 Solution

Condition on the number $M$ of stars. The independence assumptions give $\mathbb{E}[e^{it{F}_n}\mid M]=q_n(t{)}^M$, with the inverse-square one-star characteristic function $q_n$ from part (a). Averaging over the Poisson distribution therefore gives the compound Poisson distribution formula directly:Since $I_n(t)\to \sqrt{\pi m/2}|t{|}^{1/2}$, the Lévy continuity theorem gives exactly the same weak convergence of random variables as in part (a):The random number of stars introduces no additional limiting term: its effect is already contained in the exact exponential formula. This is the symmetric inverse-power Poisson field on a line at spatial intensity $1/2$ and exponent $p=2$.