Past exam of the probability, statistics and options I course of the University of Oxford / 2008 / M (Probability)

The inverse change of variables is $x=2u+v$, $y=v$, with absolute Jacobian determinant two. Positivity of $x,y$ givesThe change of variables for probability densities givesFor fixed $u$, integrate over $v>\max (0,-2u)$:Thus $\boxed{f_U(u)=\frac{1}{2}{e}^{-|u|}}$, the standard Laplace distribution.

The Markov property says that, conditional on the present state, the next state is independent of the earlier history:Two states communicate if each is reachable from the other with positive probability in finitely many steps. The equivalence classes are communicating classes; a class is closed if no positive-probability transition leaves it.

The displayed probabilities require $0\le p\le 1/2$; negative values allowed by the printed one-sided inequality are not transition probabilities. For $p=0$, every state is an absorbing singleton, so all singleton classes are closed. For $0<p\le 1/2$, neighboring states communicate in both directions, so there is one closed irreducible class $S$ and no open classes.

For $m=3$, the detailed balance relations give $2{\pi }_0={\pi }_1={\pi }_2=2{\pi }_3$. Normalization yieldsFor $0<p<1/2$, positive holding probabilities make the finite irreducible chain aperiodic, so its transition probabilities converge to ${\pi }_j$ from every initial state. At $p=1/2$, every step changes parity and the period is two; convergence from a fixed initial state fails. A stationary initial mixture still has distribution $\pi$ at every time, but this does not give convergence of the transition probabilities asked for.

A Poisson process of rate $\lambda >0$ starts from zero, has independent stationary increments, has unit jumps, and satisfies $N_t-N_s\sim \mathrm{Poisson}(\lambda (t-s))$ for $t>s$. Its inter-arrival times are independent with the exponential distribution of rate $\lambda$: any finite joint density is ${\prod }_i\lambda e^{-\lambda x_i}$ on positive coordinates.

The second arrival is a sum of two such variables. Convolution givesThis is a gamma distribution of shape two and rate $\lambda$. If $S$ is the first arrival of the independent rate-$\mu$ process, exponential independence gives