Let $X \sim N(\mu,\sigma)$ be i.i.d. random variable.
$B_i$ is a random process:
$$B_i(\mu,\sigma) \equiv \frac{ ( X_1X_2X_3\dots X_i)^2 }{ 1 + X_1^2 + (X_1X_2)^2 + (X_1X_2X_3)^2+\dots + ( X_1X_2X_3\dots X_i)^2 }$$
$B_1,B_2,B_3,\dots,B_i,\dots, B_n, \dots $
I hope to understand the asymptotic behaviour of $B_n$ when $n\to \infty$.
It is clear to me in the case when $\sigma=0$
when $\sigma=0, \mu<1$, as $n\to \infty$, we have exponential decay $\quad B_n \sim \mu^{2n} \to 0 $
when $\sigma=0, \mu=1$, as $n\to \infty$, we have power deacay $\quad \quad B_n \sim \frac{1}{n} \to 0 $
when $\sigma=0, \mu>1$, as $n\to \infty$, we have a bounded value $\quad \quad B_n \sim \frac{(\mu^2-1)\mu^{2n}}{\mu^{2n}-1} > \text{const} $
It goes beyond my knowledge when $\sigma>0$.
Recall from my Finance class
For a portflio with return=$\mu$, risk=$\sigma$, the stock price:
$\log(S_T) \sim N\big( (\mu - \sigma^2/2) (T-t_0), \sigma \sqrt{T-t_0} \big) $
$\sigma$ shifts the balance point from $\mu_c=0$ to $\mu_c=\sigma^2/2$
this is something non-trivial ! and I think it can solve my problem.
My questions is:
What would be the asymptotic expression of $B_n(\mu,\sigma)$ when $\sigma\neq 0$ ?
(what is the phase diagram on the $(\mu,\sigma)$ plane? )
I'm guessing and expecting there exist some region:
$B_n \sim \frac{1}{n^z} $ arbitary power decay
$B_n \sim e^{ -\alpha n^\beta} $ stretched exponential decay(for example $\beta = 1/2$)
I just found a clue:
$$ \frac{1}{X_{i+1}^2 B_i }+ 1 = \frac{1}{B_{i+1}} $$