# Elements of Stochastic Analysis and Stochastic Differential Equations

Chapter

## Abstract

It is often very difficult to decide about the convergence of a series ∑
be a series of independent random variables. According to the 0–1 law, the 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$$
\sum\limits_k {\xi k}
$$

_{ k }ξ_{ k }(*ω*) converges for outcomes ω ∊ Ω whose total probability is either 0 or 1.## Keywords

Correlation Function Brownian Motion Random Process Poisson Process Stochastic Differential Equation
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Springer Science+Business Media Dordrecht 1995