Total variation distance and compound poisson approximations for random sums

Abstract

We derive new upper bounds for the total variation distance between compound Poisson distributions as well as between a random sum and a compound Poisson distribution, and as a result we also obtain upper bounds for the total variation distance between compound Poisson distributions and a sum of independent random variables. These bounds are generalizations and refinements of some well-known bounds in the literature. We also derive upper bounds for the total variation distance between negative binomial distributions of order k and between the negative binomial distributions of order k and compound Poisson distributions. Upper bounds for the total variation distances between the number of success runs of length k in binary Markovian trials and its limiting distributions for several enumeration schemes, are also given.

Keywords:

• Total variation distance
• random sums
• compound poisson distribution
• negative binomial distributions of order
• k
• success runs
• binary markovian trials
• limiting distributions

Acknowledgements

The author thanks the anonymous referees for their valuable comments that helped to improve the paper.

Disclosure statement

No potential conflict of interest was reported by the author(s)