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.
  • Adler, R. and Brown, L. (1986). Tail behaviour for suprema of empirical processes. Ann. Probab. 14 1--30.
  • Albin, J. M. P. (1990). On extremal theory for stationary processes. Ann. Probab. 18 92--128.
  • Aldous, D. (1989). Probability Approximations via the Poisson Clumping Heuristic. Springer, New York.
  • Berkes, I. and Philipp, W. (1979). Approximation theorems for independent and weakly dependent random vectors. Ann. Probab. 7 29--54.
  • Berman, S. M. (1982). Sojourns and extremes of stationary processes. Ann. Probab. 10 1--46.
  • Bickel, P. and Rosenblatt, M. (1973). Two dimensional random fields. In Multivariate Analysis III (P. R. Krishnaiah, ed.) 3--13. Academic Press, New York.
  • 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.
  • 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.
  • Davydov, Y. A. (1970). The invariance principle for stationary processes. Theory. Probab. Appl. 15 487--498.
  • 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.
  • Feller, W. (1971). An Introduction to Probability Theory and its Applications 2, 2nd ed. Wiley, New York.
  • Fernique, X. (1964). Continuité des processus Gaussiens. C. R. Acad. Sci. Paris 258 6058--6060.
  • Gut, A. (1980). Convergence rates for probabilities of moderate deviations for sums of random variables with multidimensional indices. Ann. Probab. 8 298--313.
  • Harrison, M. (1985). Brownian Motion and Stochastic Flow System. Wiley, New York.
  • Ho, H. C. and Hsing, T. (1997). Limit theorems for functionals of moving averages. Ann. Probab. 25 1636--1669.
  • Hogan, M. L. and Siegmund, D. (1986). Large deviations for the maxima of some random fields. Adv. in Appl. Math. 7 2--22.
  • Hotelling, H. (1939). Tubes and spheres in $n$-spaces and a class of statistical problems. Amer. J. Math. 61 440--460.
  • Jennen, C. and Lerche, R. (1981). First exit densities of Brownian motion through one-sided moving boundaries. Z. Wahrsch. Verw. Gebiete 55 133--148.
  • 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.
  • 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.
  • 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.
  • Orey, S. and Pruitt, W. (1973). Sample functions of the $N$-parameter Wiener process. Ann. Probab. 1 138--163.
  • Philipp, W. and Stout W. F. (1975). Almost Sure Invariance Principles for Partial Sums of Weakly Dependent Random Variables. Amer. Math. Soc., Providence, RI.
  • Pickands, J. (1969). Upcrossing probabilities for stationary Gaussian processes. Trans. Amer. Math. Soc. 145 51--75.
  • Piterbarg, V. (1996). Asymptotic Methods in the Theory of Gaussian Processes and Fields. Amer. Math. Soc., Providence, RI.
  • Qualls, C. and Watanabe, H. (1972). Asymptotic properties of Gaussian processes. Ann. Math. Statist. 43 580--596.
  • Qualls, C. and Watanabe, H. (1973). Asymptotic properties of Gaussian random fields. Trans. Amer. Math. Soc. 177 155--171.
  • 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.
  • 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.
  • Wang, Q., Lin, Y. X. and Gulati, C. M. (2003). Strong approximation for long memory processes with applications. J. Theoret. Probab. 16 377--389.
  • Weyl, H. (1939). On the volume of tubes. Amer. J. Math. 61 461--472.
  • 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.