Under minimal moment conditions complete asymptotic expansions are obtained for expectations of unbounded functions of a finite family of independent random variables in the Poissonian setting when the distributions of the random variables have large atoms at zero.
"Poisson approximation for expectations of unbounded functions of independent random variables." Ann. Probab. 30 (4) 1657 - 1680, October 2002. https://doi.org/10.1214/aop/1039548369