Renewal theorems for processes with dependent interarrival times. (29th November 2018)

• Record Type: Journal Article
• Authors: Kombrink, Sabrina
• Is Part Of: Advances in applied probability. Volume 50:Number 4(2018)
• Page Start: 1193
• Page End: 1216
• Publication Date: 2018-11-29
• Subjects: Renewal theorem, -- dependent interarrival time, -- symbolic dynamics, -- key renewal theorem, -- Ruelle‒Perron‒Frobenius theory; Primary 60K05, -- 60K15, -- Secondary 28A80, -- 28A75; Probabilities -- Periodicals; Stochastic models -- Periodicals; Electronic journals; Periodicals; 519.2
• DOI: 10.1017/apr.2018.56
• Languages: English
• ISSNs: 0001-8678

Abstract

Abstract: In this paper we develop renewal theorems for point processes with interarrival times ξ( X n +1 X n …), where ( X n ) n ∈ℤ is a stochastic process with finite state space Σ and ξ:Σ A →ℝ is a Hölder continuous function on a subset Σ A ⊂Σ ℕ . The theorems developed here unify and generalise the key renewal theorem for discrete measures and Lalley's renewal theorem for counting measures in symbolic dynamics. Moreover, they capture aspects of Markov renewal theory. The new renewal theorems allow for direct applications to problems in fractal and hyperbolic geometry, for instance to the problem of Minkowski measurability of self-conformal sets.