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 >0$$lim_{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 $\hat \mu (z)=\int_{R}^d e^{i\langle z, x \rangle} \mu (dx), z \in R^d.$