Density of Generalized Verhulst Process and Bessel Process with Constant Drift

In this paper, we derive exact closed-form density functions of the generalized Verhulst process and the Bessel process with constant drift, which have applications in mathematical biology and queueing theory. We propose a generic probabilistic method for deriving exact closed-form density functions for these two diffusion processes based on a novel application of the exponential measure change in [T. Hurd and A. Kuznetsov, Explicit formulas for Laplace transforms of stochastic integrals, Markov Process. Relat. Fields, 14(2):277–290, 2008], together with formulae in [A. Borodin and P. Salminen, Handbook of Brownian Motion – Facts and Formulae, Birkhauser, Basel, 2015]. Our study generalizes several known results in the literature.

Bessel processes with constant negative drift have recently appeared as heavy-traffic limits in queueing theory. We derive a closed-form expression for the spectral representation of the transition density of the Bessel process of orderν> −1 with constant driftμ≠ 0. Whenν> -½ andμ< 0, the first term of the spectral expansion is the steady-state gamma density corresponding to the zero principal eigenvalueλ0= 0, followed by an infinite series of terms corresponding to the higher eigenvaluesλn,n= 1,2,…, as well as an integral over the continuous spectrum aboveμ2/2. When −1 <ν< -½ andμ< 0, there is only one eigenvalueλ0= 0 in addition to the continuous spectrum. As well as applications in queueing, Bessel processes with constant negative drift naturally lead to two new nonaffine analytically tractable specifications for short-term interest rates, credit spreads, and stochastic volatility in finance. The two processes serve as alternatives to the CIR process for modelling mean-reverting positive economic variables and have nonlinear infinitesimal drift and variance. On a historical note, the Sturm–Liouville equation associated with Bessel processes with constant negative drift is closely related to the celebrated Schrödinger equation with Coulomb potential used to describe the hydrogen atom in quantum mechanics. Another connection is with D. G. Kendall's pole-seeking Brownian motion.

We consider the problem of distributed detection over a network of sensors using energy measurements. Assuming that the problem is to detect the presence of stochastic signal with arbitrary temporal correlation, we exactly compute the characteristic function of the energy measurement at each sensing node. As it is shown, when the time-bandwidth product W T is large enough, the exact joint probability density function of the measurements at the sensor sites can be closely approximated by a product of non-trivial density functions that depend of some unknown parameters of the source signal and the sensor network. As those parameters can be easily estimated at each sensor node, we propose a modified generalized likelihood ratio test (GLRT). Through numerical experiments we show that this proposal has performance close to the optimal one which use the exact joint density function and with a significant reduction in the consumption of network resources as bandwidth and transmission power.

Fixed-gain AF PLNC over cascaded Nakagami-m fading channels for vehicular communications

We consider in this paper, a spectrum sharing cognitive radio system with a ratio selection scheme; where one out of N independent-and-identically-distributed transmit antennas is selected such that the ratio of the secondary transmitter (ST) to the secondary receiver (SR) channel gain to the interference from the ST to the primary receiver (PR) channel gain is maximized. Although previous works considered perfect, outdated, or partial channel state information at the transmitter, we stress that using such assumptions may lead to a feedback overhead for updating the SR with the ST-PR interference channel estimation. Considering only statistical knowledge of the ST-PR channel gain, we investigate a ratio selection scheme using a mean value (MV)-based power allocation strategy referred to as MV-based scheme. We first provide the exact statistics in terms of probability density function and cumulative distribution function of the secondary channel gain as well as of the interference channel gain. Furthermore, we derive exact cumulative density function of the received signal-to-noise ratio at the SR where the ST uses a power allocation based on instantaneous perfect channel state information (CSI) referred to as CSI-based scheme. These statistics are then used to derive exact closed form expressions of the outage probability, symbol error rate, and ergodic capacity of the secondary system when the interference channel from the primary transmitter (PT) to the SR is ignored. Furthermore, an asymptotical analysis is also carried out for the MV-based scheme as well as for the CSI-based scheme to derive the generalized diversity gain for each. Subsequently, we address the performance analysis based on exact statistics of the combined signal-to-interference-plus-noise ratio at the SR of the more challenging case; when the PT-SR interference channel is considered. Numerical results in a Rayleigh fading environment manifest that the MV-based scheme outperforms the CSI-based scheme provided that a low interference power constraint is deployed, implying that the MV-based scheme is more suitable for practical systems.

