Past exam of the mathematics course of the University of Cambridge / 2005 / ia / Paper 2 / 9F (Probability)

The expectation of a complex random quantity is defined through its real and imaginary parts. Both $\cos X$ and $\sin X$ are bounded, so all expectations below exist. Expand the addition formulae and apply the expectation of a product of independent random variables identity:Combining real and imaginary parts provesFor the simple symmetric random walk, write $W_n={\sum }_{j=1}^n{\xi }_j$, with independent increments satisfying $P({\xi }_j=1)=P({\xi }_j=-1)=1/2$. The usual casino model uses these independent, or conditionally fair, increments. Their characteristic function is $(e^{iu}+e^{-iu})/2=\cos u$, so repeated factorization givesThis is real for real $u$. Equivalently, the distribution of $W_n$ is symmetric, making $E\sin (u{W}_n)=0$.

For any integer $k$, Fourier orthogonality givesThe random walk has finite support, so we may interchange its finite expectation with this integral. ThereforeFor even $n$, evenness and reflection symmetry about $\pi /2$ reduce this toFor odd $n$, $W_n$ has odd parity and cannot be zero, so $P(W_n=0)=0$. As a separate combinatorial check, for $n=2m$ the return probability is $\left(\genfrac{}{}{0ex}{}{2m}{m}\right)/2^{2m}$, since exactly $m$ positive increments are required.