Sharp connections between Berry-Esseen characteristics and Edgeworth expansions for stationary processes

Jirak, Moritz; Wu, Wei Biao; Zhao, Ou

doi:10.1090/tran/8328

Sharp connections between Berry-Esseen characteristics and Edgeworth expansions for stationary processes
HTML articles powered by AMS MathViewer

by Moritz Jirak, Wei Biao Wu and Ou Zhao PDF

Trans. Amer. Math. Soc. 374 (2021), 4129-4183 Request permission

Abstract:

Given a weakly dependent stationary process, we describe the transition between a Berry-Esseen bound and a second order Edgeworth expansion in terms of the Berry-Esseen characteristic. This characteristic is sharp: We show that Edgeworth expansions are valid if and only if the Berry-Esseen characteristic is of a certain magnitude. If this is not the case, we still get an optimal Berry-Esseen bound, thus describing the exact transition. We also obtain (fractional) expansions given $3 < p \leq 4$ moments, where a similar transition occurs. Corresponding results also hold for the Wasserstein metric $W_1$, where a related, integrated characteristic turns out to be optimal. As an application, we establish novel weak Edgeworth expansion and CLTs in $L^p$ and $W_1$. As another application, we show that a large class of high dimensional linear statistics admits Edgeworth expansions without any smoothness constraints, that is, no non-lattice condition or related is necessary. In all results, the necessary weak-dependence assumptions are very mild. In particular, we show that many prominent dynamical systems and models from time series analysis are within our framework, giving rise to many new results in these areas.

