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DICHOTOMY RESULTS FOR EVENTUALLY ALWAYS HITTING TIME STATISTICS AND ALMOST SURE GROWTH OF EXTREMES

Holland, Mark ; Kirsebom, Maxim ; Kunde, Philipp and Persson, Tomas LU orcid (2024) In Transactions of the American Mathematical Society 377(6). p.3927-3982
Abstract

Suppose (f,X, μ) is a measure preserving dynamical system and φ: X → R a measurable function. Consider the maximum process Mn := max{X1, ,Xn}, where Xi = φ ○ fi-1 is a time series of observations on the system. Suppose that (un) is a non-decreasing sequence of real numbers, such that μ(X1 > un) → 0. For certain dynamical systems, we obtain a zero- one measure dichotomy for μ(Mn ≤ un i.o.) depending on the sequence un. Specific examples are piecewise expanding interval maps including the Gauß map. For the broader class of non-uniformly hyperbolic dynamical systems, we make significant improvements on existing literature for characterising the sequences un. Our results on the permitted sequences un are commensurate with the optimal... (More)

Suppose (f,X, μ) is a measure preserving dynamical system and φ: X → R a measurable function. Consider the maximum process Mn := max{X1, ,Xn}, where Xi = φ ○ fi-1 is a time series of observations on the system. Suppose that (un) is a non-decreasing sequence of real numbers, such that μ(X1 > un) → 0. For certain dynamical systems, we obtain a zero- one measure dichotomy for μ(Mn ≤ un i.o.) depending on the sequence un. Specific examples are piecewise expanding interval maps including the Gauß map. For the broader class of non-uniformly hyperbolic dynamical systems, we make significant improvements on existing literature for characterising the sequences un. Our results on the permitted sequences un are commensurate with the optimal sequences (and series criteria) obtained by Klass (1985) for i.i.d. processes. Moreover, we also develop new series criteria on the permitted sequences in the case where the i.i.d. theory breaks down. Our analysis has strong connections to specific problems in eventual always hitting time statistics and extreme value theory.

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Please use this url to cite or link to this publication:
author
; ; and
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
eventually always hitting points, extremal index, Extreme value theory, Robbins- Siegmund series criterion
in
Transactions of the American Mathematical Society
volume
377
issue
6
pages
56 pages
publisher
American Mathematical Society (AMS)
external identifiers
  • scopus:85195549788
ISSN
0002-9947
DOI
10.1090/tran/9102
language
English
LU publication?
yes
id
323def2a-4e08-4382-b2c2-4382bdad03dd
alternative location
https://arxiv.org/abs/2109.06314
date added to LUP
2024-08-26 09:02:42
date last changed
2025-04-04 14:21:31
@article{323def2a-4e08-4382-b2c2-4382bdad03dd,
  abstract     = {{<p>Suppose (f,X, μ) is a measure preserving dynamical system and φ: X → R a measurable function. Consider the maximum process Mn := max{X1, ,Xn}, where Xi = φ ○ fi-1 is a time series of observations on the system. Suppose that (un) is a non-decreasing sequence of real numbers, such that μ(X1 &gt; un) → 0. For certain dynamical systems, we obtain a zero- one measure dichotomy for μ(Mn ≤ un i.o.) depending on the sequence un. Specific examples are piecewise expanding interval maps including the Gauß map. For the broader class of non-uniformly hyperbolic dynamical systems, we make significant improvements on existing literature for characterising the sequences un. Our results on the permitted sequences un are commensurate with the optimal sequences (and series criteria) obtained by Klass (1985) for i.i.d. processes. Moreover, we also develop new series criteria on the permitted sequences in the case where the i.i.d. theory breaks down. Our analysis has strong connections to specific problems in eventual always hitting time statistics and extreme value theory.</p>}},
  author       = {{Holland, Mark and Kirsebom, Maxim and Kunde, Philipp and Persson, Tomas}},
  issn         = {{0002-9947}},
  keywords     = {{eventually always hitting points; extremal index; Extreme value theory; Robbins- Siegmund series criterion}},
  language     = {{eng}},
  number       = {{6}},
  pages        = {{3927--3982}},
  publisher    = {{American Mathematical Society (AMS)}},
  series       = {{Transactions of the American Mathematical Society}},
  title        = {{DICHOTOMY RESULTS FOR EVENTUALLY ALWAYS HITTING TIME STATISTICS AND ALMOST SURE GROWTH OF EXTREMES}},
  url          = {{http://dx.doi.org/10.1090/tran/9102}},
  doi          = {{10.1090/tran/9102}},
  volume       = {{377}},
  year         = {{2024}},
}