Theory of Probability and Mathematical Statistics
Limit theory for martingale transforms with heavy-tailed noise
Stelios Arvanitis and Alexandros Louka
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Abstract: A limit theorem for partial sums of martingale transforms with multiplicative noise and ergodic transform processes is established, which leads to regularly varying rates and stable limits. Its establishment is facilitated by the derivation of an extension to empirical distributions of Breiman’s Theorem along with the Principle of Conditioning. The theorem is applied for the development of the limit theory of the least squares estimator in regressions with heavy tailed noise, as well as of the limit theory of the Gaussian QMLE in GARCH-type models. Depending on the index of stability, regularly varying rates and asymptotic stable distributions, or inconsistency, are obtained.
Keywords: Martingale limit theorem, principle of conditioning, domain of attraction, α-stable distribution, regular variation, Breiman’s theorem, heavy-tailed noise regression, LSE, GARCH-type model, Gaussian QMLE, inconsistency
Bibliography: Jon Aaronson and Manfred Denker, Characteristic functions of random variables attracted to 1-stable laws, Ann. Probab. 26 (1998), no. 1, 399–415. MR 1617056, DOI 10.1214/aop/1022855426
K. Abadir and J. Magnus, 03.6.1 The central limit theorem for Student’s distribution–solution, Econometric Theory 20 (2004), 1261–1263.
Terrence M. Adams and Andrew B. Nobel, Uniform convergence of Vapnik-Chervonenkis classes under ergodic sampling, Ann. Probab. 38 (2010), no. 4, 1345–1367. MR 2663629, DOI 10.1214/09-AOP511
T. G. Andersen, R. A. Davis, J.-P. Kreiß, and T. V. Mikosch, Handbook of financial time series, Springer Science & Business Media, 2009.
Donald W. K. Andrews, Estimation when a parameter is on a boundary, Econometrica 67 (1999), no. 6, 1341–1383. MR 1720781, DOI 10.1111/1468-0262.00082
Stelios Arvanitis, Stable limit theory for the Gaussian QMLE in a non-stationary asymmetric GARCH model, Statist. Probab. Lett. 145 (2019), 166–172. MR 3873903, DOI 10.1016/j.spl.2018.09.007
Stelios Arvanitis and Alexandros Louka, Limit theory for the QMLE of the GQARCH(1,1) model, Comm. Statist. Theory Methods 44 (2015), no. 17, 3549–3575. MR 3397149, DOI 10.1080/03610926.2013.847105
Stelios Arvanitis and Alexandros Louka, Stable limits for the Gaussian QMLE in the non-stationary GARCH(1,1) model, Econom. Lett. 161 (2017), 135–137. MR 3723241, DOI 10.1016/j.econlet.2017.09.035
Stelios Arvanitis and Alexandros Louka, Inconsistency for the Gaussian QMLE in GARCH-type models with infinite variance, Comm. Statist. Theory Methods 53 (2024), no. 5, 1684–1699. MR 4687836, DOI 10.1080/03610926.2022.2107665
Katarzyna Bartkiewicz, Adam Jakubowski, Thomas Mikosch, and Olivier Wintenberger, Stable limits for sums of dependent infinite variance random variables, Probab. Theory Related Fields 150 (2011), no. 3-4, 337–372. MR 2824860, DOI 10.1007/s00440-010-0276-9
N. H. Bingham, C. M. Goldie, and J. L. Teugels, Regular variation, Encyclopedia of Mathematics and its Applications, vol. 27, Cambridge University Press, Cambridge, 1987. MR 898871, DOI 10.1017/CBO9780511721434
R. Bojanić and E. Seneta, Slowly varying functions and asymptotic relations, J. Math. Anal. Appl. 34 (1971), 302–315. MR 274676, DOI 10.1016/0022-247X(71)90114-4
Tim Bollerslev, Generalized autoregressive conditional heteroskedasticity, J. Econometrics 31 (1986), no. 3, 307–327. MR 853051, DOI 10.1016/0304-4076(86)90063-1
