Limit theorems for the volumes of excursion sets of weakly and strongly dependent heavy-tailed random fields are proved. Some generalizations to sojourn measures above moving levels and for cross-correlated scenarios are presented. Special attention is paid to Student and Fisher-Snedecor random fields. Some simulation results are also presented. Sojourn measures of Student and Fisher-Snedecor random fields3 The paper has three aims. One is to provide explicit, albeit asymptotic, formulae for the distribution of the volume of excursion sets of a class of strongly dependent random fields. The second one is to derive asymptotic results for heavy-tailed random fields. Finally, the third aim is to generalize the previous findings to sojourn measures above moving levels and for cross-correlated scenarios.There is, therefore, a need for models that are able to display strongly dependent heavytailed behaviour and yet are sufficiently simple to allow analysis. To obtain explicit results we detail the underlying structure of random fields. Namely, a basic assumption of the analysis is that we examine functionals of vector Gaussian random fields, in particular, Student and Fisher-Snedecor random fields. Consult [3,15,16,47] on excursion sets of chi-square, Student and Fisher-Snedecor random fields and their importance for image analysis and studies of brain function. Other results on sojourn measures of chi-square random fields can be found in [23,27,29,30].Minkowski functionals are widely used to characterise geometric properties of random fields, in particular in the analysis of cosmic microwave background radiation, see [36,38]. In this paper we investigate the first Minkowski functional of random fields and its expansions into multidimensional Hermite polynomials, see some one-dimensional/discrete counterparts in [18,20]. To have a complete account of results on asymptotic distributions of sojourn measures for functions of vector random fields, we also prove corresponding theorems for weakly dependent scenarios.The remainder of the paper is structured as follows. In Sections 2-4, we introduce the necessary background from the theory of random fields and briefly review some definitions and notation on the first Minkowski functional, multidimensional Hermite expansions, and Student and Fisher-Snedecor random fields. We start Sections 5 and 7 with generalizations and corrections of some classical asymptotic results to arbitrary sets and vector fields. With this in hand, we continue Sections 5 and 7 by new results for the first Minkowski functional of Student and Fisher-Snedecor random fields. In Section 7, we also show how to lift these results to sojourn measures above moving levels and for cross-correlated underlying vector fields. Sections 6 and 8 provide the proofs of all theorems and lemmata in the article. Simulation results on the limit distributions of areas of excursion sets for two types of images are given in Section 9. Short conclusions are made in Section 10.In this paper, we only consider real-valued random f...

This letter rigorously derives explicit closed-form expressions for the joint/marginal probability density functions of the uplink/downlink multipaths' time-of-arrival (TOA) and azimuth angle-of-arrival (AOA) in a wireless-communication fading channel. This derivation is based on a new "geometrical model" of omnidirectional scatterers as spatially distributed uniformly on a two-dimensional hollow-disc (i.e., a thick ring) centered upon the mobile. By varying the hollow-disc's thickness, this spatial density degenerates to the well-known uniform-ring or uniform-disc densities.Index Terms-Dispersive channels, fading channels, land mobile radio cellular systems, land mobile radio diversity systems, land mobile radio equipment, mobile communication, radio communication equipment.

The paper starts by giving a motivation for this research and justifying the considered stochastic diffusion models for cosmic microwave background radiation studies. Then it derives the exact solution in terms of a series expansion to a hyperbolic diffusion equation on the unit sphere. The Cauchy problem with random initial conditions is studied. All assumptions are stated in terms of the angular power spectrum of the initial conditions. An approximation to the solution is given and analysed by finitely truncating the series expansion. The upper bounds for the convergence rates of the approximation errors are derived. Smoothness properties of the solution and its approximation are investigated. It is demonstrated that the sample Hölder continuity of these spherical fields is related to the decay of the angular power spectrum. Numerical studies of approximations to the solution and applications to cosmic microwave background data are presented to illustrate the theoretical results.

This paper surveys Abelian and Tauberian theorems for long-range dependent random fields. We describe a framework for asymptotic behaviour of covariance functions or variances of averaged functionals of random fields at infinity and spectral densities at zero. The use of the theorems and their limitations are demonstrated through applications to some new and less-known examples of covariance functions of long-range dependent random fields.In this paper we study asymptotic properties of spectral and covariance functions of random fields. We investigate the connection between (a) the behaviour of covariance functions or variances of averaged functionals of random fields at infinity and (b) the behaviour of spectral distribution functions at zero. In the terminology of the integral transforms the statement "(b) implies (a)" is a theorem of Abelian type and the statement "(a) implies (b)" is a Tauberian theorem.Abelian and Tauberian theorems are not only of their own interest but also have numerous applications in asymptotic problems of probability theory and statistics