References

Elisa Alòs, A generalization of the Hull and White formula with applications to option pricing approximation, Finance Stoch. 10 (2006), no. 3, 353–365. MR 2244350, DOI 10.1007/s00780-006-0013-5
Jürgen Angst and Guillaume Poly, A weak Cramér condition and application to Edgeworth expansions, Electron. J. Probab. 22 (2017), Paper No. 59, 24. MR 3683368, DOI 10.1214/17-EJP77
Alexander Aue, Siegfried Hörmann, Lajos Horváth, and Matthew Reimherr, Break detection in the covariance structure of multivariate time series models, Ann. Statist. 37 (2009), no. 6B, 4046–4087. MR 2572452, DOI 10.1214/09-AOS707
Yves Benoist and Jean-François Quint, Central limit theorem for linear groups, Ann. Probab. 44 (2016), no. 2, 1308–1340. MR 3474473, DOI 10.1214/15-AOP1002
Rabi N. Bhattacharya and R. Ranga Rao, Normal approximation and asymptotic expansions, Classics in Applied Mathematics, vol. 64, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2010. Updated reprint of the 1986 edition [ MR0855460], corrected edition of the 1976 original [ MR0436272]. MR 3396213, DOI 10.1137/1.9780898719895.ch1
Patrick Billingsley, Convergence of probability measures, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR 0233396
Sergey G. Bobkov, Asymptotic expansions for products of characteristic functions under moment assumptions of non-integer orders, Convexity and concentration, IMA Vol. Math. Appl., vol. 161, Springer, New York, 2017, pp. 297–357. MR 3837276
Sergey G. Bobkov, Berry-Esseen bounds and Edgeworth expansions in the central limit theorem for transport distances, Probab. Theory Related Fields 170 (2018), no. 1-2, 229–262. MR 3748324, DOI 10.1007/s00440-017-0756-2
Philippe Bougerol and Jean Lacroix, Products of random matrices with applications to Schrödinger operators, Progress in Probability and Statistics, vol. 8, Birkhäuser Boston, Inc., Boston, MA, 1985. MR 886674, DOI 10.1007/978-1-4684-9172-2
E. Breuillard, Distributions diophantiennes et théorème limite local sur $\Bbb R^d$, Probab. Theory Related Fields 132 (2005), no. 1, 39–73 (French, with English summary). MR 2136866, DOI 10.1007/s00440-004-0388-1
Florentina Bunea, Alexandre B. Tsybakov, and Marten H. Wegkamp, Aggregation for Gaussian regression, Ann. Statist. 35 (2007), no. 4, 1674–1697. MR 2351101, DOI 10.1214/009053606000001587
P. Tchebycheff, Sur deux théorèmes relatifs aux probabilités, Acta Math. 14 (1890), no. 1, 305–315 (French). MR 1554802, DOI 10.1007/BF02413327
Louis H. Y. Chen, Larry Goldstein, and Qi-Man Shao, Normal approximation by Stein’s method, Probability and its Applications (New York), Springer, Heidelberg, 2011. MR 2732624, DOI 10.1007/978-3-642-15007-4
Cramér, H. 1928. On the composition of elementary errors. Skand. Aktuarietidskr. 11, no. 13–74, 141–180.
C. Cuny, J. Dedecker, A. Korepanov, and F. Merlevède, Rates in almost sure invariance principle for slowly mixing dynamical systems, Ergodic Theory Dynam. Systems 40 (2020), no. 9, 2317–2348. MR 4130806, DOI 10.1017/etds.2019.2
C. Cuny, J. Dedecker, A. Korepanov, and F. Merlevède, Rates in almost sure invariance principle for quickly mixing dynamical systems, Stoch. Dyn. 20 (2020), no. 1, 2050002, 28. MR 4066797, DOI 10.1142/S0219493720500021
Christophe Cuny, Jérôme Dedecker, and Florence Merlevède, On the Komlós, Major and Tusnády strong approximation for some classes of random iterates, Stochastic Process. Appl. 128 (2018), no. 4, 1347–1385. MR 3769665, DOI 10.1016/j.spa.2017.07.011
Christophe Cuny, Jérôme Dedecker, and Christophe Jan, Limit theorems for the left random walk on $\textrm {GL}_d(\Bbb R)$, Ann. Inst. Henri Poincaré Probab. Stat. 53 (2017), no. 4, 1839–1865 (English, with English and French summaries). MR 3729637, DOI 10.1214/16-AIHP773
Jérôme Dedecker and Emmanuel Rio, On mean central limit theorems for stationary sequences, Ann. Inst. Henri Poincaré Probab. Stat. 44 (2008), no. 4, 693–726 (English, with English and French summaries). MR 2446294, DOI 10.1214/07-AIHP117
Persi Diaconis and David Freedman, Iterated random functions, SIAM Rev. 41 (1999), no. 1, 45–76. MR 1669737, DOI 10.1137/S0036144598338446
Dolgopyat, D. and K. Fernando. 2019. An error term in the central limit theorem for sums of discrete random variables.
Yu. I. Ingster and I. A. Suslina, Nonparametric goodness-of-fit testing under Gaussian models, Lecture Notes in Statistics, vol. 169, Springer-Verlag, New York, 2003. MR 1991446, DOI 10.1007/978-0-387-21580-8
