Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-25T11:26:08.080Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  22 September 2009

Zakhar Kabluchko  and
Axel Munk
Zakhar Kabluchko
Affiliation:
Institut für Mathematische Stochastik, Georg-August-Universität Göttingen, Maschmühlenweg 8-10, 37073 Göttingen, Germany; [email protected]; [email protected]
Axel Munk
Affiliation:
Institut für Mathematische Stochastik, Georg-August-Universität Göttingen, Maschmühlenweg 8-10, 37073 Göttingen, Germany; [email protected]; [email protected]
Get access

Abstract

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.

Keywords

Standardized incrementsLévy's continuity modulusalmost sure limit theoremErdös-Rényi lawmultidimensional i.i.d. arraystatistical multiscale parameter selectionscan statistics.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 13 , July 2009 , pp. 409 - 416
Copyright
© EDP Sciences, SMAI, 2009

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.)

References

Bissantz, N., Mair, B. and Munk, A., A statistical stopping rule for MLEM reconstructions in PET. IEEE Nucl. Sci. Symp. Conf. Rec. 8 (2008) 4198–4200.
M. Csörgö and P. Révész, Strong approximations in probability and statistics. Academic Press, New York-San Francisco-London (1981).
Davies, P.L. and Kovac, A., Local extremes, runs, strings and multiresolution (with discussion). Ann. Statist. 29 (2001) 1–65. CrossRef
Deheuvels, P., 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. CrossRef
Dümbgen, L. and Spokoiny, V.G., Multiscale testing of qualitative hypotheses. Ann. Statist. 29 (2001) 124–152. CrossRef
Dümbgen, L. and Walther, G., Multiscale inference about a density. Preprint (Extended version: Technical report 56, Univ. of Bern). Ann. Statist. 36 (2008) 1758–1758. CrossRef
Erdös, P. and Rényi, A., On a new law of large numbers. J. Anal. Math. 23 (1970) 103–111. CrossRef
W. Feller, An introduction to probability theory and its applications. Vol. II, second edition. John Wiley and Sons, New York-London-Sydney (1971).
Hanson, D.L. and Russo, R.P., 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. CrossRef
Hinterberger, W., Hintermüller, M., Kunisch, K., von Oehsen, M. and Scherzer, O., Tube methods for BV regularization. J. Math. Imag. Vision 19 (2003) 219–235. CrossRef
Komlós, J., Major, P. and Tusnády, G., An approximation of partial sums of independent RV's, and the sample DF, Vol. I. Z. Wahrscheinlichkeitstheor. Verw. Geb. 32 (1975) 111–131. CrossRef
Lanzinger, H. and Stadtmüller, U., Maxima of increments of partial sums for certain subexponential distributions. Stoch. Process. Appl. 86 (2000) 307–322. CrossRef
Massart, P., Strong approximation for multivariate empirical and related processes, via KMT constructions. Ann. Probab. 17 (1989) 266–291. CrossRef
P. Révész, Random walk in random and non-random environments. World Scientific (1990).
Rio, E., Strong approximation for set-indexed partial sum processes via KMT constructions III. ESAIM: PS 1 (1997) 319–338. CrossRef
Shao, Q.-M., On a conjecture of Révész. Proc. Amer. Math. Soc. 123 (1995) 575–582.
Siegmund, D. and Yakir, B., Tail probabilities for the null distribution of scanning statistics. Bernoulli 6 (2000) 191–213. CrossRef
Steinebach, J., On the increments of partial sum processes with multidimensional indices. Z. Wahrscheinlichkeitstheor. Verw. Geb. 63 (1983) 59–70. CrossRef
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).
S. van de Geer and E. Mammen, Discussion of “Local extremes, strings and multiresolution.” Ann. Statist. 29 (2001) 56–59.