Philippe Bougerol and Nico Picard, Strict stationarity of generalized autoregressive processes, Ann. Probab. 20 (1992), no. 4, 1714–1730. MR 1188039
L. Breiman, On some limit theorems similar to the arc-sin law, Teor. Verojatnost. i Primenen. 10 (1965), 351–360 (English, with Russian summary). MR 184274
V. V. Buldygin, O. I. Klesov, and J. G. Steinebach, Equivalent monotone versions of PRV functions, J. Math. Anal. Appl. 401 (2013), no. 2, 526–533. MR 3018003, DOI 10.1016/j.jmaa.2012.12.040
K. C. Chanda, Strong mixing properties of linear stochastic processes, J. Appl. Probability 11 (1974), 401–408. MR 358951, DOI 10.2307/3212764
Adriana Cornea-Madeira and Russell Davidson, A parametric bootstrap for heavy-tailed distributions, Econometric Theory 31 (2015), no. 3, 449–470. MR 3348454, DOI 10.1017/S0266466614000395
James Davidson, Stochastic limit theory, Advanced Texts in Econometrics, The Clarendon Press, Oxford University Press, New York, 1994. An introduction for econometricians. MR 1430804, DOI 10.1093/0198774036.001.0001
Richard A. Davis and Tailen Hsing, Point process and partial sum convergence for weakly dependent random variables with infinite variance, Ann. Probab. 23 (1995), no. 2, 879–917. MR 1334176
Denis Denisov and Bert Zwart, On a theorem of Breiman and a class of random difference equations, J. Appl. Probab. 44 (2007), no. 4, 1031–1046. MR 2382943, DOI 10.1239/jap/1197908822
Z. Ding, C. Granger, and R. F. Engle, A long memory property of stock market returns and a new model, J. Empirical Finance 1 (1993), no. 1, 83–106.
P. Embrechts, C. Klüppelberg, and T. Mikosch, Modelling extremal events: for insurance and finance, Springer Science & Business Media, 2013.
Jianqing Fan, Lei Qi, and Dacheng Xiu, Quasi-maximum likelihood estimation of GARCH models with heavy-tailed likelihoods, J. Bus. Econom. Statist. 32 (2014), no. 2, 178–191. MR 3207827, DOI 10.1080/07350015.2013.840239
Christian Francq and Jean-Michel Zakoïan, Inference in nonstationary asymmetric GARCH models, Ann. Statist. 41 (2013), no. 4, 1970–1998. MR 3127855, DOI 10.1214/13-AOS1132
Charles M. Goldie, Implicit renewal theory and tails of solutions of random equations, Ann. Appl. Probab. 1 (1991), no. 1, 126–166. MR 1097468
A. K. Gupta, T. T. Nguyen, and W.-B. Zeng, Conditions for stability of laws with all projections stable, Sankhya A 56 (1994), no. 3, 438–443.
P. Hall and C. C. Heyde, Martingale limit theory and its application, Probability and Mathematical Statistics, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. MR 624435
Peter Hall and Qiwei Yao, Inference in ARCH and GARCH models with heavy-tailed errors, Econometrica 71 (2003), no. 1, 285–317. MR 1956860, DOI 10.1111/1468-0262.00396
Edwin Hewitt and Karl Stromberg, Real and abstract analysis. A modern treatment of the theory of functions of a real variable, Springer-Verlag, New York, 1965. MR 188387
J. B. Hill, A new moment bound and weak laws for mixingale arrays without memory or heterogeneity restrictions, with applications to tail trimmed arrays, University of North Carolina’Chapel Hill (2010).
I. A. Ibragimov and Yu. 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. MR 322926
Adam Jakubowski, Principle of conditioning in limit theorems for sums of random variables, Ann. Probab. 14 (1986), no. 3, 902–915. MR 841592
—, Principle of conditioning revisited, Demonstr. Math. 45 (2012), no. 2, 325–36.
