Let's say that I am sampling $\{x_i\}$ from a distribution with CDF $F(x)$, and the samples are always non-negative. For each new sample, let's say the $n$-th sample, provided that we have sampled $n-1$ samples beforehand. I am interested in the behavior of the quantity $q_n = \left\|\frac{x_i}{s_n} - \frac{1}{n}\right\|_\infty = \max \left\|\frac{x_i}{s_n} - \frac{1}{n}\right\|$ when $n$ goes from $0$ to $\infty$. Assume that the variation of the random variable $X$ is positive finite.
When $n=1$, we have $q_n = 0$. When $n$ grows, $q_n$ will almost surely be at somewhere positive. When $n \to \infty$, according to this answer, $q_n = \frac{M}{n\mu} - \frac{1}{n}$, where $M$ and $\mu$ are the essential supremum and the expectation of random variable $X \sim F(x)$, respectively. So $q_n$ first increases and then decreases. Is it possible to conjecture where the peak(s) will show up? Any necessary assumption on $F(x)$ is welcome. Non-trivial bounds are also welcome.
Change the infinity norm to $L_2$ norm. When $n = 1$, $q_n = 0$. When $n \to \infty$, $q_n \to \frac{1}{s_n}\sqrt{var(x_i)}$, which will also vanish to $0$. Is it possible to do the conjecture?
Thanks a lot.