Extreme value distributions for random coupon collector and birthday problems
For a random experiment with \(n\) outcomes having probabilities \(p_1,\dots,p_n\) independent trials are performed. \(N_n\) is the number of trials needed until each outcome has occurred at least \(c\) times, \(N^*_n\) is the number of trials when some unspecified outcome has occurred \(c\) times. The author considers the case when \(p_i\) are random and given by \(p_i=X_i/\sum_{j=1}^n X_j\), where \(X_j\) are i.i.d. Using the embedding in Poisson point process the author derives the asymptotic distributions of \(N_n\) and \(N^*_n\) for different types of \(X_j\) distributions. E.g. if \(\Pr(X_j\leq x)=x^\alpha L(x)\) (\(x\to 0\)), where \(L\) is slowly varying at 0 and \(n a_n^{-\alpha}L(1/a_n)\Gamma(\alpha+c)/(c+1)!\to 1\), then \[ \Pr(N_n/(na_n\mu)\leq y)\to \exp(-y^{-\alpha}),\quad y>0, \] where \(\mu=E(X)\).
scientific article; zbMATH DE number 1833862