Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 2 / 6E / d / Solution

For $f_n=n^2$, the birth increment is $2n+1$, whileWith $D=2d$, the generator identity givesAt stationarity $\mu =b/(2d)$, and solving this equation for the variance yieldsFor $b,d>0$ the chain is irreducible on the nonnegative integers, so its stationary distribution has $0<p_1<1$. Therefore

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