On log-concave and log-convex infinitely divisible sequences and densities
As is well-known, a distribution \((p_ n)_ 0^{\infty}\) on \(\{\) 0,1,2,...\(\}\) with \(p_ 0>0\) is infinitely divisible iff the \(r_ k\) defined by \[ (1)\quad (n+1)p_{n+1}=\sum^{n}_{k=0}r_ kp_{n- k}\quad (n=0,1,...), \] are all nonnegative. The author proves the following main results. If in (1) the sequence \((r_ k)\) is nonnegative and log convex (log concave), then \((p_ n)\) is log convex (log concave) iff \(r^ 2_ 0\leq r\), \((r^ 2_ 0\geq r_ 1).\) The convexity result provides an analogue to an old result for renewal sequences by \textit{N. G. de Bruijn} and \textit{P. Erdős} [J. Res. Nat. Bur. Standards 50, 161-164 (1953; Zbl 0053.369)]; the concavity result is the discrete analogue to a theorem on log concave densities by \textit{M. Yamazato} [Ann. Probab. 10, 589-601 (1982; Zbl 0495.60021)]. The proofs are brief but not easy. The results on \((p_ n)\) are transferred to infinitely divisible densities by discretization. In the final section some (counter) examples are presented.
scientific article; zbMATH DE number 4078362