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Zero biasing and a discrete central limit theorem. (English) Zbl 1111.60015

The authors propose a class of approximating distributions which have carrier space \(\mathbb Z\), thus avoiding truncation and integerization problems. These new distributions are uniquely determined by parameters \(\mu\) and \({\sigma}^2\), similarly to how the approximating normal distribution is determined in the classical central limit theorem. It is expected that any such approximating family of discrete distributions be related to the Poisson, a distribution characterized by the property of being equal to its own reduced Palm distribution. As this property is intrinsic in the study of certain Poisson approximations, and since the Palm distribution involves only the first moment of the distribution, it is of interest to determine whether there exists any counterpart to the Poisson also involving the second moment, which gives additional flexibility in approximation. One appropriate counterpart can be uncovered through the concept of zero biasing. Based on the continuous normal case, it is expected that the class of approximating distributions should arrive naturally as the unique candidates which equal their zero-biased distribution.

MSC:

60F05 Central limit and other weak theorems
60G50 Sums of independent random variables; random walks
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