For sequences of non-lattice weakly dependent random variables, we obtain asymptotic expansions for Large Deviation Principles. These expansions, commonly referred to as strong large deviation results, are in the spirit of Edgeworth Expansions for the Central Limit Theorem. We show that the results are applicable to Diophantine iid sequences, finite state Markov chains, strongly ergodic Markov chains and Birkhoff sums of smooth expanding maps & subshifts of finite type.
P.Bougerol and J.Lacroix, Products of Random Matrices with Applications to Schrödinger Operators, 1st edn, Progress in Probability and Statistics, Birkhäuser Basel, Boston, 1985.
6.
E.Breuillard, Distributions diophantiennes et théorèmes limite local sur , Probab. Theory Related Fields132(1) (2005), 39–73.
7.
D.Buraczewsk and S.Mentemeier, Precise large deviation results for products of random matrices, Ann. Inst. H. Poincaré Probab. Statist.52(3) (2016), 1474–1513. doi:10.1214/15-AIHP684.
8.
N.R.Chaganty and J.Sethuraman, Strong large deviation and local limit theorems, Ann. Probab.21(3) (1993), 1671–1690. doi:10.1214/aop/1176989136.
9.
P.L.Chebyshev, Sur deux théorèmes relatifs aux probabilités, Acta mathematica14 (1890), 305–315.
10.
H.Cramér, Sur un nouveau theorémè-limite de la thèorie des probabilités, in: Actualités Scientifiques et Industrielles, Vol. 736, Hermann & Cie, Paris, 1938, pp. 2–23.
11.
J.De Simoi and C.Liverani, Limit theorems for fast-slow partially hyperbolic systems, Invent. Math.213 (2018), 811–1016.
12.
D.Deltuviene and L.Saulis, Asymptotic expansion of the distribution density function for the sum of random variables in the series scheme in large deviation zones, Acta Applicandae Mathematicae78 (2003), 87–97. doi:10.1023/A:1025783905023.
13.
F.den Hollander, Large Deviations, Fields Institute Monographs, Vol. 14. American Mathematical Society, Providence, RI, 2000.
14.
D.Dolgopyat and K.Fernando, An error term in the Central Limit Theorem for sums of discrete random variables, preprint.
15.
C.-G.Esséen, Fourier analysis of distribution functions. A mathematical study of the Laplace–Gaussian law, Acta Math.77 (1945), 1–125. doi:10.1007/BF02392223.
16.
W.Feller, An Introduction to Probability Theory and Its Applications Vol. II, 2nd edn, John Wiley & Sons, Inc., New York–London–Sydney, 1971.
17.
K.Fernando and C.Liverani, Edgeworth expansions for weakly dependent random variables, to appear in Ann I. H. Poincare-PR.
18.
B.V.Gnedenko and A.N.Kolmogorov, Limit Distributions for Sums of Independent Random Variables, Revised edn, Addison-Wisely, Reading MA, 1967. Translated from the Russian, annonated, and revised by K. L. Chung. With Appendices by J. L. Doob and P. L. Hsu.
19.
F.Götze and C.Hipp, Asymptotic expansions for sums of weakly dependent random vectors, Z. Wahrscheinlickeitstheorie Verw.64 (1983), 211–239. doi:10.1007/BF01844607.
20.
S.Gouezel, Limit theorems in dynamical systems using the spectral method. Hyperbolic dynamics, fluctuations and large deviations, in: Proc. Sympos. Pure Math., Vol. 89, AMS, Providence, RI, 2015, pp. 161–193.
21.
I.Grama, E.Le Page and M.Peigné, Conditioned limit theorems for products of random matrices, Prob. Theory and Relat. Fields168 (2017), 601–639. doi:10.1007/s00440-016-0719-z.
22.
Y.Guivarc’h and J.Hardy, Théorèmes limites pour une classe de chaînes de Markov et applications aux difféomorphismes d’Anosov, Annales de l’I. H. P. Probabilités et statistiques24(1) (1988), 73–98.
