Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-24T03:39:20.990Z Has data issue: false hasContentIssue false
  • English
  • Français

Exact moderate and large deviations for linear random fields

Part of: Limit theorems Distribution theory Stochastic processes

Published online by Cambridge University Press:  26 July 2018

Hailin Sang  and
Yimin Xiao
Hailin Sang*
Affiliation:
The University of Mississippi
Yimin Xiao*
Affiliation:
Michigan State University
*
* Postal address: Department of Mathematics, The University of Mississippi, University, MS 38677, USA. Email address: [email protected]
** Postal address: Department of Statistics and Probability, Michigan State University, East Lansing, MI 48824, USA. Email address: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

By extending the methods of Peligrad et al. (2014), we establish exact moderate and large deviation asymptotics for linear random fields with independent innovations. These results are useful for studying nonparametric regression with random field errors and strong limit theorems.

Keywords

Large deviationmoderate deviationlinear random fieldnonparametric regressionDavis–Gut law of the iterated logarithm

MSC classification

Primary: 60F10: Large deviations 60G60: Random fields 62E20: Asymptotic distribution theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 55 , Issue 2 , June 2018 , pp. 431 - 449
Copyright
Copyright © Applied Probability Trust 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Banys, P., Davydov, Y. and Paulauskas, V. (2010). Remarks on the SLLN for linear random fields. Statist. Prob. Lett. 80, 489–496. Google Scholar
Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Cambridge University Press. 10.1017/CBO9780511721434Google Scholar
Chen, P. Y. and Wang, D. C. (2008). Convergence rates for probabilities of moderate deviations for moving average processes. Acta Math. Sinica (English Ser.) 24, 611–622. Google Scholar
Davis, J. A. (1968). Convergence rates for the law of the iterated logarithm. Ann. Math. Statist. 39, 1479–1485. Google Scholar
Davis, R. A. and Hsing, T. (1995). Point process and partial sum convergence for weakly dependent random variables with infinite variance. Ann. Prob. 23, 879–917. Google Scholar
De la Peña, V. H. and Giné, E. (1999). Decoupling: From Dependence to Independence. Springer, New York. Google Scholar
Djellout, H. and Guillin, A. (2001). Large and moderate deviations for moving average processes. Ann. Fac. Sci. Toulouse Math. (6) 10, 23–31. Google Scholar
Djellout, H., Guillin, A. and Wu, L. (2006). Moderate deviations of empirical periodogram and non-linear functionals of moving average processes. Ann. Inst. H. Poincaré Prob. Statist. 42, 393–416. Google Scholar
El Machkouri, M. (2007). Nonparametric regression estimation for random fields in a fixed-design. Statist. Inference Stoch. Process. 10, 29–47. Google Scholar
El Machkouri, M. (2014). Kernel density estimation for stationary random fields. ALEA Latin Amer. J. Prob. Math. Statist. 11, 259–279. Google Scholar
El Machkouri, M. and Stoica, R. (2010). Asymptotic normality of kernel estimates in a regression model for random fields. J. Nonparametric Statist. 22, 955–971. Google Scholar
Frolov, A. N. (2005). On probabilities of moderate deviations of sums of independent random variables. J. Math. Sci. (N.Y.) 127, 1787–1796. Google Scholar
Ghosh, S. and Samorodnitsky, G. (2009). The effect of memory on functional large deviations of infinite moving average processes. Stoch. Process. Appl. 119, 534–561. Google Scholar
Gu, W. and Tran, L. T. (2009). Fixed design regression for negatively associated random fields. J. Nonparametric Statist. 21, 345–363. Google Scholar
