# Use renewal theory for approximation.

I am confused about how to use renewal theory finding approximations required in the following questions. The questions are:

A fair 4-sided die has sides labelled 1,2,3,4. Let $$Y_n$$ equal the product of the first n rolls.
Let N be the first n for which $$Y_n$$ > $$10^{100}$$.

1. Use renewal theory to find a very good approximation to $$E(N)$$.
2. Find the approximate probability that there is no n for which 1,000,000 < $$Y_n$$ < 2,000,000.

Anyone have a good idea of how to construct a renewal process that make sense, I feel like I can't find a good way to do this.

• Note that $\log(Y_n)$ is a process with independent increments. – Ian Dec 19 '18 at 5:46
• Would you mind provide more detailed solution? – Muser_Hao Dec 20 '18 at 1:27
• The first part I think will be straightforward, since the steps are a lot smaller than the target point, so you can just approximate by the target point divided by the average step. I'm not so certain what is expected in the second part. – Ian Dec 20 '18 at 20:03