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An infinite urn scheme is defined by a probability mass function$(p_j)_{j\geq1}$ over positive integers. A random allocation consists of asample of $N$ independent drawings according to this probability distributionwhere $N$ may be deterministic or Poisson-distributed. This paper is concernedwith occupancy counts, that is with the number of symbols with $r$ or at least$r$ occurrences in the sample, and with the missing mass that is the totalprobability of all symbols that do not occur in the sample. Without any furtherassumption on the sampling distribution, these random quantities are shown tosatisfy Bernstein-type concentration inequalities. The variance factors inthese concentration inequalities are shown to be tight if the samplingdistribution satisfies a regular variation property. This regular variationproperty reads as follows. Let the number of symbols with probability largerthan $x$ be $\vec{\nu}(x)=|\{j:p_j\geq x\}|$. In a regularly varying urnscheme, $\vec{\nu}$ satisfies $\lim_{\tau \rightarrow0}\vec{\nu}(\taux)/\vec{\nu}(\tau)=x^{-\alpha}$ for $\alpha\in[0,1]$ and the variance of thenumber of distinct symbols in a sample tends to infinity as the sample sizetends to infinity. Among other applications, these concentration inequalitiesallow us to derive tight confidence intervals for the Good--Turing estimator ofthe missing mass.

Concentration inequalities in the infinite urn scheme for occupancy counts and the missing mass, with applications