Jacopo De Simoi and Carlangelo Liverani, Limit theorems for fast-slow partially hyperbolic systems, Invent. Math. 213 (2018), no. 3, 811–1016. MR 3842060, DOI 10.1007/s00222-018-0798-9
Edgeworth, F. Y. 1984. The asymmetrical probability curve. Proceedings of the Royal Society of London 56, no. 336–339, 271–272.
B. Efron, Bootstrap methods: another look at the jackknife, Ann. Statist. 7 (1979), no. 1, 1–26. MR 515681
Bradley Efron, The jackknife, the bootstrap and other resampling plans, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 38, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa., 1982. MR 659849
M. Émery and W. Schachermayer, On Vershik’s standardness criterion and Tsirelson’s notion of cosiness, Séminaire de Probabilités, XXXV, Lecture Notes in Math., vol. 1755, Springer, Berlin, 2001, pp. 265–305. MR 1837293, DOI 10.1007/978-3-540-44671-2_{2}0
Jacob Feldman and Daniel J. Rudolph, Standardness of sequences of $\sigma$-fields given by certain endomorphisms, Fund. Math. 157 (1998), no. 2-3, 175–189. Dedicated to the memory of Wiesław Szlenk. MR 1636886, DOI 10.4064/fm-157-2-3-175-189
William Feller, An introduction to probability theory and its applications. Vol. II. , 2nd ed., John Wiley & Sons, Inc., New York-London-Sydney, 1971. MR 0270403
Fernando, K. and C. Liverani. Edgeworth expansions for weakly dependent random variables. Ann. Inst. Henri Poincare B Prob. Stat., to appear.
Fernando, K. and F. Pène. 2020. Expansions in the local and the central limit theorems for dynamical systems. Available at https://arxiv.org/abs/2008.08726.
J.-P. Fouque, G. Papanicolaou, R. Sircar, and K. Solna, Singular perturbations in option pricing, SIAM J. Appl. Math. 63 (2003), no. 5, 1648–1665. MR 2001213, DOI 10.1137/S0036139902401550
F. Götze and C. Hipp, Asymptotic expansions for sums of weakly dependent random vectors, Z. Wahrsch. Verw. Gebiete 64 (1983), no. 2, 211–239. MR 714144, DOI 10.1007/BF01844607
F. Götze and C. Hipp, Asymptotic expansions for potential functions of i.i.d. random fields, Probab. Theory Related Fields 82 (1989), no. 3, 349–370. MR 1001518, DOI 10.1007/BF00339992
F. Götze and C. Hipp, Asymptotic distribution of statistics in time series, Ann. Statist. 22 (1994), no. 4, 2062–2088. MR 1329183, DOI 10.1214/aos/1176325772
Sébastien Gouëzel, Local limit theorem for nonuniformly partially hyperbolic skew-products and Farey sequences, Duke Math. J. 147 (2009), no. 2, 193–284. MR 2495076, DOI 10.1215/00127094-2009-011
C. W. J. Granger, Long memory relationships and the aggregation of dynamic models, J. Econometrics 14 (1980), no. 2, 227–238. MR 597259, DOI 10.1016/0304-4076(80)90092-5
Peter Hall, The bootstrap and Edgeworth expansion, Springer Series in Statistics, Springer-Verlag, New York, 1992. MR 1145237, DOI 10.1007/978-1-4612-4384-7
Peter Hall and Joel L. Horowitz, Bootstrap critical values for tests based on generalized-method-of-moments estimators, Econometrica 64 (1996), no. 4, 891–916. MR 1399222, DOI 10.2307/2171849
Lothar Heinrich, Nonuniform bounds for the error in the central limit theorem for random fields generated by functions of independent random variables, Math. Nachr. 145 (1990), 345–364. MR 1069041, DOI 10.1002/mana.19901450127
Loïc Hervé and Françoise Pène, The Nagaev-Guivarc’h method via the Keller-Liverani theorem, Bull. Soc. Math. France 138 (2010), no. 3, 415–489 (English, with English and French summaries). MR 2729019, DOI 10.24033/bsmf.2594
J. L. Jensen, Asymptotic expansions for strongly mixing Harris recurrent Markov chains, Scand. J. Statist. 16 (1989), no. 1, 47–63. MR 1003968
Moritz Jirak, Berry-Esseen theorems under weak dependence, Ann. Probab. 44 (2016), no. 3, 2024–2063. MR 3502600, DOI 10.1214/15-AOP1017
M. Jirak, Limit theorems for aggregated linear processes, Adv. in Appl. Probab. 45 (2013), no. 2, 520–544. MR 3102461, DOI 10.1239/aap/1370870128
Jirak, M. and M. Wahl. 2018. Relative perturbation bounds with applications to empirical covariance operators. Available at https://arxiv.org/pdf/1802.02869.
Alexey Korepanov, Rates in almost sure invariance principle for dynamical systems with some hyperbolicity, Comm. Math. Phys. 363 (2018), no. 1, 173–190. MR 3849987, DOI 10.1007/s00220-018-3234-5
I. Kontoyiannis and S. P. Meyn, Spectral theory and limit theorems for geometrically ergodic Markov processes, Ann. Appl. Probab. 13 (2003), no. 1, 304–362. MR 1952001, DOI 10.1214/aoap/1042765670
