Details

Autor(en) / Beteiligte

• Uchiyama, Kôhei

Titel

A renewal theorem for relatively stable variables

Ist Teil von

• The Bulletin of the London Mathematical Society, 2020-12, Vol.52 (6), p.1174-1190

Erscheinungsjahr

2020

Link zum Volltext

Quelle

Alma/SFX Local Collection

Beschreibungen/Notizen

• Let F{dx} be a relatively stable (r.s.) probability distribution on the whole real line and Sn the random walk started at the origin with step distribution F. We obtain an exact asymptotic form of the Green measure U{x+dy}=∑n=0∞P[Sn−x∈dy] as x→∞ when Sn is transient and Sn→∞ in probability. If F is concentrated on [0,∞), it is r.s. if and only if ℓ(x):=∫0xF{(t,∞)}dt is slowly varying (s.v.) at infinity; our result entails that if F is non‐arithmetic and r.s., then limx→∞ℓ(x)U{[x,x+h)}=h for each h>0. This surpasses the known result that assumes the stronger condition that xF{(x,∞)} is slowly varying. An obvious analogue also holds for arithmetic variables.