Typical Martingale Diverges at a Typical Point

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• 作者：Ondřej F. K. Kalenda ; Jiří Spurný
• 关键词：\(L^1\) ; bounded martingale ; \(L^p\) ; bounded martingale ; Filtration of finite \(\sigma \) ; algebras ; Oscillation ; Comeager set
• 刊名：Journal of Theoretical Probability
• 出版年：2016
• 出版时间：March 2016
• 年：2016
• 卷：29
• 期：1
• 页码：180-205
• 全文大小：535 KB
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• 作者单位：Ondřej F. K. Kalenda (1)
  Jiří Spurný (1)
  
  1. Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 , Prague 8, Czech Republic
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Probability Theory and Stochastic Processes
  Statistics
  
• 出版者：Springer Netherlands
• ISSN：1572-9230

文摘

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