On weak invariance principles for partial sums. (English) Zbl 1395.60038
Author’s abstract: Given a sequence of random functionals \(\bigl\{X_k(u)\bigr\}_{k\in \mathbb Z}\), \(u\in\mathbf{I}^d\), \(d\geq 1\), the normalized partial sums \(\check{S}_{nt}(u)=n^{-1/2}\bigl (X_1(u)+\cdots+X_{\lfloor nt\rfloor}(u)\bigr)\), \(t\in [0,1]\) and its polygonal version \(S_{nt}(u)\) are considered under a weak dependence assumption and \(p > 2\) moments. Weak invariance principles in the space of continuous functions and càdlàg functions are established. A particular emphasis is put on the process \(\check{S}_{nt}(\widehat{\theta })\), where \(\widehat{\theta}\mathop{\rightarrow}\limits^{\mathbb P}\theta\), and weaker moment conditions (\(p=2\) if \(d=1\)) are assumed.
MSC:
| 60F17 | Functional limit theorems; invariance principles |
| 60F25 | \(L^p\)-limit theorems |
| 62G10 | Nonparametric hypothesis testing |
| 60G15 | Gaussian processes |
| 60B10 | Convergence of probability measures |
Keywords:
weak invariance principle; weakly dependent processes; plug-in estimator; infinite dimension; covariance estimatorsSoftware:
FinTSReferences:
| [1] | Andrews, D.W.K., Pollard, D.: An introduction to functional central limit theorems for dependent stochastic processes. Int. Stat. Rev. / Revue Internationale de Statistique 62(1), 119-132 (1994) · Zbl 0834.60033 |
| [2] | Arcones, M.A.: Limit theorems for nonlinear functionals of a stationary Gaussian sequence of vectors. Ann. Probab. 22(4), 2242-2274 (1994) · Zbl 0839.60024 · doi:10.1214/aop/1176988503 |
| [3] | Aue, A., Berkes, I., Horváth, L.: Selection from a stable box. Bernoulli 14(1), 125-139 (2008) · Zbl 1157.60310 · doi:10.3150/07-BEJ6014 |
| [4] | Berkes, I., Hörmann, S., Horváth, L.: The functional central limit theorem for a family of GARCH observations with applications. Stat. Probab. Lett. 78(16), 2725-2730 (2008) · Zbl 1151.60323 · doi:10.1016/j.spl.2008.03.021 |
| [5] | Berkes, I., Horváth, L., Schauer, J.: Asymptotic behavior of trimmed sums. Stoch. Dyn. 12(1), 1150002 (2012) · Zbl 1262.60022 |
| [6] | Berkes, I., Philipp, W.: Approximation theorems for independent and weakly dependent random vectors. Ann. Probab. 7(1), 29-54 (1979) · Zbl 0392.60024 · doi:10.1214/aop/1176995146 |
| [7] | Bickel, P.J., Wichura, M.J.: Convergence criteria for multiparameter stochastic processes and some applications. Ann. Math. Stat. 42, 1656-1670 (1971) · Zbl 0265.60011 · doi:10.1214/aoms/1177693164 |
| [8] | Billingsley P.: Convergence of probability measures. Wiley series in probability and statistics: probability and statistics. A Wiley-Interscience Publication, 2nd edn. Wiley, New York (1999) · Zbl 0944.60003 |
| [9] | Bougerol, P., Picard, N.: Strict stationarity of generalized autoregressive processes. Ann. Probab. 20(4), 1714-1730 (1992) · Zbl 0763.60015 · doi:10.1214/aop/1176989526 |
| [10] | Bradley, R.C.: Introduction to Strong Mixing Conditions, vol. 1. Kendrick Press, Heber City (2007) · Zbl 1133.60001 |
| [11] | Brockwell, P.J., Davis, R.A.: Time Series: Theory and Methods. Springer Series in Statistics, 2nd edn. Springer, New York (1991) · Zbl 0709.62080 |
| [12] | Csörgő, M., Horváth, L.: Limit theorems in change-point analysis. Wiley series in probability and statistics. Wiley, Chichester (1997) ( With a foreword by David Kendall) · Zbl 0884.62023 |
