# Find for a given distribution a random walk

Let $\{Z_i\}$ be i.i.d. $\mathbb{R}^d$ valued random variables. Then we define $S_n:=\sum_{i=1}^n Z_i$ and call the process $\{ S_n\}$ random walk on $\mathbb{R^d}$.

Let $\mu$ be a distribution on $\mathbb{R}^d$. How can I prove that there exists a random walk on $\mathbb{R}^d$ such that $Z_n$ is $\mu$ distributed?

Any hint would be appreciated (it is not a homework)

math

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not true in general, as when $\mu$ is supported on 2 points. you might have a look at infinitely divisible distributions. –  mike Sep 26 '12 at 12:32
@mike Sorry if I'm mistaken, but I think it is true, because it is an exercise in Sato "Lévy Processes and Infinitely divisible distribution on page 5. He suggests to use Theorem 1.9. –  math Sep 26 '12 at 16:40
@math Could you write this theorem? –  Davide Giraudo Sep 27 '12 at 20:33