Does anyone know where I can find some useful information about the Aldous criterion for tightness in the space of all cadlag functions $D[0,1]$?
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There is some stuff in Billingsley "convergence of probability measures" second edition (1999).
Billingsley- Convergence of Probability Measures 2nd Ed. pp. 176
Kallenberg- Foundations of Modern Probability 2nd Ed. pp. 314
Aldous- Stopping Times and Tightness (article)
Or in the books of Ethier-Kurtz or Jacod-Shiryaev
Maybe you can find some information about this criterion on the book by Richard F. Bass, Stochastic Processes-Cambridge University Press (2011).