## Bernoulli

• Bernoulli
• Volume 21, Number 2 (2015), 851-880.

### Local limit theorems via Landau–Kolmogorov inequalities

#### Abstract

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 Erdős–Rényi 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.

#### Article information

Source
Bernoulli Volume 21, Number 2 (2015), 851-880.

Dates
First available in Project Euclid: 21 April 2015

https://projecteuclid.org/euclid.bj/1429624963

Digital Object Identifier
doi:10.3150/13-BEJ590

Mathematical Reviews number (MathSciNet)
MR3338649

Zentralblatt MATH identifier
1320.60065

#### Citation

Röllin, Adrian; Ross, Nathan. Local limit theorems via Landau–Kolmogorov inequalities. Bernoulli 21 (2015), no. 2, 851--880. doi:10.3150/13-BEJ590. https://projecteuclid.org/euclid.bj/1429624963

#### References

• [1] Barbour, A.D. (2009). Univariate approximations in the infinite occupancy scheme. ALEA Lat. Am. J. Probab. Math. Stat. 6 415–433.
• [2] Barbour, A.D. and Čekanavičius, V. (2002). Total variation asymptotics for sums of independent integer random variables. Ann. Probab. 30 509–545.
• [3] Barbour, A.D., Holst, L. and Janson, S. (1992). Poisson Approximation. Oxford Studies in Probability 2. New York: The Clarendon Press, Oxford Univ. Press.
• [4] 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.
• [5] Barbour, A.D. and Xia, A. (1999). Poisson perturbations. ESAIM Probab. Statist. 3 131–150.
• [6] Behrisch, M., Coja-Oghlan, A. and Kang, M. (2010). The order of the giant component of random hypergraphs. Random Structures Algorithms 36 149–184.
• [7] Bender, E.A., Canfield, E.R. and McKay, B.D. (1997). The asymptotic number of labeled graphs with $n$ vertices, $q$ edges, and no isolated vertices. J. Combin. Theory Ser. A 80 124–150.
• [8] Chatterjee, S. (2007). Stein’s method for concentration inequalities. Probab. Theory Related Fields 138 305–321.
• [9] Chatterjee, S., Diaconis, P. and Meckes, E. (2005). Exchangeable pairs and Poisson approximation. Probab. Surv. 2 64–106.
• [10] Chatterjee, S. and Shao, Q.-M. (2011). Nonnormal approximation by Stein’s method of exchangeable pairs with application to the Curie–Weiss model. Ann. Appl. Probab. 21 464–483.
• [11] Chen, L.H.Y., Fang, X. and Shao, Q.-M. (2013). From Stein identities to moderate deviations. Ann. Probab. 41 262–293.
• [12] Chen, L.H.Y., Goldstein, L. and Shao, Q.-M. (2011). Normal Approximation by Stein’s Method. Probability and Its Applications (New York). Heidelberg: Springer.
• [13] Davis, B. and McDonald, D. (1995). An elementary proof of the local central limit theorem. J. Theoret. Probab. 8 693–701.
• [14] Eichelsbacher, P. and Löwe, M. (2010). Stein’s method for dependent random variables occurring in statistical mechanics. Electron. J. Probab. 15 962–988.
• [15] Ellis, R.S. (1985). Entropy, Large Deviations, and Statistical Mechanics. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 271. New York: Springer.
• [16] Ellis, R.S., Newman, C.M. and Rosen, J.S. (1980). Limit theorems for sums of dependent random variables occurring in statistical mechanics. II. Conditioning, multiple phases, and metastability. Z. Wahrsch. Verw. Gebiete 51 153–169.
• [17] Gibbs, A.L. and Su, F.E. (2002). On choosing and bounding probability metrics. Internat. Statist. Rev. 70 419–435.
• [18] Goldstein, L. and Xia, A. (2006). Zero biasing and a discrete central limit theorem. Ann. Probab. 34 1782–1806.
• [19] Hardy, G.H., Landau, E. and Littlewood, J.E. (1935). Some inequalities satisfied by the integrals or derivatives of real or analytic functions. Math. Z. 39 677–695.
• [20] Karoński, M. and Łuczak, T. (2002). The phase transition in a random hypergraph. J. Comput. Appl. Math. 142 125–135.
• [21] Kordecki, W. (1990). Normal approximation and isolated vertices in random graphs. In Random Graphs ’87 (Poznań, 1987) 131–139. Chichester: Wiley.
• [22] Kwong, M.K. and Zettl, A. (1992). Norm Inequalities for Derivatives and Differences. Lecture Notes in Math. 1536. Berlin: Springer.
• [23] Mattner, L. and Roos, B. (2007). A shorter proof of Kanter’s Bessel function concentration bound. Probab. Theory Related Fields 139 191–205.
• [24] McDonald, D.R. (1979). On local limit theorem for integer valued random variables. Teor. Veroyatn. Primenen. 24 607–614.
• [25] Penrose, M.D. and Peres, Y. (2011). Local central limit theorems in stochastic geometry. Electron. J. Probab. 16 2509–2544.
• [26] Penrose, M.D. and Yukich, J.E. (2005). Normal approximation in geometric probability. In Stein’s Method and Applications. Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap. 5 37–58. Singapore: Singapore Univ. Press.
• [27] Petrov, V.V. (1975). Sums of Independent Random Variables. New York: Springer. Translated from the Russian by A.A. Brown, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 82.
• [28] Pósfai, A. (2009). An extension of Mineka’s coupling inequality. Electron. Commun. Probab. 14 464–473.
• [29] Röllin, A. (2005). Approximation of sums of conditionally independent variables by the translated Poisson distribution. Bernoulli 11 1115–1128.
• [30] Röllin, A. (2007). Translated Poisson approximation using exchangeable pair couplings. Ann. Appl. Probab. 17 1596–1614.
• [31] Ruciński, A. (1988). When are small subgraphs of a random graph normally distributed? Probab. Theory Related Fields 78 1–10.
• [32] Stepanov, V.E. (1970). The probability of the connectedness of a random graph $\mathcal{G}_{m}(t)$. Teor. Veroyatn. Primenen. 15 58–68.