I am only familiar with the very basics of random walks, so I can not judge how trivial my question is.
Assume that we have a generalised random walk where now instead of the outcomes being $\{1, -1\}$, they are $\{x_1,\ldots,x_n \}$, where $x_i \in \mathbb{Z}$ and with respective probabilities of occurrence $\{p_1,\ldots,p_n\}$. The expected value for such a generalised random walk at each iteration is: \begin{equation} EV=\sum_{i=1}^n p_i m_i . \end{equation} After $n$ iterations the expected value will be $EV(n)= EV\cdot n$. Similarly, one can calculate the variance at each step through \begin{equation} Var=\sum_{i=1}^n p_i m_i^2 -EV^2 , \end{equation} and once again after $n$ steps we have $Var(n)=n\cdot Var$. From this it follows that the standard deviation grows as $\sim\sqrt{n}$, as expected for a random walk.
So as a first question, are the above statements correct?
Assuming this is the case, my main question would then be: what is the probability that after $n$ steps I am within one standard deviation from the expected value? How do I calculate it?
Thanks in advance.