In Claude E. Shannon's paper "A Mathematical Theory of Communication", it says:
Definition: The capacity $C$ of a discrete channel is given by $$C = \lim_{t \to \infty}\frac{\log N(t)}{t}$$ If $N(t)$ represents the number of sequences of duration t we have $$N(t) = N(t - t_1) + N(t - t_2) + \dots + N(t - t_n)$$ The total number is equal to the sum of the numbers of sequences ending in $S_1, S_2, \dots, S_n$ and these are $N(t - t_1), N(t - t_2), \dots, N(t - t_n)$, respectively. According to a well-known result in finite differences, $N(t)$ is then asymptotic for large $t$ to $X_0^t$ where $X_0$ is the largest real solution of the characteristic equation: $$X^{-t_1} + X^{-t_2} + \dots + X^{-t_n} = 1$$ and therefore $C = \log X_0$
I do not understand how the last part ($C = \log X_0$) is derived. Could someone please explain this to me? Many thanks.
