Balls and Bins -- Simple Concentration Bounds

Concentration bounds are given for throwing balls into bins independently according to a distribution $p$. The probability of a $k$-loaded bin after $m$ balls is shown to be controlled on both sides by $\rho_{m,k} := m \|p\|_k / k$. This gives concentration inequalities for the maximum load as well as for the waiting time until a $k$-loaded bin.