| [13] | Dedecker, J., Doukhan, P.: A new covariance inequality and applications. Stoch. Process. Appl. 106(1), 63-80 (2003) · Zbl 1075.60513 · doi:10.1016/S0304-4149(03)00040-1 |
| [14] | Dedecker, J., Doukhan, P., Lang, G., León, R., Louhichi, S., Prieur, C.l.: Weak dependence: with examples and applications, volume 190 of Lecture Notes in Statistics. Springer, New York (2007) · Zbl 1165.62001 |
| [15] | Dedecker, J., Prieur, C.: New dependence coefficients. Examples and applications to statistics. Probab. Theory Relat. Fields 132(2), 203-236 (2005) · Zbl 1061.62058 · doi:10.1007/s00440-004-0394-3 |
| [16] | Dehling, H.: Limit theorems for sums of weakly dependent banach space valued random variables. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 63(3), 393-432 (1983) · Zbl 0496.60004 · doi:10.1007/BF00542537 |
| [17] | Dehling, H.: A note on a theorem of Berkes and Philipp. Z. Wahrsch. Verw. Gebiete 62(1), 39-42 (1983) · Zbl 0479.60013 · doi:10.1007/BF00532161 |
| [18] | Dehling, H., Durieu, O., Tusche, M.: A sequential empirical clt for multiple mixing processes with application to \[{\cal{B}}B\]-geometrically ergodic markov chains. Electron. J. Probab. 19(86), 1-26 (2014) · Zbl 1302.60047 |
| [19] | Dehling, H., Philipp, W.: Almost sure invariance principles for weakly dependent vector-valued random variables. Ann. Probab. 10(3), 689-701 (1982) · Zbl 0487.60006 · doi:10.1214/aop/1176993777 |
| [20] | Doukhan, P., Massart, P., Rio, E.: Invariance principles for absolutely regular empirical processes. Annales de l’institut Henri Poincaré (B) Probabilités et Statistiques 31(2), 393-427 (1995) · Zbl 0817.60028 |
| [21] | Doukhan, P., Wintenberger, O.: An invariance principle for weakly dependent stationary general models. Probab. Math. Stat. 27(1), 45-73 (2007) · Zbl 1124.60031 |
| [22] | Dudley, R.M.: Distances of probability measures and random variables. Ann. Math. Stat. 39(5), 1563-1572 (1968) · Zbl 0169.20602 · doi:10.1214/aoms/1177698137 |
| [23] | Eberlein, E.: An invariance principle for lattices of dependent random variables. Z. Wahrsch. Verw. Gebiete 50(2), 119-133 (1979) · Zbl 0397.60031 · doi:10.1007/BF00533633 |
| [24] | Eberlein, E.: Strong approximation of very weak Bernoulli processes. Z. Wahrsch. Verw. Gebiete 62(1), 17-37 (1983) · Zbl 0507.60016 · doi:10.1007/BF00532160 |
| [25] | Ghose, D., Kroner, K.F.: The relationship between garch and symmetric stable processes: finding the source of fat tails in financial data. J. Empir. Finance 2(3), 225-251 (1995) · doi:10.1016/0927-5398(95)00004-E |
| [26] | Haas, M.; Pigorsch, C.; Meyers, RobertA (ed.), Financial economics, fat-tailed distributions, 3404-3435 (2009), Berlin · doi:10.1007/978-0-387-30440-3_204 |
| [27] | Hannan, E.J.: Multiple Time Series. Wiley, New York (1970) · Zbl 0211.49804 · doi:10.1002/9780470316429 |
| [28] | Hannan, E.J.: Central limit theorems for time series regression. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 26, 157-170 (1973) · Zbl 0246.62086 · doi:10.1007/BF00533484 |
| [29] | Hariz, S.B.: Uniform clt for empirical process. Stoch. Process. Appl. 115(2), 339-358 (2005) · Zbl 1080.60028 · doi:10.1016/j.spa.2004.09.006 |
