I'm wondering whether the following type of problem is a standard one that has been studied by probabilists. The particular case needed (as a lemma that would help with a Polymath project) isn't quite what I'm asking here, but I'll ask a simpler case, since I don't think the case we actually need is significantly different.

Take a standard $n$-step random walk in $\mathbb Z$ (so it starts at the origin and at each step goes right with probability 1/2 and left with probablity 1/2). Let $X_m$ be where the walk has reached after $m$ steps. I am interested in the quantity $\sum_m X_m^2$. The distribution of $X_m$ is roughly normal with standard deviation of order $\sqrt m$, from which it follows easily that the expected value of this quantity has order $n^2$. I would like to prove that the probability that $\sum_mX_m^2\leq c^2n^2$ is, for suitably small $c$, extremely small. For the application I have in mind, it would be enough if "extremely small" meant "with probability at most $n^{-10}$" or something like that, and I'm prepared for $c$ to depend on $n$, as long as it doesn't tend to zero too quickly.

In fact, I think it is necessary for $c$ to tend to zero, because I think that for any fixed $c$ the probability that the $n$-step random walk is always between $-c\sqrt n$ and $c\sqrt n$ is positive. (I wouldn't mind a reference for that actually.) But a heuristic argument that I won't give here suggests that that positive probability depends exponentially on $c$, so I would expect to be OK once $c\ll 1/\log n$.

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