DEFINITION 1 A probability space (\Omega, \mathcal F, P) is a triplet of a set \Omega, a family \mathcal F of subsets \Omega, and a mapping P from \mathcal F into \mathbb R satisfying the following conditions:

  1.  \Omega \in \mathcal F, \varnothing \in \mathcal F;
  2. If A_n \in \mathcal F for n=1, 2,\cdots, then \cup_{n=1}^{\infty}A_n and \cap_{n=1}^{\infty}A_n are in \mathcal F;
  3. If A\in \mathcal F, then A^c \in \mathcal F;
  4. 0 \leq P[A] \leq 1, P[\Omega]=1, and P[\varnothing]=0;
  5. If A_n \in \mathcal F for n=1, 2,\cdots and they are disjoint, then P[\cup_{n=1}^{\infty}A_n]=\sum_{n=1}^{\infty}P[A_n]. }

DEFINITION 2 Let (\Omega, \mathcal F, P) be a probability space. A mapping X from \Omega into {\mathbb R}^d is an $latex {\mathbb R}^d$ valued random variable if it is \mathcal F-measurable, that is \{\omega:X(\omega)\in B\} is in \mathcal F for each B\in \mathcal B(\mathbb R^d). 

 

 DEFINITION 3 A stochastic process \{X_t\} on \mathbb R^d is called stochastically continuous or continuous in probability if, for every t \geq 0 and \varepsilon >0,lim_{s\to t} P[|X_t-X_s|>\varepsilon]=0.

 

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  \mathbb R^d valued variables \{X_n: n=1,2,\cdots\} is said to converge stochastically, to X if, for each \varepsilon >0lim_{n \to \infty} P[|X_n-X|>\varepsilon]=0. This is denoted by  X_n \to X in probability. If \{X_n\} converges stochastically to X and X', then X=X' a.s. 

     

 Converge almost surelyA sequence of  \mathbb R^d valued variables \{X_n: n=1,2,\cdots\} is said to converge almost surely, to X if,  P[lim_{n \to \infty} X_n=X]=1.  

 

DEFINITION 5 The characteristic function \hat \mu (z) of a probability measure \mu on R^d is \hat \mu (z)=\int_{R}^d e^{i\langle z, x \rangle} \mu (dx), z \in R^d. 

     

 

 

 

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