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

Zakhar Kabluchko; Axel Munk

ESAIM: Probability and Statistics (2009)

  • Volume: 13, page 409-416
  • ISSN: 1292-8100

Abstract

top
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.

How to cite

top

Kabluchko, Zakhar, and Munk, Axel. "Shao's theorem on the maximum of standardized random walk increments for multidimensional arrays." ESAIM: Probability and Statistics 13 (2009): 409-416. <http://eudml.org/doc/250667>.

@article{Kabluchko2009,
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. },
author = {Kabluchko, Zakhar, Munk, Axel},
journal = {ESAIM: Probability and Statistics},
keywords = {Standardized increments; Lévy's continuity modulus; almost sure limit theorem; Erdös-Rényi law; multidimensional i.i.d. array; statistical multiscale parameter selection; scan statistics.; standardized increments; almost sure convergence; Erdős-Rényi law; multidimensional i.i.d. array},
language = {eng},
month = {9},
pages = {409-416},
publisher = {EDP Sciences},
title = {Shao's theorem on the maximum of standardized random walk increments for multidimensional arrays},
url = {http://eudml.org/doc/250667},
volume = {13},
year = {2009},
}

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
DA - 2009/9//
PB - EDP Sciences
VL - 13
SP - 409
EP - 416
AB - 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.
LA - eng
KW - Standardized increments; Lévy's continuity modulus; almost sure limit theorem; Erdös-Rényi law; multidimensional i.i.d. array; statistical multiscale parameter selection; scan statistics.; standardized increments; almost sure convergence; Erdős-Rényi law; multidimensional i.i.d. array
UR - http://eudml.org/doc/250667
ER -

References

top
  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).  Zbl0539.60029
  3. P.L. Davies and A. Kovac, Local extremes, runs, strings and multiresolution (with discussion). Ann. Statist.29 (2001) 1–65.  Zbl1029.62038
  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.  Zbl0548.60027
  5. L. Dümbgen and V.G. Spokoiny, Multiscale testing of qualitative hypotheses. Ann. Statist.29 (2001) 124–152.  Zbl1029.62070
  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.  Zbl1142.62336
  7. P. Erdös and A. Rényi, On a new law of large numbers. J. Anal. Math.23 (1970) 103–111.  Zbl0225.60015
  8. W. Feller, An introduction to probability theory and its applications. Vol. II, second edition. John Wiley and Sons, New York-London-Sydney (1971).  Zbl0219.60003
  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.  Zbl0519.60030
  10. W. Hinterberger, M. Hintermüller, K. Kunisch, M. von Oehsen and O. Scherzer, Tube methods for BV regularization. J. Math. Imag. Vision19 (2003) 219–235.  Zbl1101.68927
  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.  Zbl0308.60029
  12. H. Lanzinger and U. Stadtmüller, Maxima of increments of partial sums for certain subexponential distributions. Stoch. Process. Appl.86 (2000) 307–322.  Zbl1028.60044
  13. P. Massart, Strong approximation for multivariate empirical and related processes, via KMT constructions. Ann. Probab.17 (1989) 266–291.  Zbl0675.60026
  14. P. Révész, Random walk in random and non-random environments. World Scientific (1990).  Zbl0733.60091
  15. E. Rio, Strong approximation for set-indexed partial sum processes via KMT constructions III. ESAIM: PS1 (1997) 319–338.  Zbl0930.60016
  16. Q.-M. Shao, On a conjecture of Révész. Proc. Amer. Math. Soc.123 (1995) 575–582.  Zbl0809.60036
  17. D. Siegmund and B. Yakir, Tail probabilities for the null distribution of scanning statistics. Bernoulli6 (2000) 191–213.  Zbl0976.62048
  18. J. Steinebach, On the increments of partial sum processes with multidimensional indices. Z. Wahrscheinlichkeitstheor. Verw. Geb.63 (1983) 59–70.  Zbl0511.60027
  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).  
  20. S. van de Geer and E. Mammen, Discussion of “Local extremes, strings and multiresolution.” Ann. Statist.29 (2001) 56–59.  

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.