Total variation distance and compound poisson approximations for random sums (Q6571711)

This paper deals with upper bounds for the total variation distance between the probability functions of two nonnegative integer-valued real scalar random variables. For the nonnegative integer-valued real scalar random variables \(x\) and \(y\), the total variation distance is defined as\N\[\Nd(x,y)=\frac{1}{2}\sum_{n=0}^{\infty}|Pr(x=n)-Pr(y=n)|,\N\]\Nwhere, for example, \(Pr(x=n)\) means the probability that \(x\) takes the value \(n\), and \(|(\cdot)-(\cdot\cdot)|\) means the absolute difference between \((\cdot)\) and \((\cdot\cdot)\). Upper bounds for the total variation distance \(d(x,y)\) are computed for the following cases: (i): Two compound Poisson distributions; (ii): One compound Poisson and random sum of iid (independently and identically distributed) variables; (iii): Two negative binomial distributions; (iv): A negative binomial distribution and a compound Poisson distribution; (v): Two geometric distributions; (vi): Number of success runs of length \(k\) in binary Markovian trials and its limiting distribution. In all these cases, sharper upper bounds for \(d(x,y)\) are obtained. It is shown that the upper bounds obtained in this present paper are sharper than the corresponding bounds obtained by various earlier authors. This aspect is illustrated through several numerical examples.\N\NThe present paper gives a good overview of the results obtained by various authors and hence it will be highly useful for people working in this area.