This paper has the goal of defining a class of uncertain structural systems for which it is possible to consider an approach able to give the exact response in terms of the probability density function (PDF). The uncertain structures have been identified in the discretized statically determined ones and the approach has been identified in the coupling of the approximated principal deformation modes method (APDM) and of the probability transformation method (PTM). The first one gives the explicit relationships between the response variables and the uncertainty ones, that are exact when the structures are statically determined. The second method allows to determine the explicit relationship between the PDFs of the response and of the uncertainty variables. The results of some applications have confirmed the goodness of these choices and that the proposed approach gives always exact results for both correlated and uncorrelated uncertainty random variables.

This paper investigates the performance analysis of dual selection (i.e., combining user selection and transmit antenna selection) with maximal ratio combining (MRC) over nonidentical imperfect channel estimation in a multiple-input multiple-output system. This paper derives the exact and approximate cumulative density function and probability density function (PDF) for the dual selection with MRC over nonidentical imperfect channel estimation. Using the distribution, the exact and approximate closed-form expressions of the outage probability are derived with quantification of diversity order. Further, this paper derived the exact ergodic capacity and approximate outage capacity of the the dual selection via Gaussian and Gamma approximations. The numerical results show that the capacity of the dual selection with MRC is enhanced by the number of receive antennas as well as the number of user terminals and sum of channel correlation coefficients, while the order of the receive spatial diversity for outage probability is eliminated by the nonidentical channel estimation error.

Exact distributions of MLEs of regression coefficients in GMANOVA–MANOVA model

This paper investigates the system performance for a wireless system employing maximal-ratio-combining (MRC) over Nakagami-m fading channel with multiple unequal-power co-channel interferers (CCI). It derives the exact closed-form probability density function (PDF) and moment generating function (MGF) expressions of output signal-to-interference ratio (SIR). Using the PDF and MGF of the output SIR, the system outage probability and average bit-error-rate (BER) for noncoherent and coherent modulation system are obtained. Some numerical examples are given to illustrate the effects of various parameters on the system performance. (4 pages)

This paper contributes a generic probabilistic method to derive explicit exact probability densities for stochastic volatility models. Our method is based on a novel application of the exponential measure change in [Z. Palmowski & T. Rolski (2002) A technique for exponential change of measure for Markov processes, Bernoulli 8(6), 767–785]. With this generic approach, we first derive explicit probability densities in terms of model parameters for several stochastic volatility models with nonzero correlations, namely the Heston 1993, [Formula: see text], and a special case of the [Formula: see text]-Hypergeometric stochastic volatility models recently proposed by [J. Da Fonseca & C. Martini (2016) The [Formula: see text]-Hypergeometric stochastic volatility model, Stochastic Processes and their Applications 126(5), 1472–1502]. Then, we combine our method with a stochastic time change technique to develop explicit formulae for prices of timer options in the Heston model, the [Formula: see text] model and a special case of the [Formula: see text]-Hypergeometric model.

A noble theoretical approach is presented to evaluate the bit error rate (BER) performance of an optical polarization division multiplexed (PDM) 4-multilevel quadrature amplitude modulation (4-QAM) transmission system under the combined influence of polarization mode dispersion (PMD) and group velocity dispersion (GVD) in a single mode fiber (SMF). The analysis is carried out considering a coherent homodyne receiver. The exact probability density function (pdf) fluctuation due to PMD and GVD is evaluated from its moments using a Monte-Carlo simulation technique. Average BER is evaluated by averaging the conditional BER over the pdf of the random phase fluctuation. BER performance results are evaluated for different system parameters. It is found that PDM 4-QAM coherent homodyne system doubles the data rate but suffers more power penalty than the 4-QAM system. Results show that for a BER of 10^-9 at DGD of 0.5T and GVD value of 1700 ps/nm the PDM 4-QAM system needs 2.21 dB more power than 4-QAM systems giving the leverage of doubling the data rate.

