Celebratio Mathematica — Liggett — Complete Bibliography

Complete Bibliography

Works connected to Donald L. Iglehart

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T. M. Liggett: “Total positivity and renewal theory,” pp. 141–162 in Probability, statistics, and mathematics: Papers in honor of Samuel Karlin. Edited by T. W. Anderson, K. B. Athreya, and D. L. Iglehart. Academic Press (Boston), 1989. MR 1031283 Zbl 0682.60078 incollection

Suppose that \( f(k) \) is a probability density on the positive integers, and let \( u(k) \) be the corresponding renewal sequence. Kaluza [1928] and de Bruijn and Erdős [1953] proved several results which relate convexity properties of \( f \) to convexity properties of \( u \). We first note that these convexity properties can be formulated in terms of the total positivity of certain orders of the functions \( f(i+j+1) \) and \( u(i+j) \). This observation permits us to prove an infinite collection of implications which contain the Kaluza and de Bruijn and Erdős results as special cases. In our second result, we show how the imposition of a mild total positivity assumption on \( f(k) \) permits one to give a straightforward proof of the fact that \( u(n)-u(n+1) \) is asymptotic to a constant multiple of the tail probabilities of \( f \). Continuous time versions of these results are discussed briefly. This work was motivated by a problem in interacting particle systems.

Theodore W. Anderson	Related
Donald L. Iglehart	Related
Samuel Karlin	Related
Krishna B. Athreya	Related