TITLE: Shannon's communication paper and finite differences
QUESTION [14 upvotes]: In Shannon's 1948 paper "A Mathematical Theory of Communication", early on he derives the equation $$N(t)=N(t-t_1)+N(t-t_2)+\ldots+N(t-t_n).$$
He then says "according to a well-known result in finite differences, $N(t)$ is then asymptotic to $X_0^t$  where $X_0$ is the largest solution to the equation $X_0^{-t_1}+\ldots X_0^{-t_n}=1$."
He does not cite a reference. Obviously if the $t_i$ are commensurable, this reduces to a standard constant coefficient recurrence relation, but Shannon does not explicitly make this assumption (in his examples all the $t_i$ are rational).
The result seems to be true in the case of general positive $t_i$ also if you assume some kind of regularity of $N(t)$, but here is my question:
Can anyone suggest a reference that treats this? (the books on finite differences I've seen seem to deal with the commensurable case).

REPLY [9 votes]: When the $t_i$ are incommensurable in the sense that they generate a dense subgroup, $N(t)=CX_0^t+o(X_0^t)$ for a given constant $C$. This is a consequence of the standard renewal theorem and needs no hypothesis on the monotonicity of the function $t\mapsto N(t)$.
To see this, let $(\xi_k)$ denote some i.i.d. random variables such that $P[\xi_k=t_i]=X_0^{-t_i}$ for every $k$ and $i$. Introduce $M(t)=N(t)/X_0^t$. Then 
$$
M(t)=E[M(t-\xi_1)].
$$ 
Fix $t_0$ such that $t_0\ge t_i$ for every $i$. For every positive $k$, let $S_k=\xi_1+\cdots+\xi_k$. For every $t > t_0$, consider the first time $T(t)$ such that $S_{T(t)}\ge t-t_0$. Since $T(t)$ is a stopping time, the martingale property yields
$$
M(t)=E[M(t-S_{T(t)})].
$$ 
Reversing the time axis, $t_0-(t-S_{T(t)})$ becomes the overshoot over $t-t_0$ for the renewal process based on the sequence $(\xi_k)$ and starting from $0$. In the non lattice case, the renewal theorem asserts that $t_0-(t-S_{T(t)})$ converges in distribution to a random variable $\xi_0$ when $t\to+\infty$. Being lattice means that there exists a nonzero $a$ such that the random variables $a\xi_k$ are almost surely integer valued, hence the non lattice case corresponds to non commensurate parameters $t_i$. 
Thus, when the $t_i$ are non commensurate, $N(t)/X_0^t=M(t)\to C$ wih
$$
C=E[M(t_0-\xi_0)]=X_0^{-t_0}E[N(t_0-\xi_0)X_0^{\xi_0}].
$$ 
Finally, $\xi_0$ is distributed like $u\xi'$ where $u$ and $\xi'$ are independent, $u$ is uniform on $[0,1]$ and the distribution of $\xi'$ is the size-biased distribution of the distribution of $\xi_k$, given by $P[\xi'=t_i]=t_iX_0^{-t_i}/E[\xi_k]$. Hence one can write $C$ as an explicit integral of the function $N$ over $[0,t_0]$. 
A reference is Applied Probability and Queues by Søren Asmussen.