CLT-related large deviation bounds based on Stein's method

Advances in Applied Probability

Volumn 39, Issue 3, 2007, Pages 731-752

  Raiĉ, Martin ^a  

^a UNIVERSITY OF LJUBLJANA   (Slovenia)

Author keywords

Central limit theorem; Finite population statistics; Large deviations; Random graphs

Indexed keywords

GRAPH THEORY; MATHEMATICAL TRANSFORMATIONS; POPULATION STATISTICS; RANDOM PROCESSES;

CENTRAL LIMIT THEOREM; FINITE POPULATION STATISTICS; LARGE DEVIATIONS; RANDOM GRAPHS;

PROBABILITY;

EID: 35948953514     PISSN: 00018678     EISSN: None     Source Type: Journal    
DOI: 10.1239/aap/1189518636     Document Type: Article

References (36)

1. BALDI, P., RINOTT, Y. AND STEIN, C. (1989). A normal approximation for the number of local maxima of a random function on a graph. In Probability, Statistics, and Mathematics, Academic Press, Boston, MA, pp. 59-81.
2. BARBOUR, A. D. (1982). Poisson convergence and random graphs. Math. Proc. Camb. Philos. Soc. 92, 349-359.
3. BARBOUR, A. D. (1990). Stein's method for diffusion approximations. Prob. Theory Relat. Fields 84, 297-322.
4. BARBOUR, A. D. AND CHEN, L. H. Y. (eds) (2005). An Introduction to Stein's Method. World Scientific, Singapore.
5. BARBOUR, A. D. AND CHEN, L. H. Y. (2005). Stein's Method and Applications. World Scientific, Singapore.
6. BARBOUR, A. D., KAROŃSKI, M. AND RUCIŃSKI, A. (1989). A central limit theorem for decomposable random variables with applications to random graphs. J. Combin. Theory Ser. B 47, 125-145.
7. BOLTHAUSEN, E. (1984). An estimate of the remainder in a combinatorial central limit theorem. Z. Wahrscheinlichkeitsth. 66, 379-386.
8. CHEN, L. H. Y. AND SHAO, Q.-M. (2001). A non-uniform Berry-Esseen bound via Stein's method. Prob. Theory Relat. Fields 120, 236-254.
9. CHEN, L. H. Y. AND SHAO, Q.-M. (2007). Normal approximation for nonlinear statistics using a concentration inequality approach. Bernoulli 13, 581-599.
10. CHEN, L. H. Y. AND SHAO, Q.-M. (2004). Normal approximation under local dependence. Ann. Prob. 32, 1985-2028.
11. CRAMÉR, H. (1938). Sur un nouveau théorème limite de la théorie des probabilités. In Colloque Consacre à la Théorie des Probabilités 736, Hermann, Paris, pp. 5-23.
12. DEMBO, A. AND RINOTT, Y. (1996). Some examples of normal approximations by Stein's method. In Random Discrete Structures (IMA Vol. Math. Appl. 76). Springer, New York, pp. 25-44.
13. GOLDSTEIN, L. AND RINOTT, Y. (1996). Multivariate normal approximations by Stein's method and size bias couplings. J. Appl. Prob. 33, 1-17.
14. GORCHAKOV, A. B. (1995). Upper bounds for cumulants of the sum of multi-indexed random variables. Discrete Math Appl. 5, 317-331.
15. GORCHAKOV, A. B. (1997). Explicit bounds for probabilities of large deviations of sums of random vectors with a given graph of dependencies. In Probabilistic Methods in Discrete Mathematics (Petrozavodsk, 1996), VSP, Utrecht, pp. 219-230.
16. GÖTZE, F. (1991). On the rate of convergence in the multivariate CLT. Ann. Prob. 19, 724-739.
17. HALLIN, M. AND PURI, M. L. (1992). Some asymptotic results for a broad class of nonparametric statistics. J. Statist. Planning Infer. 32, 165-196.
18. Ho, S.-T. AND CHEN, L. H. Y. (1978). An Lp bound for the remainder in a combinatorial central limit theorem. Ann. Prob. 6, 231-249.
19. JANSON, S. (1990). Poisson approximation for large deviations. Random Structures Algorithms 1, 221-229.
20. JANSON, S. (2004). Large deviations for sums of partly dependent random variables. Random Structures Algorithms 24, 234-248.
21. JANSON, S., OLESZKIEWICZ, K. AND RUCIŃSKI, A. (2004). Upper tails for subgraph counts in random graphs. Israel J. Math. 142, 61-92.
22. KALLENBERG, W. C. M. (1982). Cramér type large deviations for simple linear rank statistics. Z. Wahrscheinlichkeitsth. 60, 403-409.
23. KOKIC, P. N. AND WEBER, N. C. (1995). Large deviation probabilities for U-statistics based on samples from finite populations. In Exploring Stochastic Laws, VSP, Utrecht, pp. 175-181.
24. NANDI, H. K. AND SEN, P. K. (1963). On the properties of U-statistics when the observations are not independent. II. Unbiased estimation of the parameters of a finite population. Calcutta Statist. Assoc. Bull. 12, 124-148.
25. O'CONNELL, N. (1998). Some large deviation results for sparse random graphs. Prob. Theory Relat. Fields 110, 277-285.
26. RAIČ, M. (2004). A multivariate CLT for decomposable random vectors with finite second moments. J. Theoret. Prob. 17, 573-603.
27. RINOTT, Y. (1994). On normal approximation rates for certain sums of dependent random variables. J. Comput. Appl. Math. 55, 135-143.
28. -1/2 log n rate and applications to multivariate graph related statistics. J. Multivariate Anal. 56, 333-350.
29. RINOTT, Y. AND ROTAR, V. (1997). On coupling constructions and rates in the CLT for dependent summands with applications to the antivoter model and weighted U-statistics. Ann. Appl. Prob. 7, 1080-1105.
30. ROBINSON, J. (1977). Large deviation probabilities for samples from a finite population. Ann. Prob. 5, 913-925.
31. RUDZKIS, R., SAULIS, L. AND STATULEVIÇIUS, V. (1978). A general lemma on probabilities of large deviations. Lithuanian Math. J. 18, 226-238.
32. SAULIS,L. AND STATULEVIČUS, V. (1991). Limit Theorems for Large Deviations] (Math. Appl. (Soviet Set.) 73).
33. SCHNELLER, W. (1989). Edgeworth expansions for linear rank statistics. Ann. Statist. 17, 1103-1123.
34. STATULEVIČIUS, V. A. (1966). On large deviations. Z. Wahrscheinlichkeitsth. 6, 133-144.
35. STEIN, C. (1972). A bound for the error in the normal approximation to the distribution of a sum of dependent random variables. In Proc. Sixth Berkeley Symp. Math. Statist. Prob. (University of California, Berkeley, 1970/1971), Vol. II, Probability Theory, University of California Press, Berkeley, pp. 583-602.
36. STEIN, C. (1986). Approximate Computation of Expectations (Instit. Math. Statist. Lecture Notes Monogr. Ser. 7). Institute of Mathematical Statistics, Hayward, CA.