1. Let X1, X2, . . . be independent, L(a)-distributed random variables, and let N ∈ Po(m) be independent of X1, X2, . . . . Determine the limit dis- tribution of SN = X1 + X2 + · · · + XN (where S0 = 0) as m → ∞ and a → 0 in such a way that m · a2 → 1.
2. Let N , X1, X2, . . . be independent random variables such that N ∈ Po(λ) and Xk ∈ Po(µ), k = 1, 2, . . . . Determine the limit distribution of Y = X1 + X2 + · · · + XN as λ → ∞ and µ → 0 such that λ · µ → γ > 0. (The sum is zero for N = 0.)
3. Let X1, X2, . . . be independent Po(m)-distributed random variables, sup- pose that N ∈ Ge(p) is independent of X1, X2, . . . , and set SN = X1 + X2 + · · · + XN (and S0 = 0 for N = 0). Let m → 0 and p → 0 in such a way that p/m → α, where α is a given positive number. Show that SN converges in distribution, and determine the limit distribution.