(a) Give three definitions of a continuous-time Markov chain with a given $Q$-matrix on a finite state space: (i) in terms of holding times and jump probabilities, (ii) in terms of transition probabilities over small time intervals, and (iii) in terms of finite-dimensional distributions.

(b) A flea jumps clockwise on the vertices of a triangle; the holding times are independent exponential random variables of rate one. Find the eigenvalues of the corresponding $Q$-matrix and express transition probabilities $p_{x y}(t), t \geq 0, x, y=A, B, C$, in terms of these roots. Deduce the formulas for the sums

$$S_{0}(t)=\sum_{n=0}^{\infty} \frac{t^{3 n}}{(3 n) !}, \quad S_{1}(t)=\sum_{n=0}^{\infty} \frac{t^{3 n+1}}{(3 n+1) !}, \quad S_{2}(t)=\sum_{n=0}^{\infty} \frac{t^{3 n+2}}{(3 n+2) !}$$

in terms of the functions $e^{t}, e^{-t / 2}, \cos (\sqrt{3} t / 2)$ and $\sin (\sqrt{3} t / 2)$.

Find the limits

$$\lim _{t \rightarrow \infty} e^{-t} S_{j}(t), \quad j=0,1,2 .$$

What is the connection between the decompositions $e^{t}=S_{0}(t)+S_{1}(t)+S_{2}(t)$ and $e^{t}=\cosh t+\sinh t ?$