I. ANALYTICAL DERIVATION OF 3-D TOA/2-D-AOA DISTRIBUTIONSA signal, transmitted from a mobile user in a landmobile radiowave wireless cellular communication system, arrives at the cellular base station through multiple propagation multipaths. Each multipath carries its own propagation history of electromagnetic reflections and diffractions and corruption by multiplicative noise-a history reflected in that multipath's amplitude, Doppler, arrival angle, and arrival time delay at the receiving antenna(s). The values of these amplitudes, Doppler frequency shifts, arrival angles, and arrival time delays depend on the electromagnetic properties of and the spatial geometry among the mobile transmitter, the scatterers, and the receiving antennas. Each receiving antenna's data measurement sums these individually unobservable multipaths.The time of arrival (TOA) 1 distribution function characterizes the channel's temporal delay spread and frequency incoher- ence, which in turn determines the obtainable temporal diversity and the extent of intersymbol interference. The two-dimensional (2-D) azimuth-elevation angle of arrival (AOA) determines the angular spread and spatial decorrelation across the spatial aperture of an receiving antenna. The multipaths' nonzero elevation AOAs are most common in urban areas or over hilly terrains or with low-lying receiving antennas, where the propagating wave reflects off vertical structures like buildings or hills. The nonzero elevation AOA is critical for the use of a vertical or planar receiving antenna array, a fact recognized by the European Union's COST Action 259 [19]."Geometric modeling" idealizes the aforementioned wireless propagation environment via a geometric abstraction of the spatial relationships among the transmitter, the scatterers, and the base station. Geometric models thus attempt to embed measurable fading metrics integrally into the propagation channel's idealized geometry, such that the geometric parameters would affect these various fading metrics in an interconnected manner to reveal conceptually the channel's underlying fading dynamics. This work will maximally embed all derived statistics intrinsically within the propagation channel's geometry and to avoid a priori ad hoc statistical assumptions of the received signal's measurable statistics. See [27] for additional discussion on the nature of geometric modeling of wireless propagation.Geometric modeling contrasts with site-specific/terrain-specific/building-specific empirical measurements or ray-shooting/ ray-tracing computer simulations, which are applicable only to the one particular propagation setting under investigation but cannot be easily generalized to wider scenarios. One geometric model can apply for a wide class of propagation settings, producing the received signal's measurable fading metrics (e.g., the uplink and downlink probability density functions of the multipaths' arrival delay and 2-D arrival angle as in this paper) applicable generally within that class of channels.Geometric modeling contras...

The main result of the article is the rate of convergence to the Rosenblatt-type distributions in non-central limit theorems. Specifications of the main theorem are discussed for several scenarios. In particular, special attention is paid to the Cauchy, generalized Linnik's, and local-global distinguisher random processes and fields. Direct analytical methods are used to investigate the rate of convergence in the uniform metric.

Background: Daily paediatric asthma readmissions within 28 days are a good example of a low count time series and not easily amenable to common time series methods used in studies of asthma seasonality and time trends. We sought to model and predict daily trends of childhood asthma readmissions over time in Victoria, Australia. Methods: We used a database of 75,000 childhood asthma admissions from the Department of Health, Victoria, Australia in 1997-2009. Daily admissions over time were modeled using a semi parametric Generalized Additive Model (GAM) and by sex and age group. Predictions were also estimated by using these models. Results: N = 2401 asthma readmissions within 28 days occurred during study period. Of these, n = 1358 (57%) were boys. Overall, seasonal peaks occurred in winter (30.5%) followed by autumn (28.6%) and then spring (24.6%) (p < 0.0005). Day of the week and month were significantly associated with trends in readmission. Smooth function of time was significant (p < 0.0005) and indicated declining trends in readmissions in 2001-2002 and then increasing, returning to roughly initial levels. Predictions suggested readmissions would continue to increase by 5% per year with boys in the 2 to 5 years age group experiencing the largest increase. Conclusions: GAMs are reliable methods for low count time series such as repeat admissions. Our model implied: health services may need to be revised to accommodate for seasonal peaks in readmission especially for younger age groups.

This paper gives the exact solution in terms of the Karhunen-Loève expansion to a fractional stochastic partial differential equation on the unit sphere S 2 ⊂ R 3 with fractional Brownian motion as driving noise and with random initial condition given by a fractional stochastic Cauchy problem. A numerical approximation to the solution is given by truncating the Karhunen-Loève expansion. We show the convergence rates of the truncation errors in degree and the mean square approximation errors in time. Numerical examples using an isotropic Gaussian random field as initial condition and simulations of evolution of cosmic microwave background (CMB) are given to illustrate the theoretical results.