The characteristic function approach is particularly useful in analysis of linear combinations of independent random variables : a classical proof of the Central Limit Theorem uses characteristic functions and L�vy's continuity theorem.
2.
As in most of the definitions of Brownian motion it is required that the sample paths are continuous almost surely, one then uses kolmogorov continuity theorem to construct a continuous modification of the process constructed by Kolmogorov extension theorem.
3.
In the case of Brownian motion on \ mathbb { R } ^ n, the choice of constants \ alpha = 4, \ beta = 1, K = n ( n + 2 ) will work in the Kolmogorov continuity theorem.
4.
The limit " e " " it " ? is the characteristic function of the constant random variable ?, and hence by the L�vy continuity theorem, \ scriptstyle \ overline { X } _ n converges in distribution to ?:
5.
If a continuous-time real-valued stochastic process meets certain moment conditions on its increments, then the Kolmogorov continuity theorem says that there exists a modification of this process that has continuous sample paths with probability one, so the stochastic process has a continuous modification or version.
6.
In other words, the characteristic function converges pointwise to \ exp (-( t') ^ \ alpha ) . and therefore by L�vy's continuity theorem the sum divided by [ na ( \ ln n ) / \ alpha ] ^ { 1 / \ alpha } converges in distribution to the symmetric alpha-stable distribution with stability parameter ? and scale parameter 1.
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