I'm interested in the sequence $n_i = n_{i-1} + random(a, b)$ or, in plain English, "the next number is equal to the previous number plus a random quantity between a and b (for example $a=-1, b=1$)".
I'd like to read more about these sequences but I don't know how they're called, if they have a name at all, and how to search for them.
There's two cases to consider: is $\operatorname{random}(a,b)$ a discrete random variable equal to one of the values $\{a, a+1, \dots,b\}$ with equal probability, or a continuous random variable equal to a uniformly chosen real number in $[a,b]$?
In the first case, see a previous question: Distribution of Sum of Discrete Uniform Random Variables.
In the second case, we can normalize, defining $n_i' = \frac{n_i - ai}{b-a}$, which will now satisfy the recurrence $n_i' = n_{i-1}' + \operatorname{random}(0,1)$. This is known as the Irwin–Hall distribution or just the "uniform sum distribution".
