Non-stationary self-similar Gaussian processes as scaling limits of power law shot noise processes and generalizations of fractional Brownian motion

We study shot noise processes with Poisson arrivals and non-stationary noises. The noises are conditionally independent given the arrival times, but the distribution of each noise does depend on its arrival time. We establish scaling limits for such shot noise processes in two situations: 1) the conditional variance functions of the noises have a power law and 2) the conditional noise distributions are piecewise. In both cases, the limit processes are self-similar Gaussian with nonstationary increments. Motivated by these processes, we introduce new classes of self-similar Gaussian processes with non-stationary increments, via the time-domain integral representation, which are natural generalizations of fractional Brownian motions.Published versio

Directionally Convex Ordering of Random Measures, Shot Noise Fields and Some Applications to Wireless Communications

Directionally convex ($dcx$) ordering is a tool for comparison of dependence structure of random vectors that also takes into account the variability of the marginal distributions. When extended to random fields it concerns comparison of all finite dimensional distributions. Viewing locally finite measures as non-negative fields of measure-values indexed by the bounded Borel subsets of the space, in this paper we formulate and study the $dcx$ ordering of random measures on locally compact spaces. We show that the $dcx$ order is preserved under some of the natural operations considered on random measures and point processes, such as deterministic displacement of points, independent superposition and thinning as well as independent, identically distributed marking. Further operations such as position dependent marking and displacement of points though do not preserve the $dcx$ order on all point processes, are shown to preserve the order on Cox point processes. We also examine the impact of $dcx$ order on the second moment properties, in particular on clustering and on Palm distributions. Comparisons of Ripley's functions, pair correlation functions as well as examples seem to indicate that point processes higher in $dcx$ order cluster more. As the main result, we show that non-negative integral shot-noise fields with respect to $dcx$ ordered random measures inherit this ordering from the measures. Numerous applications of this result are shown, in particular to comparison of various Cox processes and some performance measures of wireless networks, in both of which shot-noise fields appear as key ingredients. We also mention a few pertinent open questions.Comment: Accepted in Advances in Applied Probability. Propn. 3.2 strengthened and as a consequence Cor 6.1,6.2,6.

Networks of ⋅/G/∞\cdot/G/\infty Server Queues with Shot-Noise-Driven Arrival Intensities

We study infinite-server queues in which the arrival process is a Cox process (or doubly stochastic Poisson process), of which the arrival rate is given by shot noise. A shot-noise rate emerges as a natural model, if the arrival rate tends to display sudden increases (or: shots) at random epochs, after which the rate is inclined to revert to lower values. Exponential decay of the shot noise is assumed, so that the queueing systems are amenable for analysis. In particular, we perform transient analysis on the number of customers in the queue jointly with the value of the driving shot-noise process. Additionally, we derive heavy-traffic asymptotics for the number of customers in the system by using a linear scaling of the shot intensity. First we focus on a one dimensional setting in which there is a single infinite-server queue, which we then extend to a network setting

Large deviations for multidimensional state-dependent shot noise processes

Shot noise processes are used in applied probability to model a variety of physical systems in, for example, teletraffic theory, insurance and risk theory and in the engineering sciences. In this work we prove a large deviation principle for the sample-paths of a general class of multidimensional state-dependent Poisson shot noise processes. The result covers previously known large deviation results for one dimensional state-independent shot noise processes with light tails. We use the weak convergence approach to large deviations, which reduces the proof to establishing the appropriate convergence of certain controlled versions of the original processes together with relevant results on existence and uniqueness

Statistical properties and economic implications of Jump-Diffusion Processes with Shot-Noise effects

This paper analyzes the Shot-Noise Jump-Diffusion model of Altmann, Schmidt and Stute (2008), which introduces a new situation where the effects of the arrival of rare, shocking information to the financial markets may fade away in the long run. We analyze several economic implications of the model, providing an analytical expression for the process distribution. We also prove that certain specifications of this model can provide negative serial persistence. Additionally, we find that the degree of serial autocorrelation is related to the arrival and magnitude of abnormal information. Finally, a GMM framework is proposed to estimate the model parameters

On the use of shot noise for photon counting

Lieu et al. (2015) have recently claimed that it is possible to substantially improve the sensitivity of radio astronomical observations. In essence, their proposal is to make use of the intensity of the photon shot noise as a measure of the photon arrival rate. Lieu et al. (2015) provide a detailed quantum-mechanical calculation of a proposed measurement scheme that uses two detectors and conclude that this scheme avoids the sensitivity degradation that is associated with photon bunching. If correct, this result could have a profound impact on radio astronomy. Here I present a detailed analysis of the sensitivity attainable using shot-noise measurement schemes that use either one or two detectors, and demonstrate that neither scheme can avoid the photon bunching penalty. I perform both semiclassical and fully quantum calculations of the sensitivity, obtaining consistent results, and provide a formal proof of the equivalence of these two approaches. These direct calculations are furthermore shown to be consistent with an indirect argument based on a correlation method that establishes an independent limit to the sensitivity of shot-noise measurement schemes. Collectively, these results conclusively demonstrate that the photon bunching sensitivity penalty applies to shot noise measurement schemes just as it does to ordinary photon counting, in contradiction to the fundamental claim made by Lieu et al. (2015). The source of this contradiction is traced to a logical fallacy in their argument.Comment: 34 pages, 9 figures; submitted to Ap