The Annals of Probability

Maxima of asymptotically Gaussian random fields and moderate deviation approximations to boundary crossing probabilities of sums of random variables with multidimensional indices

Hock Peng Chan and Tze Leung Lai

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Abstract

Several classical results on boundary crossing probabilities of Brownian motion and random walks are extended to asymptotically Gaussian random fields, which include sums of i.i.d. random variables with multidimensional indices, multivariate empirical processes, and scan statistics in change-point and signal detection as special cases. Some key ingredients in these extensions are moderate deviation approximations to marginal tail probabilities and weak convergence of the conditional distributions of certain “clumps” around high-level crossings. We also discuss how these results are related to the Poisson clumping heuristic and tube formulas of Gaussian random fields, and describe their applications to laws of the iterated logarithm in the form of the Kolmogorov–Erdős–Feller integral tests.

Article information

Source
Ann. Probab., Volume 34, Number 1 (2006), 80-121.

Dates
First available in Project Euclid: 17 February 2006

Permanent link to this document
https://projecteuclid.org/euclid.aop/1140191533

Digital Object Identifier
doi:10.1214/009117905000000378

Mathematical Reviews number (MathSciNet)
MR2206343

Zentralblatt MATH identifier
1106.60022

Subjects
Primary: 60F10: Large deviations 60G60: Random fields
Secondary: 60F20: Zero-one laws 60G15: Gaussian processes

Keywords
Multivariate empirical processes moderate deviations random fields integral tests boundary crossing probability

Citation

Chan, Hock Peng; Lai, Tze Leung. Maxima of asymptotically Gaussian random fields and moderate deviation approximations to boundary crossing probabilities of sums of random variables with multidimensional indices. Ann. Probab. 34 (2006), no. 1, 80--121. doi:10.1214/009117905000000378. https://projecteuclid.org/euclid.aop/1140191533