23.
Y.Guivarc’h and E.Le Page, Spectral gap properties for linear random walks and Pareto’s asymptotics for affine stochastic recursions, Ann. Inst. H. Poincaré Probab. Statist.52(2) (2016), 503–574. doi:10.1214/15-AIHP668.
24.
P.Hebbar, L.Koralov and J.Nolen, Branching diffusion processes in periodic media, 2019, preprint.
25.
H.Hennion and L.Hervé, Limit Theorems for Markov Chains and Stochastic Properties of Dynamical Systems by Quasi-Compactness, 1st edn, Lecture Notes in Mathematics, Springer-Verlag, Berlin Heidelberg, 2001.
26.
L.Hervé and F.Pène, The Nagaev–Guivarc’h method via the Keller–Liverani theorem, Bull. Soc. Math. France138(3) (2010), 415–489. doi:10.24033/bsmf.2594.
27.
I.A.Ibragimov and Y.V.Linnik, Independent and Stationary Sequences of Random Variables, Wolters-Noordhoff Publishing, Groningen, 1971, With a supplementary chapter by I. A. Ibragimov and V. V. Petrov. Translation from the Russian edited by J. F. C. Kingman.
28.
C.Joutard, Strong large deviations for arbitrary sequences of random variables, Ann. Inst. Stat. Math.65(1) (2013), 49–67. doi:10.1007/s10463-012-0361-1.
29.
T.Kato, Perturbation Theory for Linear Operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995, reprint of the 1980 edition.
30.
H.Kesten, Random difference equations and renewal theory for products of random matrices, Acta Math.131 (1973), 207–248. doi:10.1007/BF02392040.
31.
I.Kontoyiannis and S.P.Meyn, Spectral theory and limit theorems for geometrically ergodic Markov chains, Ann. App. Prob.13(1) (2003), 304–362. doi:10.1214/aoap/1042765670.
32.
E.Le Page, Théorèmes limites pour les produits de matrices alèatoires, in: Probability Measures on Groups, Springer, Berlin Heidelberg, 1982, pp. 258–303.
33.
S.V.Nagaev, Some limit theorems for stationary Markov chains, Theory Probab. Appl.2(4) (1959), 378–406. doi:10.1137/1102029.
34.
S.V.Nagaev, More exact statement of limit theorems for homogeneous Markov chain, Theory Probab. Appl.6(1) (1961), 62–81. doi:10.1137/1106005.
35.
W.Parry and M.Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, in: Astérisque, Société Mathématique de France, 1990, pp. 187–188.
36.
F.Pène, Mixing and decorrelation in infinite measure: The case of periodic Sinai billiard, Ann. Inst. H. Poincaré Probab. Statist.5(1) (2019), 378–411. doi:10.1214/18-AIHP885.
37.
V.V.Petrov, Sums of Independent Random Variables, Springer-Verlag, 1975.
38.
C.Pham, Conditioned limit theorems for products of positive random matrices, ALEA, Lat. Am. J. Probab. Math. Stat.15 (2018), 67–100. doi:10.30757/ALEA.v15-04.
39.
L.Saulis and V.A.Statulevicius, Limit Theorems for Large Deviations, Kluwer Academic Publishers, 1991.
40.
V.G.Sprindzuk, Metric Theory of Diophantine Approximations, Scripta Series in Math., V. H. Winston & Sons, Washington, DC, 1979.
41.
T.Tsuchida, Long-time asymptotics of heat kernels for one-dimensional elliptic operators with periodic coefficients, Proc. London Math. Soc.97(2) (2008), 450–476. doi:10.1112/plms/pdn014.
42.
L.-S.Young, Some large deviation results for dynamical systems, Trans. of the AMS318(2) (1990), 525–543. doi:10.1090/S0002-9947-1990-0975689-7.
