Let's consider a one-dimensional random walk, $X_t = \sum\limits_{i=i}^{t}\varepsilon_i$ with $\varepsilon_i \sim D(\mu, \sigma^2)$, i.e. all $\varepsilon_i$ should be drawn from a particular symmetrical distribution $D$ (e.g. a normal or an exponential normal distribution) with a mean of $\mu$ and a variance of $\sigma^2$.
Now I'd like to calculate the probability that the random walk process ends at a value of $x$ or less after the $t$th step, i.e. I'm looking for the formula to calculate the probabilities $P(X_t \leq x)$.
I'd be very grateful if you'd point out this formula and could give me a clue if such a formula can be generalized for symmetric distributions (as I'd expect it). Thanks a lot in advance for your help!