Past exam of the options II course of the University of Oxford / 2007 / M (Probability)

The central limit theorem states that for independent identically distributed variables of finite mean $\mu$ and positive finite variance ${\sigma }^2$,For a continuous strictly increasing cumulative distribution function, its inverse is defined on $(0,1)$. Thus $\mathbb{P}(F(X)\le y)=\mathbb{P}(X\le F^{-1}(y))=y$ for $0<y<1$, proving the uniform law.

If $T=-\log (1-Y)$, then for $t\ge 0$, $\mathbb{P}(T\le t)=\mathbb{P}(Y\le 1-e^{-t})=1-e^{-t}$. Hence $f_T(t)=e^{-t}{1}_{(0,\infty )}(t)$, an exponential density of rate one. Its first two moments are one and two, by integration by parts, so $\mathbb{E}T=1$ and $\mathrm{Var}T=1$.

Although the original variables may have different distributions, their separate transforms $Y_k=F_k(X_k)$ are independent and all uniform. Consequently $T_k=-\log (1-Y_k)$ are independent identically distributed exponentials of rate one, and the statistic is exactlyThe stated central limit theorem therefore yields $\boxed{Z_n\stackrel{d}{\to }N(0,1)}$.

A Poisson process of rate $\lambda$ is a counting process starting at zero with independent stationary increments, with $N(t)-N(s)$ Poisson of mean $\lambda (t-s)$ for $t\ge s$, and right-continuous paths with unit jumps. Equivalently the small-interval probabilities are $\lambda h+o(h)$ for one event and $o(h)$ for two or more.

Independence of the two processes makes their joint increments over disjoint intervals independent. On an interval of length $h$, the sum of their independent Poisson increments has probability generating functionso it is Poisson with mean $(\alpha +\beta )h$. Simultaneous arrivals have probability zero, so the sum retains unit jumps. It is therefore a Poisson process of rate $\alpha +\beta$.

The first arrival times are independent exponential variables of rates $\alpha$ and $\beta$. For positive rates,

Stationarity at zero gives ${\pi }_0=q{\pi }_0+q{\pi }_1$, hence ${\pi }_1=(p/q){\pi }_0$. At one, ${\pi }_1=p{\pi }_0+q{\pi }_2$; substituting $p{\pi }_0=q{\pi }_1$ gives $q{\pi }_2=p{\pi }_1$. More generally the interior equations are ${\pi }_i=p{\pi }_{i-1}+q{\pi }_{i+1}$. Induction givesSince $p/q<1$, normalization of this geometric series givesThese probabilities satisfy detailed balance ${\pi }_{i}p={\pi }_{i+1}q$, including the boundary balance with the holding probability. Thus they do define a stationary distribution. Irreducibility then gives its uniqueness.