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The area of a spectrally positive stable process stopped at zero

Letemplier J., Simon T.

Probability and Mathematical Statistics

2018

Vol. 38, Fasc. 1

27--37

A multiplicative identity in law for the area of a spectrally positive Lévy α-stable process stopped at zero is established. Extending that of Lefebvre for Brownian motion, it involves an inverse beta random variable and the square of a positive stable random variable. This simple identity makes it possible to study precisely the behaviour of the density at zero, which is Fréchet-like.

Tail asymptotics of signal-to-interference ratio distribution in spatial cellular network models

Miyoshi N., Shirai T.

Probability and Mathematical Statistics

2017

Vol. 37, Fasc. 2

431--453

We consider a spatial stochastic model of wireless cellular networks, where the base stations (BSs) are deployed according to a simple and stationary point process on Rd, d ≥ 2. In this model, we investigate tail asymptotics of the distribution of signal-to-interference ratio (SIR), which is a key quantity in wireless communications. In the case where the pathloss function representing signal attenuation is unbounded at the origin, we derive the exact tail asymptotics of the SIR distribution under an appropriate sufficient condition. While we show that widely-used models based on a Poisson point process and on a determinantal point process meet the sufficient condition, we also give a counterexample violating it. In the case of bounded path-loss functions, we derive a logarithmically asymptotic upper bound on the SIR tail distribution for the Poisson-based and α-Ginibre-based models. A logarithmically asymptotic lower bound with the same order as the upper bound is also obtained for the Poisson-based model.

Extremes of chi-square processes with trend

Liu P., Ji L.

Probability and Mathematical Statistics

2016

Vol. 36, Fasc. 1

1--20

This paper studies the supremum of chi-square processes with trend over a threshold-dependent-time horizon. Under the assumptions that the chi-square process is generated from a centered self-similar Gaussian process and the trend function is modeled by a polynomial function, we obtain the exact tail asymptotics of the supremum of the chi-square proces with trend. These results are of interest in applications in engineering, insurance, queueing and statistics, etc. Some possible extensions of our results are also discussed.