Hostname: page-component-7dd5485656-zlgnt Total loading time: 0 Render date: 2025-10-28T12:30:09.681Z Has data issue: false hasContentIssue false
  • English
  • Français

Cramér type moderate deviations for random fields

Part of: Limit theorems Distribution theory Stochastic processes

Published online by Cambridge University Press:  12 July 2019

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

Abstract

We study the Cramér type moderate deviation for partial sums of random fields by applying the conjugate method. The results are applicable to the partial sums of linear random fields with short or long memory and to nonparametric regression with random field errors.

Keywords

Cramér type moderate deviationlong range dependencenonparametric regressionspatial linear processrandom field

MSC classification

Primary: 60F10: Large deviations 60G60: Random fields 62E20: Asymptotic distribution theory

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 56 , Issue 1 , March 2019 , pp. 223 - 245
Copyright
© Applied Probability Trust 2019 

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.)

Article purchase

Temporarily unavailable

References

Amosova, N. N. (1979). On probabilities of moderate deviations for sums of independent random variables. Teor. Veroyatn. Primen. 24, 858–865.Google Scholar
Asmussen, S. and Albrecher, H. (2010). Ruin Probabilities. World Scientific, Hackensack, NJ.CrossRefGoogle Scholar
Babu, G. J. and Singh, K. (1978a). Probabilities of moderate deviations for some stationary strong-mixing processes. Sankhyā Ser. A 40, 38–43.Google Scholar
Babu, G. J. and Singh, K. (1978b). On probabilities of moderate deviations for dependent processes. Sankhyā Ser. A 40, 28–37.Google Scholar
Bahadur, R. R. and Rao, R. R. (1960). On deviations of the sample mean. Ann. Math. Statist. 31, 1015–1027.CrossRefGoogle Scholar
Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Cambridge University Press.CrossRefGoogle Scholar
Cramér, H. (1938). Sur un nouveau théorème-limite de la théorie des probabilités. Actual. Sci. Ind. 736, 5–23.Google Scholar
Fan, X., Grama, I. and Liu, Q. (2013). Cramér large deviation expansions for martingales under Bernstein’s condition. Stoch. Process. Appl. 123, 3919–3942.CrossRefGoogle Scholar
Feller, W. (1943). Generalization of a probability limit theorem of Cramér. Trans. Amer. Math. Soc. 54, 361–372.Google Scholar
Frolov, A. N. (2005). On the probabilities of moderate deviations of sums for independent random variables. J. Math. Sci. (New York) 127, 1787–1796.CrossRefGoogle Scholar
Ghosh, M. (1974). Probabilities of moderate deviations under m-dependence. Canad. J. Statist. 2, 157–168.CrossRefGoogle Scholar
Ghosh, M. and Babu, G. J. (1977). Probabilities of moderate deviations for some stationary φ-mixing processes. Ann. Prob. 5, 222–234.CrossRefGoogle Scholar
Grama, I. G. (1997). On moderate deviations for martingales. Ann. Prob. 25, 152–183.Google Scholar
Grama, I. and Haeusler, E. (2000). Large deviations for martingales via Cramér’s method. Stoch. Process. Appl. 85, 279–293.CrossRefGoogle Scholar
Grama, I. G. and Haeusler, E. (2006). An asymptotic expansion for probabilities of moderate deviations for multivariate martingales. J. Theoret. Prob 19, 1–44.CrossRefGoogle Scholar
Granger, C. and Joyeux, R. (1980). An introduction to long memory time series models and fractional differencing. J. Time Series Anal. 1 15–29.CrossRefGoogle Scholar
Gu, W. and Tran, L. T. (2009). Fixed design regression for negatively associated random fields. J. Nonparametric Statist. 21, 345–363.Google Scholar
Heinrich, L. (1990). Some bounds of cumulants of m-dependent random fields. Math. Nachr. 149, 303–317.Google Scholar
Hosking, J. R. M. (1981). Fractional differencing. Biometrika 68 165–176.CrossRefGoogle Scholar
Johnstone, I. M. (1999). Wavelet shrinkage for correlated data and inverse problems: adaptivity results. Statist. Sinica 9, 51–83.Google Scholar
Joutard, C. (2006). Sharp large deviations in nonparametric estimation. J. Nonparametr. Statist. 18, 293–306.CrossRefGoogle Scholar
