Let $\{\xi_k\}$ is a sequence of iid random variables with $E(\xi_1)=0$ and $E(\xi_1)^2=\sigma^2<\infty$. Define the random walk $Y_n=\sum_{k=1}^n \xi_k$. Is it necessarily true that the conditional expectation $E(Y_n|Y_n>0)=\mathcal{O}(\sqrt{n})$ as $n\rightarrow\infty$?
From the classical CLT, we know that $\frac{Y_n}{\sqrt{n}}\Rightarrow N(0,\sigma^2)$, in which the arrow denotes weak convergence as $n\rightarrow\infty$. This seems to heavily support the suggested limiting behavior, as do the results of some computations I've done with specific choices for $\{\xi_k\}$. However, I'm stuck on how to prove the bound concretely. Is there some twist on CLT that I'm just not seeing, or is there more to the picture? Maybe I need to assume more on the random walk in question.
Any assistance provided will be much appreciated.
Let $W_n=n^{-1/2}Y_n$. Then
$$ n^{-1/2}\mathsf{E}[Y_n\mid Y_n>0]=\mathsf{E}[W_n\mid W_n>0]\le \frac{\sqrt{\mathsf{E}|W_n|^2}}{\mathsf{P}(W_n>0)}=\frac{\sigma}{\mathsf{P}(W_n>0)}, $$
and $\mathsf{P}(W_n>0)\to 1/2$ (by the CLT), i.e., for any $\epsilon>0$, $\mathsf{P}(W_n>0)\ge \frac{1}{2}-\epsilon$ for $n$ large enough.