Typical Martingale Diverges at a Typical Point

Ondřej F. K. Kalenda () and Jiří Spurný ()
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Ondřej F. K. Kalenda: Charles University
Jiří Spurný: Charles University

Journal of Theoretical Probability, 2016, vol. 29, issue 1, 180-205

Abstract: Abstract We investigate convergence of martingales adapted to a given filtration of finite $$\sigma $$ σ -algebras. To any such filtration, we associate a canonical metrizable compact space $$K$$ K such that martingales adapted to the filtration can be canonically represented on $$K$$ K . We further show that (except for trivial cases) typical martingale diverges at a comeager subset of $$K$$ K . ‘Typical martingale’ means a martingale from a comeager set in any of the standard spaces of martingales. In particular, we show that a typical $$L^1$$ L 1 -bounded martingale of norm at most one converges almost surely to zero and has maximal possible oscillation on a comeager set.

Keywords: $$L^1$$ L 1 -bounded martingale; $$L^p$$ L p -bounded martingale; Filtration of finite $$\sigma $$ σ -algebras; Oscillation; Comeager set; 60G42; 54E52; 54E70 (search for similar items in EconPapers)
Date: 2016
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DOI: 10.1007/s10959-014-0567-7

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