Shao's theorem on the maximum of standardized random walk increments for multidimensional arrays

Kabluchko, Zakhar; Munk, Axel

ESAIM: Probability and Statistics, Tome 13 (2009), pp. 409-416.

We generalize a theorem of Shao [Proc. Amer. Math. Soc. 123 (1995) 575-582] on the almost-sure limiting behavior of the maximum of standardized random walk increments to multidimensional arrays of i.i.d. random variables. The main difficulty is the absence of an appropriate strong approximation result in the multidimensional setting. The multiscale statistic under consideration was used recently for the selection of the regularization parameter in a number of statistical algorithms as well as for the multiscale signal detection.

[1] N. Bissantz, B. Mair and A. Munk, A statistical stopping rule for MLEM reconstructions in PET. IEEE Nucl. Sci. Symp. Conf. Rec. 8 (2008) 4198-4200.

[2] M. Csörgö and P. Révész, Strong approximations in probability and statistics. Academic Press, New York-San Francisco-London (1981). | MR | Zbl

[3] P.L. Davies and A. Kovac, Local extremes, runs, strings and multiresolution (with discussion). Ann. Statist. 29 (2001) 1-65. | MR | Zbl

[4] P. Deheuvels, On the Erdös-Rényi theorem for random fields and sequences and its relationships with the theory of runs and spacings. Z. Wahrscheinlichkeitstheor. Verw. Geb. 70 (1985) 91-115. | MR | Zbl

[5] L. Dümbgen and V.G. Spokoiny, Multiscale testing of qualitative hypotheses. Ann. Statist. 29 (2001) 124-152. | MR | Zbl

[6] L. Dümbgen and G. Walther, Multiscale inference about a density. Preprint (Extended version: Technical report 56, Univ. of Bern). Ann. Statist. 36 (2008) 1758-1758. | MR | Zbl

[7] P. Erdös and A. Rényi, On a new law of large numbers. J. Anal. Math. 23 (1970) 103-111. | MR | Zbl

[8] W. Feller, An introduction to probability theory and its applications. Vol. II, second edition. John Wiley and Sons, New York-London-Sydney (1971). | MR | Zbl

[9] D.L. Hanson and R.P. Russo, Some results on increments of the Wiener process with applications to lag sums of i.i.d. random variables. Ann. Probab. 11 (1983) 609-623. | MR | Zbl

[10] W. Hinterberger, M. Hintermüller, K. Kunisch, M. Von Oehsen and O. Scherzer, Tube methods for BV regularization. J. Math. Imag. Vision 19 (2003) 219-235. | MR | Zbl

[11] J. Komlós, P. Major and G. Tusnády, An approximation of partial sums of independent RV's, and the sample DF, Vol. I. Z. Wahrscheinlichkeitstheor. Verw. Geb. 32 (1975) 111-131. | MR | Zbl

[12] H. Lanzinger and U. Stadtmüller, Maxima of increments of partial sums for certain subexponential distributions. Stoch. Process. Appl. 86 (2000) 307-322. | MR | Zbl

[13] P. Massart, Strong approximation for multivariate empirical and related processes, via KMT constructions. Ann. Probab. 17 (1989) 266-291. | MR | Zbl

[14] P. Révész, Random walk in random and non-random environments. World Scientific (1990). | Zbl

[15] E. Rio, Strong approximation for set-indexed partial sum processes via KMT constructions III. ESAIM: PS 1 (1997) 319-338. | Numdam | MR | Zbl

[16] Q.-M. Shao, On a conjecture of Révész. Proc. Amer. Math. Soc. 123 (1995) 575-582. | MR | Zbl

[17] D. Siegmund and B. Yakir, Tail probabilities for the null distribution of scanning statistics. Bernoulli 6 (2000) 191-213. | MR | Zbl

[18] J. Steinebach, On the increments of partial sum processes with multidimensional indices. Z. Wahrscheinlichkeitstheor. Verw. Geb. 63 (1983) 59-70. | MR | Zbl

[19] J. Steinebach, On a conjecture of Révész and its analogue for renewal processes, in Asymptotic methods in probability and statistics, Barbara Szyszkowicz Ed., A volume in honour of Miklós Csörgö. ICAMPS '97, an international conference at Carleton Univ., Ottawa, Canada. Elsevier, North-Holland, Amsterdam (1997). | Zbl

[20] S. Van De Geer and E. Mammen, Discussion of “Local extremes, strings and multiresolution.” Ann. Statist. 29 (2001) 56-59.