| [30] | Huskova, M.: Tests and estimators for the change point problem based on M-statistics. Stat. Risk Model. 14(2), 115-136 (1996) · Zbl 0864.62008 |
| [31] | Jacod, J., Shiryaev, A.N.: Limit theorems for stochastic processes, volume 288 of Grundlehren der Mathematischen Wissenschaften [fundamental principles of mathematical sciences], 2nd edn. Springer, Berlin (2003) · Zbl 1018.60002 |
| [32] | Kallenberg, O.: Foundations of Modern Probability. Probability and its Applications (New York), 2nd edn. Springer, New York (2002) · Zbl 0996.60001 |
| [33] | Karatzas, I., Shreve, S.E.: Brownian motion and stochastic calculus, volume 113 of graduate texts in mathematics, 2nd edn. Springer, New York (1991) |
| [34] | Kuelbs, J., Philipp, W.: Almost sure invariance principles for partial sums of mixing \[BB\]-valued random variables. Ann. Probab. 8(6), 1003-1036 (1980) · Zbl 0451.60008 · doi:10.1214/aop/1176994565 |
| [35] | Levental, S.: A uniform clt for uniformly bounded families of martingale differences. J. Theor. Probab. 2(3), 271-287 (1989) · Zbl 0681.60023 · doi:10.1007/BF01054016 |
| [36] | Mandelbrot, B.: New methods in statistical economics. J. Political Econ. 71, 421 (1963) · Zbl 0121.15105 · doi:10.1086/258792 |
| [37] | Marcus, M.B., Philipp, W.: Almost sure invariance principles for sums of \[BB\]-valued random variables with applications to random Fourier series and the empirical characteristic process. Trans. Am. Math. Soc. 269(1), 67-90 (1982) · Zbl 0485.60030 |
| [38] | Merlevède, F., Peligrad, M., Utev, S.: Recent advances in invariance principles for stationary sequences. Probab. Surv. 3, 1-36 (2006). ( electronic) · Zbl 1189.60078 |
| [39] | Peligrad, M., Utev, S.: Invariance principle for stochastic processes with short memory. In: High dimensional probability, volume 51 of IMS Lecture Notes Monogr. Ser., pp. 18-32. Inst. Math. Statist., Beachwood (2006) · Zbl 1122.60038 |
| [40] | Philipp, W.: Almost sure invariance principles for sums of \[BB\]-valued random variables. In: Probability in Banach spaces, II (Proc. Second Internat. Conf., Oberwolfach, 1978), volume 709 of Lecture Notes in Math., pp. 171-193. Springer, Berlin (1979) · Zbl 0911.60010 |
| [41] | Philipp, W.: Weak and \[L^p\] Lp-invariance principles for sums of \[BB\]-valued random variables. Ann. Probab. 8(1), 68-82 (1980) · Zbl 0426.60033 · doi:10.1214/aop/1176994825 |
| [42] | Rio, E.: Processus empiriques absolument réguliers et entropie universelle. Probab. Theory Relat. Fields 111(4), 585-608 (1998) · Zbl 0911.60010 · doi:10.1007/s004400050179 |
| [43] | Tsay, R.S.: Analysis of Financial Time Series. Wiley Series in Probability and Statistics Wiley-Interscience, 2nd edn. Wiley, Hoboken (2005) · Zbl 1086.91054 |
| [44] | van der Vaart, A.W.: Asymptotic statistics. Cambridge University Press, Cambridge (1998) · Zbl 0910.62001 · doi:10.1017/CBO9780511802256 |
| [45] | Wu, W.B.: Nonlinear system theory : another look at dependence. Proc Natl Acad Sci USA 102, 14150-14154 (2005) · Zbl 1135.62075 · doi:10.1073/pnas.0506715102 |
| [46] | Wu, W.B.: Strong invariance principles for dependent random variables. Ann. Probab. 35(6), 2294-2320 (2007) · Zbl 1166.60307 · doi:10.1214/009117907000000060 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
