Let $(X_k)_{k=1}^\infty$ be a sequence of i.i.d. random variables, with $\mathbb{E}(X_i)$ finite and negative. Define $S_n := X_1 + ... + X_n$, and $M_n := \max(S_1,S_2,...,S_n)$.
It follows from the final bullet point of Sangchul Lee's answer to this question Is the expectation of the supremum of a random walk with negative drift finite? that if $X_i$ have infinite variance, then $\mathbb{E}(M_n) \rightarrow +\infty$ as $n \rightarrow +\infty$. However I am curious how quickly this growth to infinity occurs?
I am not sure if there is a general answer in terms of the distribution of $X_i$, so I started by trying to use a simple test example. I defined the $X_i$ variables to be absolutely continuous with density function $f(x) := \frac{2}{(x+3)^3}$ for $x \in [-2,\infty)$.
This is the simpliest example I could think of which would have a finite negative mean (in this case the mean is $-1$), yet infinite variance. However, with this example, I am struggling to work out how quickly $\mathbb{E}(M_n)$ diverges.
By a very rough numerical calculation, it seems as though the density function of $M_2$ decays like $O(\frac{1}{x^{2.8}})$ (maybe the true exponent is $e$?). I'm guessing that as $n \rightarrow +\infty$ the density of $M_n$ tends to a decay rate of $O(\frac{1}{x^2})$, but again I have no idea how to attack this.
Does anyone know what kind of analytic tools I can use to tackle this problem? Cheers.
