Quenched asymptotics for Brownian motion of renormalized Poisson potential and for the related parabolic Anderson models

Let $B_s$ be a $d$-dimensional Brownian motion and $\omega(dx)$ be an independent Poisson field on $\mathbb{R}^d$. The almost sure asymptotics for the logarithmic moment generating function [\log\math bb{E}_0\exp\biggl{\pm\theta\int_0^t\bar{V}(B_s) ds\biggr}\qquad (t\to\infty)] are investigated in connection with the renormalized Poisson potential of the form [\bar{V}(x)=\int_{\mathbb{R}^d}{\frac{1}{|y-x|^p}}[\omega(dy)-dy],\qquad x\in\mathbb{R}^d.] The investigation is motivated by some practical problems arising from the models of Brownian motion in random media and from the parabolic Anderson models.Comment: Published in at http://dx.doi.org/10.1214/11-AOP655 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Exponential asymptotics and law of the iterated logarithm for intersection local times of random walks

Let \alpha ([0,1]^p) denote the intersection local time of p independent d-dimensional Brownian motions running up to the time 1. Under the conditions p(d-2)<d and d\ge 2, we prove lim_{t\to\infty}t^{-1}\log P\bigl{\alpha([0,1]^p)\ge t^{(d(p-1))/2}\bigr}=-\gamma_{\alpha}(d,p) with the right-hand side being identified in terms of the the best constant of the Gagliardo-Nirenberg inequality. Within the scale of moderate deviations, we also establish the precise tail asymptotics for the intersection local time I_n=#{(k_1,...,k_p)\in [1,n]^p;S_1(k_1)=... =S_p(k_p)} run by the independent, symmetric, Z^d-valued random walks S_1(n),...,S_p(n). Our results apply to the law of the iterated logarithm. Our approach is based on Feynman-Kac type large deviation, time exponentiation, moment computation and some technologies along the lines of probability in Banach space. As an interesting coproduct, we obtain the inequality \bigl(EI_{n_1+... +n_a}^m\bigr)^{1/p}\le \sum_{k_1+... +k_a=m\limits_{k_1,...,k_a\ge 0}}\frac{m!}{k_1!... k_a!}\bigl(EI_{n_1}^{k_1}\bigr)^{1/p}... \bigl(EI_{n_a}^{k_a}\bigr)^{1/p} in the case of random walks.Comment: Published at http://dx.doi.org/10.1214/009117904000000513 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

An elastic net orthogonal forward regression algorithm

In this paper we propose an efficient two-level model identification method for a large class of linear-in-the-parameters models from the observational data. A new elastic net orthogonal forward regression (ENOFR) algorithm is employed at the lower level to carry out simultaneous model selection and elastic net parameter estimation. The two regularization parameters in the elastic net are optimized using a particle swarm optimization (PSO) algorithm at the upper level by minimizing the leave one out (LOO) mean square error (LOOMSE). Illustrative examples are included to demonstrate the effectiveness of the new approaches

Modeling of complex-valued Wiener systems using B-spline neural network

In this brief, a new complex-valued B-spline neural network is introduced in order to model the complex-valued Wiener system using observational input/output data. The complex-valued nonlinear static function in the Wiener system is represented using the tensor product from two univariate Bspline neural networks, using the real and imaginary parts of the system input. Following the use of a simple least squares parameter initialization scheme, the Gauss–Newton algorithm is applied for the parameter estimation, which incorporates the De Boor algorithm, including both the B-spline curve and the first-order derivatives recursion. Numerical examples, including a nonlinear high-power amplifier model in communication systems, are used to demonstrate the efficacy of the proposed approaches