# Understanding the Kesten Multiplicative Process

I read in D. Sornette's Critical Phenomena in Natural Sciences about the Kesten Multiplicative process:

$$X_{n+1} = a_n X_n + b_n$$ Where $a_n$ and $b_n$ are stochastic variables drawn from the pdfs $P_{a_n}(a_n)$ and $P_{b_n}(b_n)$.

Sornette writes: "the random multiplicative coefficients $a_n$ lead to non-trivial intermittent behavior for a large class of distributions for $a_n$...it provides a simple and general mechanism for generating power law ditributions."

Is it possible to prove that the Kesten Process can be used to generate a power law distribution? Or, if I wanted to find the class of $a_n$ distributions that yeild power law behavior, are there methods other than experimentation of doing that?

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I think this is in Kesten's original paper, which was in the annals of prob in the early 70's, unfortunately prior to when you can get it from JSTOR. The basic idea is that you can solve the recursion explicitly to find that $X_n$ resembles ( but is but is not exactly equal to) $\sum_i \prod^i a_j$. You can show that this r.v. does not have moments of all orders, and in fact the condition for the rth moment is that $\mathbb E(a_j^r) < \infty$, at least if $r \ge 1$. To get its tail behavior is basically a large deviation calculation on $\sum^i log( a_j)$. If you do this you are again messing around with $\mathbb E (a_j^r)$ which is the mgf for $log(a_j)$. It is interesting that you can get the tail behavior of these guys, which arise in arch models, but it seems like a bit much just to get power law distributions.