Renewal theorems for processes with dependent interarrival times
In this paper we develop renewal theorems for point processes with interarrival times ξ(Xn+1Xn…), where (Xn)n∈ℤ is a stochastic process with finite state space Σ and ξ:ΣA→ℝ is a Hölder continuous function on a subset ΣA⊂Σℕ. The theorems developed here unify and generalise the key renewal theorem for...
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| Published in | Advances in applied probability Vol. 50; no. 4; pp. 1193 - 1216 |
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| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
Cambridge, UK
Cambridge University Press
01.12.2018
Applied Probability Trust |
| Subjects | |
| Online Access | Get full text |
| ISSN | 0001-8678 1475-6064 |
| DOI | 10.1017/apr.2018.56 |
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| Abstract | In this paper we develop renewal theorems for point processes with interarrival times ξ(Xn+1Xn…), where (Xn)n∈ℤ is a stochastic process with finite state space Σ and ξ:ΣA→ℝ is a Hölder continuous function on a subset ΣA⊂Σℕ. 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. |
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| AbstractList | In this paper we develop renewal theorems for point processes with interarrival times ξ(Xn+1Xn…), where (Xn)n∈ℤ is a stochastic process with finite state space Σ and ξ:ΣA→ℝ is a Hölder continuous function on a subset ΣA⊂Σℕ. 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. 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. In this paper we develop renewal theorems for point processes with interarrivai times ξ(Xn+1Xn· · ·), where (Xn)nϵℤ is a stochastic process with finite state space Σ and ξ: ΣA → ℝ is a Holder continuous function on a subset ΣA ⊂ Σℕ. 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. |
| Author | Kombrink, Sabrina |
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| Keywords | dependent interarrival time Secondary 28A80 28A75 Renewal theorem Primary 60K05 key renewal theorem 60K15 symbolic dynamics Ruelle‒Perron‒Frobenius theory |
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| Snippet | In this paper we develop renewal theorems for point processes with interarrival times ξ(Xn+1Xn…), where (Xn)n∈ℤ is a stochastic process with finite state space... In this paper we develop renewal theorems for point processes with interarrivai times ξ(Xn+1Xn· · ·), where (Xn)nϵℤ is a stochastic process with finite state... 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... |
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| Title | Renewal theorems for processes with dependent interarrival times |
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