DEFINITION 1 A probability space is a triplet of a set , a family of subsets , and a mapping from into satisfying the following conditions:

- ;
- If for then and are in ;
- If , then ;
- ;
- If for and they are disjoint, then }

DEFINITION 2 Let be a probability space. A mapping from into is an $latex {\mathbb R}^d$ valued random variable if it is -measurable, that is is in for each

DEFINITION 3 A stochastic process on is called stochastically continuous or continuous in probability if, for every and ,.

The most basic stochastic process modeled for continuous motions is the Brownian motion and that for jumping random motions is the Poisson process. These two belongs to a class called L\’evy processes.

Converge stochastically (converge in Probability)

A sequence of valued variables is said to converge stochastically, to X if, for each , . This is denoted by in probability. If converges stochastically to X and , then

Converge almost surelyA sequence of valued variables is said to converge almost surely, to X if, .

DEFINITION 5 The characteristic function of a probability measure on is

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