Premium
Harmonic Renewal Sequences and the First Positive Sum
Author(s) -
Grübel Rudolf
Publication year - 1988
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/s2-38.1.179
Subject(s) - mathematics , sequence (biology) , combinatorics , lattice (music) , distribution (mathematics) , harmonic , random walk , renewal theory , mathematical analysis , physics , statistics , quantum mechanics , chemistry , biochemistry , acoustics
With a probability distribution p on the integers, regarded as a sequence, we associate its harmonic renewal sequence v ( p ) = ∑ 1 ∞n − 1p ∗ n. Let h + = v (δ 1 ), h = v (½(δ -1 + δ 1 )), where δ x is the unit mass at x . We characterize summability of v ( p )− h + , v ( p )− h in terms of p . This enables us to compare the asymptotic behaviour of p and v ( p ) and to obtain expansions. Harmonic renewal sequences are important for lattice‐type random walks. Our results lead to tail expansions for ladder height distributions.