Elements of Stochastic Analysis and Stochastic Differential Equations

  • Yu. A. Rozanov
Part of the Mathematics and Its Applications book series (MAIA, volume 344)


It is often very difficult to decide about the convergence of a series ∑ k x k in the case when ∑ k |x k | = ∞ and the ± signs of x k , k = 1, 2,... , do not form a regular pattern; of course one can apply the general criterion ∑ k=m n x k → 0, m, n → ∞, but actually nothing else. In such a case, Stochastic Analysis can be helpful provided the signs of the summands follow a typical ‘head’ or ‘tail’ sequences in a series of independent coin tossings. For example, let
$$ \sum\limits_k {\xi k} $$
be a series of independent random variables. According to the 0–1 law, the series ∑ k ξ k (ω) converges for outcomes ω ∊ Ω whose total probability is either 0 or 1.


Correlation Function Brownian Motion Random Process Poisson Process Stochastic Differential Equation 
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Copyright information

© Springer Science+Business Media Dordrecht 1995

Authors and Affiliations

  • Yu. A. Rozanov
    • 1
  1. 1.Steklov Mathematical InstituteMoscowRussia

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