Joutard, C. (2013). Strong large deviations for arbitrary sequences of random variables. Ann. Inst. Statist. Math. 65, 49–67.Google Scholar
Khinchin, A. I. (1929). Über einen neuen Grenzwertsatz der Wahrscheinlichkeitsrechnung. Math. Ann. 101, 745–752 (in German).CrossRefGoogle Scholar
Koul, H. L., Mimoto, N. and Surgailis, D. (2016). A goodness-of-fit test for marginal distribution of linear random fields with long memory. Metrika 79, 165–193.CrossRefGoogle Scholar
Lahiri, S. N. and Robinson, P. M. (2016). Central limit theorems for long range dependent spatial linear processes. Bernoulli 22, 345–375.CrossRefGoogle Scholar
Lee, S.-H., Tan, V. Y. F. and Khisti, A. (2016). Streaming data transmission in the moderate deviations and central limit regimes. IEEE Trans. Inform. Theory 62, 6816–6830.CrossRefGoogle Scholar
Lee, S.-H., Tan, V. Y. F. and Khisti, A. (2017). Exact moderate deviation asymptotics in streaming data transmission. IEEE Trans. Inform. Theory 63, 2726–2736.Google Scholar
Michel, R. (1976). Nonuniform central limit bounds with applications to probabilities of deviations. Ann. Prob. 4, 102–106.CrossRefGoogle Scholar
Nagaev, S. V. (1965). Some limit theorems for large deviations. Teor. Veroyatn. Primen. 10, 231–254.Google Scholar
Nagaev, S. V. (1979). Large deviations of sums of independent random variables. Ann. Prob. 7, 745–789.CrossRefGoogle Scholar
Peligrad, M., Sang, H., Zhong, Y. and Wu, W. B. (2014a). Exact moderate and large deviations for linear processes. Statist. Sinica 24, 957–969.Google Scholar
Peligrad, M., Sang, H., Zhong, Y. and Wu, W. B. (2014b). Supplementary material for the paper “Exact moderate and large deviations for linear processes”. Statist. Sinica, 15 pp. Available at http://www3.stat.sinica.edu.tw/statistica/j24n2/24-2.htmlGoogle Scholar
Petrov, V. V. (1954). A generalization of the Cramér limit theorem. Uspehi Mat. Nauk 9, 195–202 (in Russian).Google Scholar
Petrov, V. V. (1965). On the probabilities of large deviations for sums of independent random variables. Theory Prob. Appl. 10, 287–298.CrossRefGoogle Scholar
Petrov, V. V. (1975). Sums of Independent Random Variables. Springer, Heidelberg.CrossRefGoogle Scholar
Petrov, V. V. (1995). Limit Theorems of Probability Theory. Oxford University Press.Google Scholar
Petrov, V. V., Robinson, J. (2006). On large deviations for sums of independent random variables. Available at http://www.maths.usyd.edu.au/u/pubs/publist/preprints/2007/petrov-2.pdf.Google Scholar
Picard, D. and Tribouley, K. (2000). Adaptive confidence interval for pointwise curve estimation. Ann. Statist. 28, 298–335.Google Scholar
Privalov, I. I. (1984). Introduction to the Theory of Functions of a Complex Variable, 13th edn. Nauka, Moscow (in Russian).Google Scholar
Rubin, H. and Sethuraman, J. (1965). Probabilities of moderate deviations. Sankhyā Ser. A 27, 325–346.Google Scholar
Sang, H. and Xiao, Y. (2018). Exact moderate and large deviations for linear random fields. J. Appl. Prob. 55, 431–449.CrossRefGoogle Scholar
Seneta, E. (1976). Regularly Varying Functions (Lecture Notes Math. 508). Springer, Berlin.CrossRefGoogle Scholar
Slastnikov, A. D. (1978). Limit theorems for probabilities of moderate deviations. Teor. Veroyatn. Primen. 23, 340–357.Google Scholar
Statulevičius, V.A. (1966). On large deviations. Z. Wahrscheinlichkeitsth. 6, 133-144.CrossRefGoogle Scholar
Surgailis, D. (1982). Zones of attraction of self-similar multiple integrals. Lithuanian Math. J. 22 327–340.CrossRefGoogle Scholar
Talagrand, M. (2014). Upper and Lower Bounds for Stochastic Processes. Modern Methods and Classical Problems. Springer, Heidelberg.CrossRefGoogle Scholar
Van Der Vaart, A. W. (1998). Asymptotic Statistics. Cambridge University Press.CrossRefGoogle Scholar
Wu, W. B. and Zhao, Z. (2008). Moderate deviations for stationary processes. Statist. Sinica 18, 769–782.Google Scholar
Zhang, S. and Wong, M.-Y. (2003). Wavelet threshold estimation for additive regression models. Ann. Statist. 31, 152–173.Google Scholar