It is well known that under certain conditions, binomial distributions converge to Poisson distributions. Simons and Johnson (1971) showed that the convergence is actually much stronger than in the usual sense. In this note the author shows that the result of Simons and Johnson is also true for Poisson binomial distributions which include binomial distributions as special cases.
"On the Convergence of Poisson Binomial to Poisson Distributions." Ann. Probab. 2 (1) 178 - 180, February, 1974. https://doi.org/10.1214/aop/1176996766