. Probab., 34, 349-374.
• [11] Chen, L.H.Y. (1975). Poisson approximation for dependent trials. Ann. Probab., 3, 534-545.
• [12] Chen, L.H.Y. and Xia, A. (2004) Stein´s method, Palm theory and Poisson process approximation. Ann. Probab., 32, 2545-2569.
• [13] Dembo, A. and Karlin, S. (1992) Poisson approximations for r-scan processes. Ann. Appl. Probab., 2, 329-357.
• [14] Freedmen, D. (1974) The Poisson approximation for dependent events. Ann. Probab., 2, 256-269.
• [15] Rootzén, H., Leadbetter, M.R. and De Haan, L. (1998) On the distribution of tail array sums for strongly mixing stationary sequences. Ann. Appl. Probab., 8, 868-885.
• [16] Serfling, R.J. (1975) A general Poisson approximation theorem. Ann. Probab., 3, 726-731.
• [17] Serfling, R.J. (1978) Some elementary results on Poisson approximations in a sequence of Bernoulli trials. SIAM Rev., 20, 567-579.
• [18] Serfozo, R. (1986) Compound Poisson approximation for sums of random variables Ann. Probab., 14, 1391-1398.
• [19] Vellaisamy, P. and Chaudhuri, B. (1999) On compound Poisson approximation for sums of random variables. Statist. Probab. Lett., 41, 179-189.
• [20] Wang, Y.H. (1986) Coupling methods in approximations. Canad. J. Statist., 14, 69-74.
• [21] Wang, Y.H. (1989) From Poisson to compound Poisson approximations. Math. Sci., 14, 38-49.