Gut, A. (1980). Convergence rates for probabilities of moderate deviations for sums of random variables with multidimensional indices. Ann. Prob. 8, 298–313. Google Scholar
Hallin, M., Lu, Z. and Tran, L. T. (2004a). Kernel density estimation for spatial processes: the L 1 theory. J. Multivariate Anal. 88, 61–75. Google Scholar
Hallin, M., Lu, Z. and Tran, L. T. (2004b). Local linear spatial regression. Ann. Statist. 32, 2469–2500. Google Scholar
Jiang, T. F., Rao, M. B. and Wang, X. C. (1995). Large deviations for moving average processes. Stoch. Process. Appl. 59, 309–320. Google Scholar
Klesov, O. (2014). Limit Theorems for Multi-Indexed Sums of Random Variables (Prob. Theory Stoch. Modelling 71). Springer, Heidelberg. 10.1007/978-3-662-44388-0Google Scholar
Li, D. L. (1991). Convergence rates of law of iterated logarithm for B-valued random variables. Sci. China Ser. A 34, 395–404. Google Scholar
Li, D. and Rosalsky, A. (2007). A supplement to the Davis–Gut law. J. Math. Anal. Appl. 330, 1488–1493. Google Scholar
Li, L., Liu, J. and Xiao, Y. (2009). Wavelet regression with long memory infinite moving average errors. J. Appl. Prob. Statist. 4, 183–211. Google Scholar
Mallik, A. and Woodroofe, M. (2011). A central limit theorem for linear random fields. Statist. Prob. Lett. 81, 1623–1626. 10.1016/j.spl.2011.06.007Google Scholar
Marinucci, D. and Poghosyan, S. (2001). Asymptotics for linear random fields. Statist. Prob. Lett. 51, 131–141. Google Scholar
Mikosch, T. and Samorodnitsky, G. (2000). The supremum of a negative drift random walk with dependent heavy-tailed steps. Ann. Appl. Prob. 10, 1025–1064. Google Scholar
Mikosch, T. and Wintenberger, O. (2013). Precise large deviations for dependent regularly varying sequences. Prob. Theory Relat. Fields 156, 851–887. Google Scholar
Nagaev, A. V. (1969a). Integral limit theorems taking large deviations into account when Cramér's condition does not hold. I. Theory Prob. Appl. 14, 51–64. Google Scholar
Nagaev, A. V. (1969b). Integral limit theorems taking large deviations into account when Cramér's condition does not hold. II. Theory Prob. Appl. 14, 193–208. Google Scholar
Nagaev, A. V. (1969c). Limit theorems for large deviations when Cramér's condition is violated. Izv. Akad. Nauk Uzbek. SSR Ser. Fiz.-Mat. Nauk 6, 17–22 (in Russian). Google Scholar
Nagaev, S. V. (1979). Large deviations of sums of independent random variables. Ann. Prob. 7, 745–789. Google Scholar
Paulauskas, V. (2010). On Beveridge-Nelson decomposition and limit theorems for linear random fields. J. Multivariate Anal. 101, 621–639. 10.1016/j.jmva.2009.10.001Google Scholar
Peligrad, M., Sang, H., Zhong, Y. and Wu, W. B. (2014). Exact moderate and large deviations for linear processes. Statistica Sinica 24, 957–969. Google Scholar
Saulis, L. and Statulevičius, V. (2000). Limit theorems on large deviations. In Limit Theorems of Probability Theory, Springer, Berlin, pp. 185–266. Google Scholar
Seneta, E. (1976). Regularly Varying Functions (Lecture Notes Math. 508). Springer, Berlin. Google Scholar
Surgailis, D. (1982). Zones of attraction of self-similar multiple integrals. Lithuanian Math. J. 22, 327–340. Google Scholar
Tran, L. T. (1990). Kernel density estimation on random fields. J. Multivariate Anal. 34, 37–53. Google Scholar
Wang, Y. and Woodroofe, M. (2014). On the asymptotic normality of kernel density estimators for causal linear random fields. J. Multivariate Anal. 123, 201–213. Google Scholar
Wu, W. B. and Min, W. (2005). On linear processes with dependent innovations. Stoch. Process. Appl. 115, 939–958. Google Scholar
Wu, W. B. and Zhao, Z. (2008). Moderate deviations for stationary processes. Statistica Sinica 18, 769–782. Google Scholar