This problem is a variation on my previously asked question.
Consider a sequence of random variables $x_1, x_2, \ldots, x_N$ defined as follows: $$x_0 = \alpha, \qquad x_n \sim \mathcal{LogN}(\log x_{n-1}, \sigma^2)$$ Here $\alpha$ is a positive constant.
Question: how to estimate the pdf/cdf of $x_N$ for an arbitrary $N \in \mathbb{N}$ large enough?
The question is motivated by the following problem: assume that we tune a ukulele (or any other string instrument) "by ear".
- we tune the $1$st string by tuner (so it's tuned precisely).
- to tune the $2$nd string, we play the second string with the finger on $5$th fret and try to match the frequency of the first string. Since the ear is not perfect, assume that instead of hearing the sound at frequency $h$ Hz we hear a distorted sound at frequency $u \sim \mathcal{LogN}( \log h, \sigma^2)$ for some given $\sigma$ small enough (we can replace this r.v. with a one with any reasonable, say Gamma, distribution).
- to tune the $3$rd string we match a note on the $4$th fret of the 3rd string with the open 2nd string
- finally tune the $4$th string using $3$rd.
How much the $4$th string is distorted comparing with the true frequency? The question is then formalized as mentioned above: start with a frequency of the $1$st string $x_0$, tune the $2$nd string at the frequency $x_1 \sim \mathcal{LogN}(\log x_0, \sigma^2)$, repeat. The only method that come to mind to estimate the probability $\mathbb{P}(X_N > \alpha)$ is the Monte-Carlo simulation (just sample $x_1, x_2, \ldots, x_N$ one by one till convergence). Are the any hints/ways to calculate given probability exactly?
Let $(Z_n)_{n \in \mathbb{N}}$ be IID standard normal random variables. Your variables can be written as $$X_n=e^{\ln X_{n-1}+\sigma Z_n}=X_{n-1}e^{\sigma Z_n}$$ But then we have a recursive relationship: $$X_n=\alpha\prod_{1\leq k \leq n}e^{\sigma Z_k}=\alpha e^{\sigma \sum_{k\leq n}Z_n}=e^{\ln(\alpha)+\sigma \sum_{k\leq n}Z_k}$$ And we conclude: $$\sigma \sum_{k\leq n}Z_k\sim \mathcal{N}(0,n\sigma^2)\implies X_n \sim \textrm{Lognorm}(\ln(\alpha),n\sigma^2)$$