We derive the exact asymptotics of P (supu≤tX(u) > x) if x and t tend to infinity with x/t constant, for a Lévy process X that admits exponential moments. The proof is based on a renewal argument and a two-dimensional renewal theorem of Höglund (1990).

For a Markov renewal process where the time parameter is discrete, we present a novel method for calculating the asymptotic variance. Our approach is based on the key renewal theorem and is applicable even when the state space of the Markov chain is countably infinite.

The tail decay of M/G/1-type Markov renewal processes is studied. The Markov renewal process is transformed into a Markov chain so that the problem of tail decay is reformulated in terms of the decay of the coefficients of a suitable power series. The latter problem is reduced to analyze the analyticity domain of the power series.

Model Renewal processes have been a frequent object of analysis in early studies of stochastic processes, see Cox (1962), for instance. Only recently the idea of parallel renewal processes receives more attention, see Borgelt and Picado-Muino (2012), Gaigalas (2003), Kai et al. (2014), CRC (1994), Kallen et al. (2010), Truccolo (2005), Modir et al. (2010). However, little emphasis has been give...

For some ambiguous scenes perceptual conflict arises between integration and segregation. Initially, all stimulus features seem integrated. Then abruptly, perhaps after a few seconds, a segregated percept emerges. For example, segregation of acoustic features into streams may require several seconds. In behavioral experiments, when a subject's reports of stream segregation are averaged over rep...

The purpose of this thesis is to introduce a cyclic renewal process as an extension of an alternating renewal process, where each of underlying i.i.d. nonnegative random increments is composed of multiple stages, i.e. Yn = ∑J j=1 Xj , n 1, where Yn denotes the lifetime of the n-th cycle and Yn’s are i.i.d. with respect to n. Such a process may be appropriate for analyzing optimal preventive mai...

This paper investigates the ruin probabilities of a renewal risk model with stochastic investment returns and dependent claim sizes. The investment is described as a portfolio of one risk-free asset and one risky asset whose price process is an exponential Lévy process. The claim sizes are assumed to follow a one-sided linear process with independent and identically distributed step sizes. When...

Spontaneous activities of pigeon auditory nerve fibres were tested for a renewal process. The experiments falsified mutually statistically independent lengths of time intervals between successive actions potentials. However, the generation of action potentials on the eighth nerve by a complex system as the inner ear might not be described by a simple renewal process. The superposition of many d...

Renewal-like results and stability theorems relating to the largetime behaviour of a random walk Sn reflected in its maximum, Rn = max0≤j≤n Sj − Sn, are proved. Mainly, we consider the behaviour of the exit time, τ(r), where τ(r) = min{n ≥ 1 : Rn > r}, r > 0, and the exit position, Rτ(r), as r grows large, with particular reference to the cases when Sn has finite variance and/or finite mean. Th...