# question regarding multiplicative random walk and uniform distribution

Let $$V_n$$ be a value after $$n$$ time, and at $$n = 0, V_0 = 1$$, and then it changed by a factor $$U_n$$ in $$n$$-th interval. which means that $$V_n = U_1 *...* U_n$$ and each $$U$$ is a independent uniform distributed in $$(0,k)$$ with parameter $$k$$.

Now by the law of large number that $$(V_n)^{1/n}$$ approaches to a number $$L$$ when $$n\to\infty$$.

How can we evaluate $$L$$ with $$k$$?

I tried:

to separate the $$V_n$$ with the log into $$U$$'s but not sure how to proceed the next step.

Thank you! ## 1 Answer

You asked a follow-up question, which I was planning to answer when you deleted it. What I was intending to say was:

• $$\log(K)-\log(U_n) \sim Exp(1)$$

• so $$\log(U_n)$$ has mean $$\log(K)-1$$ and variance and standard deviation $$1$$

• so, since $$\log(V_0)=0$$, we have $$\frac1n \log(V_n) \xrightarrow{LLN} \log(K)-1$$ and $$V_n^{1/n} \xrightarrow{LLN} \frac{K}{e}$$ and this is $$L$$

• and we have $$\frac1{\sqrt{n}}\left(\log(V_n) -n(\log(K)-1)\right) \xrightarrow{d} \mathcal N(0,1)$$ in distribution using the CLT

• so $$\left(\frac{V_n}{L^n}\right)^{1/n} \xrightarrow{d} Lognormal(0,1)$$ in distribution

• meaning $$W \sim Lognormal(0,1)$$ with mean $$\sqrt{e}$$ and variance $$e^2-e$$ and $$E[W^n]=e^{n^2/2}$$