Dependent Gaussian Random Variables Research Articles

An efficient and spectral accurate numerical method for computing SDE driven by multivariate Gaussian variables

There are many previous studies on designing efficient and high-order numerical methods for stochastic differential equations (SDEs) driven by Gaussian random variables. They mostly focus on proposing numerical methods for SDEs with independent Gaussian random variables and rarely solving SDEs driven by dependent Gaussian random variables. In this paper, we propose a Galerkin spectral method for solving SDEs with dependent Gaussian random variables. Our main techniques are as follows: (1) We design a mapping transformation between multivariate Gaussian random variables and independent Gaussian random variables based on the covariance matrix of multivariate Gaussian random variables. (2) First, we assume the unknown function in the SDE has the generalized polynomial chaos expansion and convert it to be driven by independent Gaussian random variables by the mapping transformation; second, we implement the stochastic Galerkin spectral method for the SDE in the Gaussian measure space; and third, we obtain deterministic differential equations for the coefficients of the expansion. (3) We employ a spectral method solving the deterministic differential equations numerically. We apply the newly proposed numerical method to solve the one-dimensional and two-dimensional stochastic Poisson equations and one-dimensional and two-dimensional stochastic heat equations, respectively. All the presented stochastic equations are driven by two Gaussian random variables, and they are dependent and have multivariate normal distribution of their joint probability density.

Wiener–Hermite polynomial expansion for multivariate Gaussian probability measures

This paper introduces a new, transformation-free, generalized polynomial chaos expansion (PCE) comprising multivariate Hermite orthogonal polynomials in dependent Gaussian random variables. The second-moment properties of Hermite polynomials reveal a weakly orthogonal system when obtained for a general Gaussian probability measure. Still, the exponential integrability of norm allows the Hermite polynomials to constitute a complete set and hence a basis in a Hilbert space. The completeness is vitally important for the convergence of the generalized PCE to the correct limit. The optimality of the generalized PCE and the approximation quality due to truncation are discussed. New analytical formulae are proposed to calculate the mean and variance of a generalized PCE approximation of a general output variable in terms of the expansion coefficients and statistical properties of Hermite polynomials. However, unlike in the classical PCE, calculating the coefficients of the generalized PCE requires solving a coupled system of linear equations. Besides, the variance formula of the generalized PCE contains additional terms due to statistical dependence among Gaussian variables. The additional terms vanish when the Gaussian variables are statistically independent, reverting the generalized PCE to the classical PCE. Numerical examples illustrate the generalized PCE approximation in estimating the statistical properties of various output variables.

Characteristic function of a quadratic form formed by correlated complex Gaussian variables

An exact closed-form analytical expression is obtained for a one-dimensional characteristic function of an arbitrary quadratic form formed by statistically dependent Gaussian random variables with nonzero means. Necessary and sufficient conditions under which an arbitrary quadratic form has the generalized χ2 distribution are formulated. Parameters of the distribution of an arbitrary quadratic form are analyzed for particular cases.

On the Decision Threshold of Eigenvalue Ratio Detector Based on Moments of Joint and Marginal Distributions of Extreme Eigenvalues

Eigenvalue Ratio (ER) detector based on the two extreme eigenvalues of the received signal covariance matrix is currently one of the most effective solution for spectrum sensing. However, the analytical results of such scheme often depend on asymptotic assumptions since the distribution of the ratio of two extreme eigenvalues is exceptionally complex to compute. In this paper, a non-asymptotic spectrum sensing approach for ER detector is introduced to approximate the marginal and joint distributions of the two extreme eigenvalues. The two extreme eigenvalues are considered as dependent Gaussian random variables such that their joint probability density function (PDF) is approximated by a bivariate Gaussian distribution function for any number of cooperating secondary users and received samples. The PDF approximation approach is based on the moment matching method where we calculate the exact analytical moments of joint and marginal distributions of the two extreme eigenvalues. The decision threshold is calculated by exploiting the statistical mean and the variance of each of the two extreme eigenvalues and the correlation coefficient between them. The performance analysis of our newly proposed approximation approach is compared with the already published asymptotic Tracy-Widom approximation approach. It has been shown that our results are in perfect agreement with the simulation results for any number of secondary users and received samples.

