I'm trying to quantify
- for what values of $\theta$ and
- at what speed
a product $Y$ of $n$ independent uniform random variables $x_i$ between zero and $\theta$, decays to zero. That is $$Y=\prod_1^n x_i, \quad x_i\sim U[0,\theta].$$ Empirically it seems that the tipping point from convergence to zero and divergence is somewhere around $\theta=2.75$ but I am having a hard time finding out why. Obviously, the smaller $\tau$ the faster the deday. But I also wonder how fast exactly (say in expectation).
Thanks a lot in advance :)!
Presumably the $x_i$ are independent. Then $\mathbb E[Y_n] = \prod_{i=1}^n \mathbb E[x_i] = (\theta/2)^n$, so the threshold for expected value $\to 0$ is at $\theta = 2$. But if you want an "almost surely" result, use logarithms to turn the product into a sum, where you can use the Laws of Large Numbers. $$ \ln(Y_n) = \sum_{i=1}^n \ln(x_i)$$ Now $\mathbb E[\ln(x_i)] = \ln(\theta)-1$. The Strong Law of Large Numbers says that with probability $1$, $\ln(Y_n)/n \to \ln(\theta) - 1$ as $n \to \infty$. In particular, if $\ln(\theta) < 1$, i.e. $\theta < e$, this implies $\ln(Y_n) \to -\infty$ i.e. $Y_n \to 0$, while if $\theta > e$ it implies $Y_n \to \infty$.