P. Jeganathan, A solution of the martingale central limit problem. I, II, Sankhyā Ser. A 44 (1982), no. 3, 299–318, 319–340. MR 705457
Stefanos Kechagias and Vladas Pipiras, Definitions and representations of multivariate long-range dependent time series, J. Time Series Anal. 36 (2015), no. 1, 1–25. MR 3300202, DOI 10.1111/jtsa.12086
Claudia Klüppelberg, Subexponential distributions and integrated tails, J. Appl. Probab. 25 (1988), no. 1, 132–141. MR 929511, DOI 10.2307/3214240
RafałKulik and Philippe Soulier, Heavy-tailed time series, Springer Series in Operations Research and Financial Engineering, Springer, New York, [2020] ©2020. MR 4174389, DOI 10.1007/978-1-0716-0737-4
Michael Stanislaus Loretan, Testing covariance stationarity of heavy-tailed economic time series, ProQuest LLC, Ann Arbor, MI, 1991. Thesis (Ph.D.)–Yale University. MR 2687497
Thomas Mikosch and Daniel Straumann, Stable limits of martingale transforms with application to the estimation of GARCH parameters, Ann. Statist. 34 (2006), no. 1, 493–522. MR 2275251, DOI 10.1214/009053605000000840
Stefan Mittnik, Svetlozar T. Rachev, and Marc S. Paolella, Stable Paretian modeling in finance: some empirical and theoretical aspects, A practical guide to heavy tails (Santa Barbara, CA, 1995) Birkhäuser Boston, Boston, MA, 1998, pp. 79–110. MR 1652284
Ilya Molchanov, Theory of random sets, Probability and its Applications (New York), Springer-Verlag London, Ltd., London, 2005. MR 2132405
Daniel B. Nelson, Conditional heteroskedasticity in asset returns: a new approach, Econometrica 59 (1991), no. 2, 347–370. MR 1097532, DOI 10.2307/2938260
Liang Peng and Qiwei Yao, Least absolute deviations estimation for ARCH and GARCH models, Biometrika 90 (2003), no. 4, 967–975. MR 2024770, DOI 10.1093/biomet/90.4.967
Peter C. B. Phillips and Tassos Magdalinos, Limit theory for moderate deviations from a unit root, J. Econometrics 136 (2007), no. 1, 115–130. MR 2328587, DOI 10.1016/j.jeconom.2005.08.002
S. Rachev and S. Mittnik, Stable Paretian models in finance, series in financial economics and quantitative analysis, John Wiley & Sons Ltd., 1988.
R. T. Rockafellar, Variational analysis, Springer, 1997.
Gennady Samorodnitsky and Murad S. Taqqu, Stable non-Gaussian random processes, Stochastic Modeling, Chapman & Hall, New York, 1994. Stochastic models with infinite variance. MR 1280932
E. Sentana, Quadratic ARCH models, Rev. Econ. Stud. 62 (1995), no. 4, 639–661.
Daniel Straumann, Estimation in conditionally heteroscedastic time series models, Lecture Notes in Statistics, vol. 181, Springer-Verlag, Berlin, 2005. MR 2142271
Daniel Straumann and Thomas Mikosch, Quasi-maximum-likelihood estimation in conditionally heteroscedastic time series: a stochastic recurrence equations approach, Ann. Statist. 34 (2006), no. 5, 2449–2495. MR 2291507, DOI 10.1214/009053606000000803
Donatas Surgailis, A quadratic
model with long memory and Lévy stable behavior of squares, Adv. in Appl. Probab. 40 (2008), no. 4, 1198–1222. MR 2488538, DOI 10.1239/aap/1231340170
A. W. van der Vaart, Asymptotic statistics, Cambridge Series in Statistical and Probabilistic Mathematics, vol. 3, Cambridge University Press, Cambridge, 1998. MR 1652247, DOI 10.1017/CBO9780511802256
Qiying Wang, Martingale limit theorem revisited and nonlinear cointegrating regression, Econometric Theory 30 (2014), no. 3, 509–535. MR 3205605, DOI 10.1017/S026646661300039X
Olivier Wintenberger, Continuous invertibility and stable QML estimation of the EGARCH(1,1) model, Scand. J. Stat. 40 (2013), no. 4, 846–867. MR 3145121, DOI 10.1111/sjos.12038
O. Wintenberger and S. Cai, Parametric inference and forecasting in continuously invertible volatility models, arXiv preprint arXiv:1106.4983 (2011).
