A Compound Poisson Convergence Theorem

In 1971, Simons and Johnson showed that the classical theorem of binomial to Poisson convergence is actually stronger than in the usual sense. Their result was proved valid also for the distributions of sums of independent, but not necessarily identically distributed, Bernoulli random variables by Chen in 1974. Here we show that their result is indeed true for a much larger class of random variables. The limiting distribution is generalized to a compound Poisson distribution.