The coupon collector's problem revisited: asymptotics of the variance (Q2879912)

Let there be \(N\) types of coupons and the probability to sample the \(j\)-th type coupon in one trial is \(p_j=f(j)/A_N\), where \(f\) is a fixed function, \(A_N=\sum_{j=1}^N f(j)\). The authors investigate asymptotic behavior of the random number \(T_N\) of trials it takes until all \(N\) types are collected at least once (as \(N\to\infty\)); e.g., if \(f(x)\to\infty\), \(f'(x)/f(x)\to 0\), \(f''(x)f(x)/(f'(x))^2=O(1)\), \(f'''(x)(f(x))^2/(f'(x))^3=O(1)\) as \(x\to\infty\), then NEWLINE\[NEWLINE \text{Var}(T_N)\sim{\pi^2\over 6}A_N^2(f(N))^2. NEWLINE\]NEWLINE The asymptotic distribution of suitably normalized \(T_N\) is also obtained. The proofs are based on the Euler-Maclaurin sum formula and the Laplace method for sums.