Past exam of the probability, statistics and options I course of the University of Oxford / 2009 / M / 3 (M3)

The iid central limit theorem states that if $Y_1,Y_2,\dots$ are independent and identically distributed with finite mean $\mu$ and variance $0<{\sigma }^2<\infty$, thenEquivalently, its cumulative distribution functions converge at every real argument to the standard normal cumulative distribution function.

The joint density of $X,Y$ is $(2\pi {)}^{-1}{e}^{-(x^2+y^2)/2}$. The polar-coordinate Jacobian is $r$, soIntegration over the angular coordinate gives $f_R(r)=r{e}^{-r^2/2}$ for $r\ge 0$ and zero otherwise. Thus $R$ has the Rayleigh distribution, while $\Theta$ is uniform and independent of $R$.

For $w\ge 0$, $\mathbb{P}(W\le w)=\mathbb{P}(R\le \sqrt{w})=1-e^{-w/2}$. Differentiation givesThis is an exponential distribution of rate $1/2$, with mean two and variance four.

The independent variables $W_i$ have mean two and variance four by part (b). Their standardization isThe central limit theorem therefore proves the claimed convergence to the standard normal law, whose continuous cumulative distribution function is $(2\pi {)}^{-1/2}{\int }_{-\infty }^x{e}^{-u^2/2}du$ at every real $x$.