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

Large deviation estimates are derived for sums of random variables with certain dependence structures, including finite population statistics and random graphs. The argument is based on Stein's method, but with a novel modification of Stein's equation inspired by the Cramér transform.

[1]  Qi-Man Shao,et al.  Normal approximation for nonlinear statistics using a concentration inequality approach , 2007, 0708.4272.

[2]  Andrew D. Barbour,et al.  Stein's Method And Applications , 2005 .

[3]  Louis H. Y. Chen,et al.  An Introduction to Stein's Method , 2005 .

[4]  S. Janson,et al.  Upper tails for subgraph counts in random graphs , 2004 .

[5]  Martin Raič A Multivariate CLT for Decomposable Random Vectors with Finite Second Moments , 2004 .

[6]  Louis H. Y. Chen,et al.  Normal approximation under local dependence , 2004, math/0410104.

[7]  Svante Janson,et al.  Large deviations for sums of partly dependent random variables , 2004, Random Struct. Algorithms.

[8]  V. V. Petrov On Probabilities of Moderate Deviations , 2002 .

[9]  Louis H. Y. Chen,et al.  Uniform and Non-uniform Bounds in Normal Approximation for Nonlinear Statistics , 2002 .

[10]  Qi-Man Shao,et al.  A non-uniform Berry–Esseen bound via Stein's method , 2001 .

[11]  Large deviation results for a U-statistical sum with product kernel , 2001, Bulletin of the Australian Mathematical Society.

[12]  JI f. Some Large Deviation Results for Sparse Random Graphs , 2001 .

[13]  Marco Scarsini,et al.  On the Number of Pure Strategy Nash Equilibria in Random Games , 2000, Games Econ. Behav..

[14]  Y. Rinott,et al.  Normal approximations by Stein's method , 2000 .

[15]  Robert W. Keener,et al.  Tail probability approximations for U-statistics , 1998 .

[16]  Y. Rinott,et al.  On coupling constructions and rates in the CLT for dependent summands with applications to the antivoter model and weighted $U$-statistics , 1997 .

[17]  G. Reinert,et al.  Stein's method and the zero bias transformation with application to simple random sampling , 1997, math/0510619.

[18]  Yosef Rinott,et al.  Multivariate normal approximations by Stein's method and size bias couplings , 1996 .

[19]  Y. Rinott,et al.  A Multivariate CLT for Local Dependence withn -1/2 log nRate and Applications to Multivariate Graph Related Statistics , 1996 .

[20]  Amir Dembo,et al.  Some Examples of Normal Approximations by Stein’s Method , 1996 .

[21]  A. Gorchakov Upper estimates for semi-invariants of the sum of multi-indexed random variables , 1995 .

[22]  Harald Cram'er,et al.  Sur un nouveau théorème-limite de la théorie des probabilités , 2018 .

[23]  T. Ledwina,et al.  Cramér-Type Large Deviations for Some U-Statistics , 1994 .

[24]  Y. Rinott On normal approximation rates for certain sums of dependent random variables , 1994 .

[25]  F. Götze,et al.  The Rate of Convergence for Multivariate Sampling Statistics , 1993 .

[26]  M. Puri,et al.  Some asymptotic results for a broad class of nonparametric statistics , 1992 .

[27]  F. Götze On the Rate of Convergence in the Multivariate CLT , 1991 .

[28]  A. Sakhanenko,et al.  Berry-esseen type estimates for large deviation probabilities , 1991 .

[29]  A. K. Aleshkevichiene Probabilities of large deviations for U-statistics and von Mises functionals , 1991 .

[30]  L. Saulis,et al.  Limit theorems for large deviations , 1991 .

[31]  Jean-Marie Dufour,et al.  Simple exact bounds for distributions of linear signed rank statistics , 1992 .

[32]  A. Barbour Stein's method for diffusion approximations , 1990 .

[33]  Svante Janson,et al.  Poisson Approximation for Large Deviations , 1990, Random Struct. Algorithms.

[34]  V. Bentkus,et al.  Probabilities of large deviations for L-statistics , 1990 .

[35]  Andrzej Rucinski,et al.  A central limit theorem for decomposable random variables with applications to random graphs , 1989, J. Comb. Theory B.

[36]  W. Schneller Edgeworth Expansions for Linear Rank Statistics , 1989 .

[37]  Pierre Baldi,et al.  A Normal Approximation for the Number of Local Maxima of a Random Function on a Graph , 1989 .

[38]  M. Puri,et al.  On the Rate of Convergence in Normal Approximation and Large Deviation Probabilities for a Class of Statistics , 1988 .

[39]  C. Stein Approximate computation of expectations , 1986 .

[40]  Cramer Type Large Deviations for Generalized Rank Statistics , 1985 .

[41]  E. Bolthausen An estimate of the remainder in a combinatorial central limit theorem , 1984 .

[42]  M. Vandemaele On Large Deviation Probabilities for U-Statistics , 1983 .

[43]  N. N. Amosova,et al.  Probabilities of moderate deviations , 1982 .

[44]  A. Barbour Poisson convergence and random graphs , 1982 .

[45]  W. Kallenberg Cramér type large deviations for simple linear rank statistics , 1982 .

[46]  N. Amosova Local limit theorems for probabilities of moderate deviations , 1980 .

[47]  Large Deviation Probabilities for U-Statistics , 1979 .

[48]  L. Saulis,et al.  A general lemma on probabilities of large deviations , 1978 .

[49]  Louis H. Y. Chen,et al.  An $L_p$ Bound for the Remainder in a Combinatorial Central Limit Theorem , 1978 .

[50]  J. Robinson Large Deviation Probabilities for Samples from a Finite Population , 1977 .

[51]  Louis H. Y. Chen Poisson Approximation for Dependent Trials , 1975 .

[52]  S. Hodge,et al.  Statistics and Probability , 1972 .

[53]  C. Stein A bound for the error in the normal approximation to the distribution of a sum of dependent random variables , 1972 .

[54]  W. Rudin Real and complex analysis , 1968 .

[55]  Pranab Kumar Sen,et al.  On the Properties of U-Statistics When the Observations are not Independent , 1963 .

[56]  Pranab Kumar Sen,et al.  On the Properties of U-Statistics when the Observations are not Independent , 1963 .

[57]  W. Hoeffding A Class of Statistics with Asymptotically Normal Distribution , 1948 .