# Convergence in distribution of Wiener Process

I am considering this in the sense that I know according to the central limit theorem, for an i.i.d. process $X_n$ (with mean $m$ and variance $σ^2$), the corresponding normalized sum process is: $$Z_n = \frac{S_n-nm}{σ\sqrt{n}}$$ with $S_n = X_1+X_2+ . . . + X_n$. I know that this does indeed converge in distribution to a zero-mean unit-variance Gaussian.

My problem is to perform analysis on the Wiener process using the Taylor expansion of the characteristic function of $Z_n$:

$$E\left[\exp\left(-\frac{j \omega}{\sigma \sqrt{n}}(X_1-m)\right)\right]=\displaystyle\sum\limits_{k=0}^∞ \frac{1}{k!}\left(-\frac{j \omega}{\sigma \sqrt{n}}\right)^k E \left[(X_1-m)^k\right]$$

and examining the terms $k=3,4,5...$ to verify that higher-order terms become negligible and for large $n$ and therefore the process converges in distribution to a Gaussian. I am having trouble determining the mean and variance of $X_1$ and consequently the $kth$ central moments.

Any help would be greatly appreciated. Thanks!

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Where do you see a Wiener process here? Do you know what is a Wiener process? –  Did Dec 14 '12 at 23:59