The limits of sequences of iterated overshoot distribution functions

Beek, P. van; Braat, J.J.M.

doi:10.1016/0304-4149(73)90014-8

Cited by 12 publications

(8 citation statements)

References 2 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…then, necessarily lim n→∞ c n+1 /c n = l ∈ [1, ∞) and lim n→∞ E[Z k n ] = E[Z k ∞ ] ∈ (0, ∞) for every k ∈ N. Many authors were motivated by this problem and studied the set of possible distribution for Z ∞ , see the works of Arratia, Goldstein and Kochman [2], van Beek and Braat [3], Garcia [6], Shantaram and Harkness [16], Pakes [11,12], Vardi, Shepp and Logan [18] for instance. Their approach was mainly based on the fact that the distribution of Z ∞ necessarily satisfies…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Refined properties for the Mellin transform, with applications to convergence of families obtained by biasing or by the stationary excess operator

Jedidi,

Bouzeffour,

Harthi

2016

Preprint

View full text Add to dashboard Cite

We first provide some properties of the Mellin transform of nonnegative random variables, such that monotonicity, injectivity and effect of size biasing. Convergence of Mellin transforms is also entirely formalized through convergence in distribution and uniform integrability. As an application, we study a problem raised by Harkness and Shantaram (1969) who obtained, under sufficient conditions, a limit theorem for sequences of nonnegative random variables build with the iterated stationary excess operator. We reformulate this problem through the concept of multiply monotone functions and through the convergence of the families build by the continuous time version of the iterated stationary excess operator and also by size biasing. The latter allows us to show that in our context, continuous time convergence is equivalent to discrete time convergence, that the conditions of Harkness and Shantaram are actually necessary and that the only possible limits in distribution are mixture of exponential with log-normal distributions.

show abstract

Section: Introduction and Main Resultsmentioning

confidence: 99%

Refined properties for the Mellin transform, with applications to convergence of families obtained by biasing or by the stationary excess operator

Jedidi,

Bouzeffour,

Harthi

2016

Preprint

View full text Add to dashboard Cite

show abstract

“…Remark 1 The distribution function in Eq. (19) can be interpreted as the stationary distribution of the mth iterated overshoot of a renewal process with inter-renewal distribution F(·): see [23]. That is, the residual of the residual of the residual.…”

Section: Markov Chains Of Residual Lifetimesmentioning

confidence: 99%

On the scalability and message count of Trickle-based broadcasting schemes

et al. 2015

View full text Add to dashboard Cite

As the use of wireless sensor networks increases, the need for efficient and reliable broadcasting algorithms grows. Ideally, a broadcasting algorithm should have the ability to quickly disseminate data, while keeping the number of transmissions low.In this paper, we analyze the popular Trickle algorithm, which has been proposed as a suitable communication protocol for code maintenance and propagation in wireless sensor networks. We show that the broadcasting process of a network using Trickle can be modeled by a Markov chain and that this chain falls under a class of Markov chains, closely related to residual lifetime distributions. It is then shown that this class of Markov chains admits a stationary distribution of a special form. These results are used to analyze the Trickle algorithm and its message count. Our results prove conjectures made in the literature concerning the effect of a listen-only period. Besides providing a mathematical analysis of the algorithm, we propose a generalized version of Trickle, with an additional parameter defining the length of a listen-only period.

show abstract

“…The relevance of this operator is for the most part due to the fact that F 0 is the limit distribution as time tends to infinity of both the forward and backward recurrence times in a renewal process with inter-renewal distribution function F (see, e.g., [2,4] or [6]). Investigations of the stationary excess operator and related topics can be found in [1,3,5,[7][8][9][10][11][12][13][14][15][16][17][18][19] and [20].…”

Section: The Stationary Excess Operatormentioning

confidence: 99%

“…Next we are going to investigate the behavior of F (n) λ as n → ∞. It turns out that using a simple transformation we can transfer known convergence results from [9] (see also [18]) to our situation. To this end we define for a distribution function F the min-transform of F to be the distribution function…”

Section: Theorem 5 the Laplace-stieltjes Transform Of Xmentioning

confidence: 99%