Consider a geometric scaled walk,

\begin{align*} X_k &= X_0\exp[\frac{1}{\sqrt{k}}(S_k)]\\ &= X_0\exp[\frac{1}{\sqrt{k}}(2T_k - k)] \end{align*}

where $T_k \space \tilde{} \space Binomial(k, 0.5)$

How can I use the Central Limit Theorem to show that the cdf of $\frac{S_k}{\sqrt{k}}$ converges to that of the standard normal as $k \rightarrow \infty$? Is this is the same as showing that a random walk converges to the Brownian motion in continuous time?

I can't seem to find a formal proof of this result anywhere so any guidance would be much appreciated.

  • $\begingroup$ The commonest approach is to show that the characteristic functions converge to the characteristic function of a standard normal. $\endgroup$
    – Did
    Aug 17, 2016 at 10:47
  • $\begingroup$ Possible duplicate of Proving the normal approximation to the binomial $\endgroup$ Aug 17, 2016 at 12:09


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