On the optimality of multivariate Poisson approximation

Let {[xi]i}1 [less-than-or-equals, slant] i [less-than-or-equals, slant] n be a sequence of independent Bernoulli point processes defined on a complete separable metric space (E, [delta]) (i.e., [xi]i is a random measure with P([xi]i(B) = 1) = 1 - P([xi]i(B) = 0) = E[xi]i(B) for each relatively compact Borel set B [subset, double equals] E), [xi] = [summation operator]ni = 1 [xi]i a Poisson-Bernoulli point process and the success probabilities pi = E[xi]i(E), 1 [less-than-or-equals, slant] i [less-than-or-equals, slant] n, be ordered according to their magnitude. The problem under consideration is the optimal fit of a Poisson point process [eta] = [summation operator]ni = 1 [eta]i to [xi] with respect to the total variation distance where the Poisson point processes [eta]i are independent with intensity measures proportional to those of the Bernoulli point processes. In case that the supports of the Bernoulli point processes are mutually disjoint it is proved under rather weak constraints that the optimal approximation is achieved by distributing the total intensity or mass E[eta](E) = -[summation operator]ni = 1 log(1 - pi) emphasizing those point processes with the smallest intensities. To be more precise, E[eta]i(E) = pi/(1 - pi), k [less-than-or-equals, slant] i [less-than-or-equals, slant] n, for some suitable k, and the remaining mass is distributed equally to the rest of the Poisson point processes disregarding the differences of the success probabilities. For the above case the exact minimal total variation is calculated which is an upper bound, in general. As the solution of the problem is well known in case of identically distributed Bernoulli point processes (coinciding supports) and the supports can be chosen to overlap to an arbitrary extent this evaluates the whole bandwidth of possibly best approximations providing sharp upper and lower bounds for the total variation distance. The result is applied to Poisson approximations to the multinomial distribution and the record counting process (i.e., the process counting the records of an i.i.d. sequence of random variables with continuous distribution).