A central limit theorem for triangular arrays of weakly dependent random variables, with applications in statistics

We derive a central limit theorem for triangular arrays of possibly nonstationary random variables satisfying a condition of weak dependence in the sense of Doukhan and Louhichi [Stoch. Proc. Appl. 84 (1999) 313–342]. The proof uses a new variant of the Lindeberg method: the behavior of the partial sums is compared to that of partial sums of dependent Gaussian random variables. We also discuss a few applications in statistics which show that our central limit theorem is tailor-made for statistics of different type.

Convergence of dependent walks in a random scenery to fBm-local time fractional stable motions

It is classical to approximate the distribution of fractional Brownian motion by a renormalized sum $ S_n $ of dependent Gaussian random variables. In this paper we consider such a walk $ Z_n $ that collects random rewards $ \xi_j $ for $ j \in \mathbb Z$, when the ceiling of the walk $ S_n $ is located at $ j$. The random reward (or scenery) $ \xi_j $ is independent of the walk and with heavy tail. We show the convergence of the sum of independent copies of $ Z_n$ suitably renormalized to a stable motion with integral representation, whose kernel is the local time of a fractional Brownian motion (fBm). This work extends a previous work where the random walk $ S_n$ had independent increments limits.

Strong result for real zeros of random algebraic polynomials

An estimate is given for the lower bound of real zeros of random algebraic polynomials whose coefficients are non‐identically distributed dependent Gaussian random variables. Moreover, our estimated measure of the exceptional set, which is independent of the degree of the polynomials, tends to zero as the degree of the polynomial tends to infinity.

The good, bad and ugly: distributed detection of a known signal in dependent Gaussian noise

Most results about quantized detection rely strongly on an assumption of independence among random variables. With this assumption removed, little is known. Thus, in this paper, Bayes-optimal binary quantization for the detection of a shift in mean in a pair of dependent Gaussian random variables is studied. This is arguably the simplest meaningful problem one could consider. If results and rules are to be found, they ought to make themselves plain in this problem. For certain problem parametrizations (meaning the signals and correlation coefficient), optimal quantization is achievable via a single threshold applied to each observation-the same as under independence. In other cases, one observation is best ignored or is quantized with two thresholds; neither behavior is seen under independence. Further, and again in distinction from the case of independence, it is seen that in certain situations, an XOR fusion rule is optimal, and in these cases, the implied decision rule is bizarre. The analysis is extended to the multivariate Gaussian problem.

Asymptotic variance of M-estimators for dependent Gaussian random variables

This paper discusses the asymptotic behavior of M-estimators for dependent Gaussian random variables. We show that for a Gaussian distribution, the asymptotic variance of an M-estimator of scale is minimal in the independent case and must necessarily increase for dependent data. This is not true for location estimation where the asymptotic variance can increase or decrease for dependent observations, depending on the sign of the correlation. Several examples are analyzed, showing that the asymptotic variance of the maximum likelihood estimator varies widely under dependencies.

On the lower bound of the number of real roots of a random algebraic equation

We estimate a lower bound of the number of real roots of a random algebraic equation whose random coefficients are dependent Gaussian random variables.

Multiple stochastic integrals with dependent integrators

Let μ be a σ-finite measure, R = ( r ij ) be a covariance matrix, and B 1,…, B n be dependent Gaussian measures satisfying EB i(A 1) B j(A 2) = r ijμ(A 1 ⌢ A 2) . Multiple integrals of the form I n ( f) = ∫ f( x 1,…, x n ) dB 1( x 1) … dB n ( x n ), with f ∈ L 2( μ n ) are investigated. A diagram formula is established and a class of functions which play the role of the Hermite polynomials for these more general integrals is introduced. Cumulants of double integrals are evaluated and the following result is established: if { X j } and { Y j } are correlated stationary sequences of strongly dependent Gaussian random variables, then Σ j=1 [ Nt] X j Y j , adequately normalized, converges in D[0, 1] to I 2( f i ).

Synthesis of multivariate Gaussian random processes with a preassigned covariance (Corresp.)

Evaluation of complex systems in a laboratory environment requires the generation of inputs to the system sensors that are representative of the operational environment. It is therefore necessary to synthesize input test signals that reflect the mutual dependencies found in situ. For multivariate Gaussian inputs, algorithms are derived allowing 1) the transformation of dependent Gaussian random variables into independent variables; 2) the generation of jointly Gaussian random variables with a constant covariance matrix; and 3) the synthesis of stationary multivariate Gaussian random processes. These algorithms have simple electronic hardware and computer software implementations that will facilitate the laboratory evaluation and digital computer simulation of complex systems.