In Shannon's 1948 paper "[A Mathematical Theory of Communication](http://cm.bell-labs.com/cm/ms/what/shannonday/shannon1948.pdf)", 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). 



Shannon's communication paper and finite differences
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](http://books.google.com/books?hl=fr&lr=&id=BeYaTxesKy0C&oi=fnd&pg=PR5&dq=Applied+Probability+and+Queues&ots=QxdOx7Pq4a&sig=xYy2TVuwbr-VGz5wb8oxtPP6V3w#v=onepage&q&f=false) by Søren Asmussen.