Baumgartner et al. (1998) proposed a novel statistical test for the null hypothesis that two independently drawn samples of data originate from the same population, and Murakami (2006) generalized the test statistic for more than two samples. Whereas the expressions of the exact density and distribution functions of the generalized Baumgartner statistic are not yet found, the characteristic function of its limiting distribution has been obtained. Due to the development of computational power, the Fourier series approximation can be readily utilized to accurately and efficiently approximate its density function based on its Laplace transform. Numerical examples show that the Fourier series method provides an accurate approximation for statistical quantities of the generalized Baumgartner statistic.

We consider an exactly soluble model consisting of two electrons attracted to a common center by harmonic-oscillator forces with mutual Coulomb repulsion between the electrons. The exact wave function and energy for the singlet ground state is obtained by separating the Schr\"odinger equation in center-of-mass and relative coordinates and numerically integrating the relative-coordinate eigenvalue equation. The exact Kohn-Sham (KS) orbital for the two equivalent electrons with opposite spins is constructed from the exact density, and the Kohn-Sham equation is inverted, yielding the exact exchange-correlation potential as a function of position.The exact correlation potential, as a function of position, is then obtained by subtracting the exact exchange potential from the exact exchange-correlation potential. The exact ground-state energy, exchange potential, correlation potential, KS single-particle energy eigenvalues ${\ensuremath{\epsilon}}_{\mathrm{KS}}$, exchange energy, and correlation energy are compared to the results given by density-functional theory employing the local-density approximation (LDA) plus correction terms and the exact KS orbitals for values of the harmonic-oscillator spring constant that vary over 4 orders of magnitude. We find that although total energies are accurately given by LDA plus correction terms, the values of ${\ensuremath{\epsilon}}_{\mathrm{KS}}$ are significantly in error. We show that this is mainly due to errors in the exchange potential.In addition we show analytically that in a singlet ground state of a two-electron system, the exact expectation value of the exchange potential equals the exact exchange energy, whereas the LDA expectation value of the exchange potential is only (2/3) the LDA exchange energy. Furthermore, we show that although the gradient expansion approximations for the exchange energy are significant improvements over the LDA, neither of these approximations significantly decreases the error in the LDA for the expectation value of the exchange potential. Both the correlation energy and the expectation value of the correlation potential in the LDA plus corrections are significantly in error when compared with the results of the exact calculation, the most accurate results being given by the LDA with self-interaction correction.

The ground state at 4/11 filling factor is very well understood, Mukherjee et al. [Phys. Rev. Lett. 112, 016801 (2014)] in terms of the 1/3-filled second effective Landau level of the composite fermions whose correlations resemble that of electrons in the ground state of a two-body Haldane pseudo-potential of relative angular momentum 3, ${V}_{3}$. We here propose a closed-form ground-state wave function for ${V}_{3}$ at 1/3 filling factor. We successfully compare it with the exact wave function for systems with a few electrons by calculating their mutual overlap, pair-correlation function, and entanglement spectra. By numerical exact diagonalization for a few electron systems, we find a window of nonzero ${V}_{3}$ is essential together with ${V}_{1}$ for being 4/11 state incompressible. The constructed wave function for the 4/11 state using this proposed wave function has satisfactorily high overlap with previously studied composite-fermion-diagonalized ground-state wave function.

A novel approximation to the probability density function for the product of arbitrary n independent Rayleigh random variables is proposed. The approximation is based on a transformed Nakagami-m distribution. Unlike the existing exact density function in the literature, the new approximation is compact and only employs simple functions that are very easy to calculate and to manipulate. New maximum likelihood estimators for the distribution parameters are also derived that are otherwise impossible using the existing exact density function. A new estimator using the logarithm of the sample and a new estimator without knowledge of the number of variables are also developed. Numerical results show that the new approximation has very high accuracy in most cases considered. Numerical results also show that the new estimators using maximum likelihood and sample logarithm outperform the existing estimators.