A 1-D random walker strarting at time $t=0$ and location $x=0$, moves to the right ($x+1$) or the left ($x-1$) according to independent random variables $R_1,R_2,\ldots$ and $L_1,L_2,\ldots$, such that the $k^{\mathrm{th}}$ jump to the right occurs at the time $\sum_{i=1}^{k} R_i$ and the $k^{\mathrm{th}}$ jump to the left occurs at the time $\sum_{i=1}^{k} L_i$. Assume $R_i$s and $L_i$s are samples of the same probability density function $f(x)$. Show that the probability that the location of the random walker remains $x\leq M$ after the first $N$ steps to the right, tends to $1-\delta$, for all $\delta>0$, as $N, M \to\infty$, as long as $M=\mathcal{O}(\sqrt{N})$.
My Solution: 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.