Limit Theorems for the Complete Feedback Card Guessing Game

We study the optimal score $T_{n,m}$ in a complete feedback card guessing game. The deck has $n$ types and $m$ cards of each type. Its predictable compensator $S_{n,m}$ is the sum of the conditional success probabilities. We prove that Var$(S_{n,m})\leq C((\log\log n)^2+m/\log n)$, uniformly in $m$