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.
Mots clés : standardized increments, Lévy's continuity modulus, almost sure limit theorem, Erdös-Rényi law, multidimensional i.i.d. array, statistical multiscale parameter selection, scan statistics
@article{PS_2009__13__409_0,
author = {Kabluchko, Zakhar and Munk, Axel},
title = {Shao's theorem on the maximum of standardized random walk increments for multidimensional arrays},
journal = {ESAIM: Probability and Statistics},
pages = {409--416},
publisher = {EDP-Sciences},
volume = {13},
year = {2009},
doi = {10.1051/ps:2008020},
mrnumber = {2554963},
zbl = {1188.60014},
language = {en},
url = {http://www.numdam.org/articles/10.1051/ps:2008020/}
}
TY - JOUR AU - Kabluchko, Zakhar AU - Munk, Axel TI - Shao's theorem on the maximum of standardized random walk increments for multidimensional arrays JO - ESAIM: Probability and Statistics PY - 2009 SP - 409 EP - 416 VL - 13 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ps:2008020/ DO - 10.1051/ps:2008020 LA - en ID - PS_2009__13__409_0 ER -
%0 Journal Article %A Kabluchko, Zakhar %A Munk, Axel %T Shao's theorem on the maximum of standardized random walk increments for multidimensional arrays %J ESAIM: Probability and Statistics %D 2009 %P 409-416 %V 13 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/ps:2008020/ %R 10.1051/ps:2008020 %G en %F PS_2009__13__409_0
Kabluchko, Zakhar; Munk, Axel. Shao's theorem on the maximum of standardized random walk increments for multidimensional arrays. ESAIM: Probability and Statistics, Tome 13 (2009), pp. 409-416. doi : 10.1051/ps:2008020. http://www.numdam.org/articles/10.1051/ps:2008020/
[1] , and , A statistical stopping rule for MLEM reconstructions in PET. IEEE Nucl. Sci. Symp. Conf. Rec. 8 (2008) 4198-4200.
[2] and , Strong approximations in probability and statistics. Academic Press, New York-San Francisco-London (1981). | MR | Zbl
[3] and , Local extremes, runs, strings and multiresolution (with discussion). Ann. Statist. 29 (2001) 1-65. | MR | Zbl
[4] , 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] and , Multiscale testing of qualitative hypotheses. Ann. Statist. 29 (2001) 124-152. | MR | Zbl
[6] and , Multiscale inference about a density. Preprint (Extended version: Technical report 56, Univ. of Bern). Ann. Statist. 36 (2008) 1758-1758. | MR | Zbl
[7] and , On a new law of large numbers. J. Anal. Math. 23 (1970) 103-111. | MR | Zbl
[8] , An introduction to probability theory and its applications. Vol. II, second edition. John Wiley and Sons, New York-London-Sydney (1971). | MR | Zbl
[9] and , 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] , , , and , Tube methods for BV regularization. J. Math. Imag. Vision 19 (2003) 219-235. | MR | Zbl
[11] , and , 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] and , Maxima of increments of partial sums for certain subexponential distributions. Stoch. Process. Appl. 86 (2000) 307-322. | MR | Zbl
[13] , Strong approximation for multivariate empirical and related processes, via KMT constructions. Ann. Probab. 17 (1989) 266-291. | MR | Zbl
[14] , Random walk in random and non-random environments. World Scientific (1990). | Zbl
[15] , Strong approximation for set-indexed partial sum processes via KMT constructions III. ESAIM: PS 1 (1997) 319-338. | Numdam | MR | Zbl
[16] , On a conjecture of Révész. Proc. Amer. Math. Soc. 123 (1995) 575-582. | MR | Zbl
[17] and , Tail probabilities for the null distribution of scanning statistics. Bernoulli 6 (2000) 191-213. | MR | Zbl
[18] , On the increments of partial sum processes with multidimensional indices. Z. Wahrscheinlichkeitstheor. Verw. Geb. 63 (1983) 59-70. | MR | Zbl
[19] , 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] and , Discussion of “Local extremes, strings and multiresolution.” Ann. Statist. 29 (2001) 56-59.
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