Series Criteria for Growth Rates of Partial Maxima of Iterated Ergodic Map Values

M. J. Appel ()
Additional contact information
M. J. Appel: Mortgage Guaranty Insurance Corporation

Journal of Theoretical Probability, 2002, vol. 15, issue 1, 153-159

Abstract: Abstract Birkhoff's well-known ergodic theorem states that the simple averages of a sequence of real (integrable) function values on successive iterates of a measure-preserving mapping T converge a.s. to the conditional expected value of the function conditioned on the invariant sigma-field. If the mapping is in addition ergodic, then the limit is simply the unconditional expected value: $$\frac{1}{n}\sum\limits_{k = 0}^{n - 1} {f \circ T^k \to \int_\Omega {f\;dP,{ a}{.s as }n \to \infty } } { (0}{.1)}$$ In this article, we discuss the analogous result for sequences of partial maxima: given a measurable f, if T is measure-preserving and ergodic then $$M_n = \mathop {\max }\limits_{k\; \leqslant \;n} f \circ T^k \uparrow {ess}\;\sup f,{ a}{.s as }n \to \infty {(0}{.2)}$$ Series criteria are provided which characterize the a.s. maximal and minimal growth rates of the sequence of partial maxima.

Keywords: extrema; essential suprema; maxima; ergodic theory; strict sense stationarity; series criteria (search for similar items in EconPapers)
Date: 2002
References: Add references at CitEc
Citations:

Downloads: (external link)
http://link.springer.com/10.1023/A:1013843518677 Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:jotpro:v:15:y:2002:i:1:d:10.1023_a:1013843518677

Ordering information: This journal article can be ordered from
https://www.springer.com/journal/10959

DOI: 10.1023/A:1013843518677

Access Statistics for this article

Journal of Theoretical Probability is currently edited by Andrea Monica

More articles in Journal of Theoretical Probability from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().