  In this article, we obtain, for the total variance distance, the error bounds
between Poisson and convolution of power series distributions via Stein's
method. This provides a unified approach to many known discrete distributions.
Several Poisson limit theorems follow as corollaries from our bounds. As
applications, we compare the Poisson approximation results with the negative
binomial approximation results, for the sums of Bernoulli, geometric, and
logarithmic series random variables.

Probability