Local limit theorems via Landau–Kolmogorov inequalities

In this article, we prove new inequalities between some common probability metrics. Using these inequalities, we obtain novel local limit theorems for the magnetization in the Curie-Weiss model at high temperature, the number of triangles and isolated vertices in Erd\H{o}s-R\'{e}nyi random graphs, as well as the independence number in a geometric random graph. We also give upper bounds on the rates of convergence for these local limit theorems and also for some other probability metrics. Our proofs are based on the Landau-Kolmogorov inequalities and new smoothing techniques.

[1]  Adrian Röllin.  Translated Poisson approximation using exchangeable pair couplings. , 2007 .

[2]  Zero biasing and a discrete central limit theorem , 2005, math/0509444.

[3]  Milton Abramowitz,et al.  Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables , 1964 .

[4]  Richard H. Liang.  Stein ’ s method for concentration inequalities , 2007 .

[5]  Y. Peres,et al.  Local Central Limit Theorems in Stochastic Geometry , 2010, 1006.3523.

[6]  A. Barbour,et al.  Poisson Approximation , 1992 .

[7]  Edward A. Bender,et al.  Central and Local Limit Theorems Applied to Asymptotic Enumeration , 1973, J. Comb. Theory A.

[8]  Andrzej Ruciflski.  When are small subgraphs of a random graph normally distributed , 1988 .

[9]  Man Kam Kwong,et al.  Norm Inequalities for Derivatives and Differences , 1993 .

[10]  P. Eichelsbacher,et al.  Stein's method for dependent random variables occurring in Statistical Mechanics , 2009, 0908.1909.

[11]  K. B. Oldham,et al.  An Atlas of Functions. , 1988 .

[12]  A. Barbour,et al.  Total variation asymptotics for sums of independent integer random variables , 2002 .

[13]  C. Newman,et al.  Limit theorems for sums of dependent random variables occurring in statistical mechanics , 1978 .

[14]  Q. Shao,et al.  Stein's Method of Exchangeable Pairs with Application to the Curie-Weiss Model , 2009, 0907.4450.

[15]  Brendan D. McKay,et al.  The Asymptotic Number of Labeled Graphs withnVertices, qEdges, and No Isolated Vertices , 1997, J. Comb. Theory, Ser. A.

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

[17]  Hsien-Kuei Hwang,et al.  LARGE DEVIATIONS OF COMBINATORIAL DISTRIBUTIONS II. LOCAL LIMIT THEOREMS , 1998 .

[18]  V. V. Petrov.  Sums of Independent Random Variables , 1975 .

[19]  Mathew D. Penrose,et al.  Normal Approximation in Geometric Probability , 2004 .

[20]  D. McDonald,et al.  An elementary proof of the local central limit theorem , 1995 .

[21]  Univariate approximations in the infinite occupancy scheme , 2009, 0902.0879.

[22]  P. Eichelsbacher,et al.  Stein's Method for Dependent Random Variables Occuring in Statistical Mechanics , 2010 .

[23]  David R. McDonald,et al.  On Local Limit Theorem for Integer-Valued Random Variables , 1980 .

[24]  Gregory F. Lawler,et al.  Random Walk: A Modern Introduction , 2010 .

[25]  Alison L Gibbs,et al.  On Choosing and Bounding Probability Metrics , 2002, math/0209021.

[26]  V. E. Stepanov.  On the Probability of Connectedness of a Random Graph $\mathcal{G}_m (t)$ , 1970 .

[27]  Tomasz Łuczak,et al.  The phase transition in a random hypergraph , 2002 .

[28]  Anna Pósfai.  An extension of Mineka's coupling inequality , 2009 .

[29]  Bero Roos,et al.  A shorter proof of Kanter’s Bessel function concentration bound , 2006 .

[30]  J. Littlewood,et al.  Some inequalities satisfied by the integrals or derivatives of real or analytic functions , 1935 .

[31]  Louis H. Y. Chen,et al.  Normal Approximation by Stein's Method , 2010 .

[32]  Exchangeable pairs and Poisson approximation , 2004, math/0411525.

[33]  Qi-Man Shao,et al.  From Stein identities to moderate deviations , 2009, 0911.5373.

[34]  Adrian Röllin,et al.  Approximation of sums of conditionally independent variables by the translated Poisson distribution , 2005 .

[35]  Amin Coja-Oghlan,et al.  The order of the giant component of random hypergraphs , 2007, Random Struct. Algorithms.

[36]  R. Ellis,et al.  Entropy, large deviations, and statistical mechanics , 1985 .