If you're interested in asymptotic behavior, it may be of interest to approximate this process as a product of uniform random distribution. 

The product of $t$ uniform random variable has probability density

$$p_t(x) = \frac{~|\log x|^{t-1}}{(t-1)!}$$

In particular

$$\int_{0}^{1/n} p_t(x) \textrm{d}x = \frac{\Gamma(t,\log n)}{(t-1)!}$$