Log-convexity and the cycle index polynomials with relation to compound Poisson distributions

2016/09/22 by Matsui, Muneya
#05A20 #60C05 #60E07 #60E10 #Combinatorics (math.CO) #FOS: Mathematics #Probability (math.PR) #Statistics Theory (math.ST)

paper · doi:10.48550/arxiv.1609.06875

Abstract

We extend the exponential formula by Bender and Canfield (1996), which relates log-concavity and the cycle index polynomials. The extension clarifies the log-convexity relation. The proof is by noticing the property of a compound Poisson distribution together with its moment generating function. We also give a combinatorial proof of extended "log-convex part" referring Bender and Canfield's approach, where the formula by Bruijn and Erdös (1953) is additionally exploited. The combinatorial approach yields richer structural results more than log-convexity. Furthermore, we consider normal and binomial convolutions of sequences which satisfy the exponential formula. The operations generate interesting examples which are not included in well known laws about log-concavity/convexity.

Related

• Concentration inequalities for log-concave sequences
• On the entropy and log-concavity of compound Poisson measures
• Poisson processes and a log-concave Bernstein theorem
• The index of coincidence for the binomial distribution is log-convex
• Log-concavity and log-convexity of series containing multiple Pochhammer symbols