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In this article, we obtain, for the total variance distance, the error boundsbetween Poisson and convolution of power series distributions via Stein'smethod. This provides a unified approach to many known discrete distributions.Several Poisson limit theorems follow as corollaries from our bounds. Asapplications, we compare the Poisson approximation results with the negativebinomial approximation results, for the sums of Bernoulli, geometric, andlogarithmic series random variables.

Poisson Approximation to the Convolution of Power Series Distributions