Suppose $X_1, X_2, \ldots$ is an iid sequence of integer valued inter-arrival times in days, let $S_n = \sum_{i=1}^n X_i$ be the corresponding arrival times, and let $N(n) = N(0,n]$ denote the number of arrivals in an n day period. The distribution of the $X_i$'s does not satisfy the memoryless property and is in fact bimodal.
I'm interested in predicting the expected number of arrivals in the following year. The prediction occurs say $r$ days before the start of the following year, with time $0$ identified at $x$ days ago. The central limit theorem for renewal processes implies that $$N(x+r,x+r+365] = N(0,x+r+365] - N(0,x+r] \sim \text{N}\left( \frac{365}{\mu}, 2 \cdot 365\frac{\sigma^2}{\mu^3} \right) $$ approximately holds for $x+r$ large enough, where $\mu$ and $\sigma^2$ are the mean and variance of $X_i$. Using this (and assuming that replacing the mean and variance with their estimates is valid) I can get a crude prediction interval for $N(x+r,x+r+365]$, the number of arrivals in the following year.
However, I'd like to make use of the knowledge about the previous arrival at time $S_{N(x+r)}$ to improve the prediction. I can't seem to find any theoretical results about conditional distributions of the form $$ P[N(s+t) = k | S_{N(s)} = u] $$ and I rather dread trying to work out approximations from my sample data. I feel as if this must be a problem which others have encountered before and likely solved. Is this the case?