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References

  • Adler, R. (2000). On excursion sets, tube formulas and maxima of random fields. Ann. Appl. Probab. 10 1--74.
    Mathematical Reviews (MathSciNet): MR1765203
    Project Euclid: euclid.aoap/1019737664
  • Adler, R. and Brown, L. (1986). Tail behaviour for suprema of empirical processes. Ann. Probab. 14 1--30.
    Mathematical Reviews (MathSciNet): MR815959
    JSTOR: links.jstor.org
  • Albin, J. M. P. (1990). On extremal theory for stationary processes. Ann. Probab. 18 92--128.
    Mathematical Reviews (MathSciNet): MR1043939
    JSTOR: links.jstor.org
  • Aldous, D. (1989). Probability Approximations via the Poisson Clumping Heuristic. Springer, New York.
    Mathematical Reviews (MathSciNet): MR969362
    Zentralblatt MATH: 0679.60013
  • Berkes, I. and Philipp, W. (1979). Approximation theorems for independent and weakly dependent random vectors. Ann. Probab. 7 29--54.
    Mathematical Reviews (MathSciNet): MR515811
    JSTOR: links.jstor.org
  • Berman, S. M. (1982). Sojourns and extremes of stationary processes. Ann. Probab. 10 1--46.
    Mathematical Reviews (MathSciNet): MR637375
    JSTOR: links.jstor.org
  • Bickel, P. and Rosenblatt, M. (1973). Two dimensional random fields. In Multivariate Analysis III (P. R. Krishnaiah, ed.) 3--13. Academic Press, New York.
    Mathematical Reviews (MathSciNet): MR348832
    Zentralblatt MATH: 0297.60020
  • Chan, H. P. and Lai, T. L. (2003). Saddlepoint approximations and nonlinear boundary-crossing probabilities of Markov random walks. Ann. Appl. Probab. 13 394--428.
    Mathematical Reviews (MathSciNet): MR1970269
    Digital Object Identifier: doi:10.1214/aoap/1050689586
    Project Euclid: euclid.aoap/1050689586
    Zentralblatt MATH: 1029.60058
  • Chan, H. P. and Lai, T. L. (2004). Maxima of random fields and strong limit theorems for sums of linear processes with long-range dependence. Technical report, Dept. Statistics, Stanford Univ.
  • Chan, H. P. and Lai, T. L. (2004). Excursion sets of asymptotically Gaussian random fields and applications to signal detection. Technical report, Dept. Statistics, Stanford Univ.
  • Chan, H. P. and Lai, T. L. (2004). Tail probabilities of scan statistics and asymptotically efficient tests and on-line detection on structural changes. Technical report, Dept. Statistics, Stanford Univ.
  • Chung, K. L. (1949). An estimate concerning the Kolmogoroff limit distribution. Trans. Amer. Math. Soc. 67 36--50.
    Mathematical Reviews (MathSciNet): MR34552
  • Davydov, Y. A. (1970). The invariance principle for stationary processes. Theory. Probab. Appl. 15 487--498.
    Mathematical Reviews (MathSciNet): MR283872
  • Dudley, R. M. and Philipp, W. (1983). Invariance principles for sums of Banach space valued random elements and empirical processes. Z. Wahrsch. Verw. Gebiete 62 509--552.
    Mathematical Reviews (MathSciNet): MR690575
    Digital Object Identifier: doi:10.1007/BF00534202
  • Feller, W. (1971). An Introduction to Probability Theory and its Applications 2, 2nd ed. Wiley, New York.
    Zentralblatt MATH: 0219.60003
  • Fernique, X. (1964). Continuité des processus Gaussiens. C. R. Acad. Sci. Paris 258 6058--6060.
    Mathematical Reviews (MathSciNet): MR164365
  • Gut, A. (1980). Convergence rates for probabilities of moderate deviations for sums of random variables with multidimensional indices. Ann. Probab. 8 298--313.
    Mathematical Reviews (MathSciNet): MR566595
    JSTOR: links.jstor.org
  • Harrison, M. (1985). Brownian Motion and Stochastic Flow System. Wiley, New York.
    Mathematical Reviews (MathSciNet): MR798279
    Zentralblatt MATH: 0659.60112
  • Ho, H. C. and Hsing, T. (1997). Limit theorems for functionals of moving averages. Ann. Probab. 25 1636--1669.
    Mathematical Reviews (MathSciNet): MR1487431
    Project Euclid: euclid.aop/1023481106
  • Hogan, M. L. and Siegmund, D. (1986). Large deviations for the maxima of some random fields. Adv. in Appl. Math. 7 2--22.
    Mathematical Reviews (MathSciNet): MR834217
    Digital Object Identifier: doi:10.1016/0196-8858(86)90003-5
    Zentralblatt MATH: 0612.60029
  • Hotelling, H. (1939). Tubes and spheres in $n$-spaces and a class of statistical problems. Amer. J. Math. 61 440--460.
    Mathematical Reviews (MathSciNet): MR1507387
  • Jennen, C. and Lerche, R. (1981). First exit densities of Brownian motion through one-sided moving boundaries. Z. Wahrsch. Verw. Gebiete 55 133--148.
    Mathematical Reviews (MathSciNet): MR608013
    Digital Object Identifier: doi:10.1007/BF00535156
  • Komlós, J., Major, P. and Tusnády, G. (1975). An approximation of partial sums of independent RV's and the DF I. Z. Wahrsch. Verw. Gebiete 32 111--131.
    Mathematical Reviews (MathSciNet): MR375412
  • Lai, T. L. and Shan, J. Z. (1999). Efficient recursive algorithms for detection of abrupt changes in signals and control systems. IEEE Trans. Automat. Control 44 952--966.
    Mathematical Reviews (MathSciNet): MR1690539
    Digital Object Identifier: doi:10.1109/9.763211
    Zentralblatt MATH: 0956.93060
  • Marcus, M. B. and Shepp, L. A. (1972). Sample behavior of Gaussian processes. Proc. Sixth Berkeley Symp. Math. Statist. Probab. 2 423--441. Univ. California Press, Berkeley.
    Mathematical Reviews (MathSciNet): MR402896
    Zentralblatt MATH: 0379.60040
  • Orey, S. and Pruitt, W. (1973). Sample functions of the $N$-parameter Wiener process. Ann. Probab. 1 138--163.
    Mathematical Reviews (MathSciNet): MR346925
    Digital Object Identifier: doi:10.1214/aop/1176997030
  • Philipp, W. and Stout W. F. (1975). Almost Sure Invariance Principles for Partial Sums of Weakly Dependent Random Variables. Amer. Math. Soc., Providence, RI.
    Zentralblatt MATH: 0361.60007
  • Pickands, J. (1969). Upcrossing probabilities for stationary Gaussian processes. Trans. Amer. Math. Soc. 145 51--75.
    Mathematical Reviews (MathSciNet): MR250367
  • Piterbarg, V. (1996). Asymptotic Methods in the Theory of Gaussian Processes and Fields. Amer. Math. Soc., Providence, RI.
    Mathematical Reviews (MathSciNet): MR1361884
    Zentralblatt MATH: 0841.60024
  • Qualls, C. and Watanabe, H. (1972). Asymptotic properties of Gaussian processes. Ann. Math. Statist. 43 580--596.
    Mathematical Reviews (MathSciNet): MR307318
    Digital Object Identifier: doi:10.1214/aoms/1177692638
  • Qualls, C. and Watanabe, H. (1973). Asymptotic properties of Gaussian random fields. Trans. Amer. Math. Soc. 177 155--171.
    Mathematical Reviews (MathSciNet): MR322943
  • Rio, E. (1993). Strong approximations for set-indexed partial sums via KMT constructions I, II. Ann. Probab. 21 759--790, 1706--1727.
  • Smirnov, N. V. (1944). Approximate laws of distribution of random variables from empirical data. Uspekhi Mat. Nauk 10 179--206.
    Mathematical Reviews (MathSciNet): MR12387
  • Strassen, V. (1967). Almost sure behavior of independent random variables and martingales. Proc. Fifth Berkeley Symp. Math. Statist. Probab. 2 315--343. Univ. California Press, Berkeley.
    Mathematical Reviews (MathSciNet): MR214118
    Zentralblatt MATH: 0201.49903
  • Wang, Q., Lin, Y. X. and Gulati, C. M. (2003). Strong approximation for long memory processes with applications. J. Theoret. Probab. 16 377--389.
    Mathematical Reviews (MathSciNet): MR1982033
    Digital Object Identifier: doi:10.1023/A:1023570510824
    Zentralblatt MATH: 1020.60017
  • Weyl, H. (1939). On the volume of tubes. Amer. J. Math. 61 461--472.
    Mathematical Reviews (MathSciNet): MR1507388
  • Wichura, M. (1985). Boundary crossing probabilities for Brownian motion. Technical report, Dept. Statistics, Univ. Chicago.
  • Zimmerman, G. (1972). Some sample function properties of the two-parameter Gaussian process. Ann. Math. Statist. 43 1235--1246.
    Mathematical Reviews (MathSciNet): MR317401
    Digital Object Identifier: doi:10.1214/aoms/1177692475

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