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- Title
Functional strong law of large numbers for Betti numbers in the tail.
- Authors
Owada, Takashi; Wei, Zifu
- Abstract
The objective of this paper is to investigate the layered structure of topological complexity in the tail of a probability distribution. We establish the functional strong law of large numbers for Betti numbers, a basic quantifier of algebraic topology, of a geometric complex outside an open ball of radius R n , such that R n → ∞ as the sample size n increases. The nature of the obtained law of large numbers is determined by the decay rate of a probability density and how rapidly R n diverges. In particular, if R n diverges sufficiently slowly, the limiting function in the law of large numbers is crucially affected by the emergence of arbitrarily large connected components supporting topological cycles in the limit.
- Subjects
DISTRIBUTION (Probability theory); ALGEBRAIC topology; BETTI numbers; EXTREME value theory; LAW of large numbers
- Publication
Extremes, 2022, Vol 25, Issue 4, p655
- ISSN
1386-1999
- Publication type
Article
- DOI
10.1007/s10687-022-00441-x