A random walker starting at time $t=0$ and location $x=0$ moves to the right ($x+1$) or the left ($x-1$). The $k^{\mathrm{th}}$ moves to the right and left occure at the times $\sum_{i=1}^{k} R_i$ and $\sum_{i=1}^{k} L_i$, respectively, where $R_i$s and $L_i$s are independent samples of the probability density function $f(x)$. Show that for all $\delta>0$, there exists $m=\Theta(\sqrt{n})$ such that the probability that the location of the random walker remains $x\leq m$ from $t=0$ to the time of the $n^{\mathrm{th}}$ step to the right, is lower bounded by $1-\delta$, as $n\to\infty$.
My Effort: If $f(x)=\lambda e^{-\lambda x}$, the memorylessness of exponential random variables makes this problem equivalent to a symmetric random walk, then we can find the survival probability of a random walk and use the Brownian motion limit to prove this (see Survival Probability in here). How about the general $f(x)$?
I think we make it equivalent to another Brownian motion, I don't know how to find the parameters of that Brownian motion.