Soumendra Nath Lahiri, Refinements in asymptotic expansions for sums of weakly dependent random vectors, Ann. Probab. 21 (1993), no. 2, 791–799. MR 1217565
Soumendra Nath Lahiri, Asymptotic expansions for sums of random vectors under polynomial mixing rates, Sankhyā Ser. A 58 (1996), no. 2, 206–224. MR 1662519
S. N. Lahiri, Resampling methods for dependent data, Springer Series in Statistics, Springer-Verlag, New York, 2003. MR 2001447, DOI 10.1007/978-1-4757-3803-2
Li, P. T. J. Hastie, and K. W. Church. 2006. Very sparse random projections. Proceedings of the 12th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD ’06, New York, NY, USA, pp. 287–296. ACM.
Weidong Liu, Han Xiao, and Wei Biao Wu, Probability and moment inequalities under dependence, Statist. Sinica 23 (2013), no. 3, 1257–1272. MR 3114713
Michael Maxwell and Michael Woodroofe, Central limit theorems for additive functionals of Markov chains, Ann. Probab. 28 (2000), no. 2, 713–724. MR 1782272, DOI 10.1214/aop/1019160258
Per Aslak Mykland, Asymptotic expansions and bootstrapping distributions for dependent variables: a martingale approach, Ann. Statist. 20 (1992), no. 2, 623–654. MR 1165585, DOI 10.1214/aos/1176348649
Per Aslak Mykland, Asymptotic expansions for martingales, Ann. Probab. 21 (1993), no. 2, 800–818. MR 1217566
Per Aslak Mykland, Martingale expansions and second order inference, Ann. Statist. 23 (1995), no. 3, 707–731. MR 1345195, DOI 10.1214/aos/1176324617
S. V. Nagaev, Some limit theorems for stationary Markov chains, Teor. Veroyatnost. i Primenen. 2 (1957), 389–416 (Russian, with English summary). MR 0094846
Donald S. Ornstein, An example of a Kolmogorov automorphism that is not a Bernoulli shift, Advances in Math. 10 (1973), 49–62. MR 316682, DOI 10.1016/0001-8708(73)90097-2
Françoise Pène, Rate of convergence in the multidimensional central limit theorem for stationary processes. Application to the Knudsen gas and to the Sinai billiard, Ann. Appl. Probab. 15 (2005), no. 4, 2331–2392. MR 2187297, DOI 10.1214/105051605000000476
V. V. Petrov, Sums of independent random variables, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 82, Springer-Verlag, New York-Heidelberg, 1975. Translated from the Russian by A. A. Brown. MR 0388499
Benedikt M. Pötscher and Ingmar R. Prucha, Dynamic nonlinear econometric models, Springer-Verlag, Berlin, 1997. Asymptotic theory. MR 1468737, DOI 10.1007/978-3-662-03486-6
L. A. Shepp, A local limit theorem, Ann. Math. Statist. 35 (1964), 419–423. MR 166817, DOI 10.1214/aoms/1177703766
A. Tikhomirov, ‘On the convergence rate in the central limit theorem for weakly dependent random variables’, Theory of Probability and Its Applications 25(4) (1980), 790–809.
A. M. Vershik, Theory of decreasing sequences of measurable partitions, Algebra i Analiz 6 (1994), no. 4, 1–68 (Russian); English transl., St. Petersburg Math. J. 6 (1995), no. 4, 705–761. MR 1304093
Dalibor Volný, Michael Woodroofe, and Ou Zhao, Central limit theorems for superlinear processes, Stoch. Dyn. 11 (2011), no. 1, 71–80. MR 2771342, DOI 10.1142/S0219493711003164
Michael Woodroofe, Very weak expansions for sequentially designed experiments: linear models, Ann. Statist. 17 (1989), no. 3, 1087–1102. MR 1015139, DOI 10.1214/aos/1176347257
Wei Biao Wu, Nonlinear system theory: another look at dependence, Proc. Natl. Acad. Sci. USA 102 (2005), no. 40, 14150–14154. MR 2172215, DOI 10.1073/pnas.0506715102
Wei Biao Wu, Strong invariance principles for dependent random variables, Ann. Probab. 35 (2007), no. 6, 2294–2320. MR 2353389, DOI 10.1214/009117907000000060
Wei Biao Wu, Asymptotic theory for stationary processes, Stat. Interface 4 (2011), no. 2, 207–226. MR 2812816, DOI 10.4310/SII.2011.v4.n2.a15
Wei Biao Wu and Xiaofeng Shao, Limit theorems for iterated random functions, J. Appl. Probab. 41 (2004), no. 2, 425–436. MR 2052582, DOI 10.1239/jap/1082999076
Alex Zhai, A high-dimensional CLT in $\mathcal W_2$ distance with near optimal convergence rate, Probab. Theory Related Fields 170 (2018), no. 3-4, 821–845. MR 3773801, DOI 10.1007/s00440-017-0771-3
Danna Zhang and Wei Biao Wu, Asymptotic theory for estimators of high-order statistics of stationary processes, IEEE Trans. Inform. Theory 64 (2018), no. 7, 4907–4922. MR 3819347, DOI 10.1109/tit.2017.2764480