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This paper gives answers to some questions posed in Hanson and Wright, Z. Wahrscheinlichkeitstheor. Verw. Geb. , 19 (1971), on rates of convergence in probability to zero for weighted sums of independent random variables.

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The problem of random allocation is that of placing n balls independently with equal probability to N boxes. For several domains of increasing numbers of balls and boxes, the final number of empty boxes is known to be asymptotically either normally or Poissonian distributed. In this paper we first derive a certain two-index transfer theorem for mixtures of the domains by considering random numbers of balls and boxes. As a consequence of a well known invariance principle this enables us to prove a corresponding general almost sure limit theorem. Both theorems inherit a mixture of normal and Poisson distributions in the limit. Applications of the general almost sure limit theorem for logarithmic weights complement and extend results of Fazekas and Chuprunov [10] and show that asymptotic normality dominates.

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