Let an, n≥ 1 be a sequence of independent standard normal random variables. Consider the randomtrigonometric polynomial Tn（θ）=∑^n_i=1 aj cos（j θ）, 0≤θ≤π and let Nn be the number of real roots of Tn（θ）in （0, 2π）. In this paper it is proved that limn→∞ Var（Nn）/n=co,where 0 〈 co〈 ∞.
In this paper,we consider the limiting spectral distribution of the information-plus-noise type sample covariance matrices Cn=1/N（Rn＋σXn）（Rn＋σXn）,under the assumption that the entries of Xn are independent but non-identically distributed random variables.It is proved that,almost surely,the empirical spectral distribution of Cn converges weakly to a non-random distribution whose Stieltjes transform satisfies a certain equation.Our result extends the previous one with the entries of Xn are i.i.d.random varibles to a more general case.The proof of the result mainly employs the Stein equation and the cumulant expansion formula of independent random variables.
We consider efficient methods for the recovery of block sparse signals from underdetermined system of linear equations. We show that if the measurement matrix satisfies the block RIP with δ2s 〈 0.4931, then every block s-sparse signal can be recovered through the proposed mixed l2/ll-minimization approach in the noiseless case and is stably recovered in the presence of noise and mismodeling error. This improves the result of Eldar and Mishali （in IEEE Trans. Inform. Theory 55： 5302-5316, 2009）. We also give another sufficient condition on block RIP for such recovery method： 58 〈 0.307.
This paper focuses on the dilute real symmetric Wigner matrix Mn=1/√n（aij）n×n, whose offdiagonal entries aij （1 ≤ em ≠ j ≤ n） have mean zero and unit variance, Eaij4 =θnα （θ 〉 0） and the fifth moments of aij satisfy a Lindeberg type condition. When the dilute parameter 0 〈 α ≤ 1/3 and the test function satisfies some regular conditions, it proves that the centered linear eigenvalue statistics of Mn obey the central limit theorem.
Let Xn be a standard real symmetric （complex Hermitian） Wigner matrix, Y1,Y2,...,Yn a sequence of independent real random variables independent of Xn. Consider the deformed Wigner matrix Hn,a = n^（-1/2）Xn ＋ n^（-a/2）diag（y1,...,yn）, where 0 〈 a 〈 1. It is well known that the average spectral distribution is the classical Wigner semicircle law, i.e., the Stieltjes transform mn,a（z） converges in probability to the corresponding Stieltjes transform rn（z）. In this paper, we shall give the asymptotic estimate for the expectation Emn,a （z） and variance Var（mn,a （z））, and establish the central limit theorem for linear statistics with sufficiently regular test function. A basic tool in the study is Stein＇s equation and its generalization which naturally leads to a certain recursive equation.
In this paper,we study the nonparametric estimation of the second infinitesimal moment by using the reweighted Nadaraya-Watson (RNW) approach of the underlying jump diffusion model.We establish strong consistency and asymptotic normality for the estimate of the second infinitesimal moment of continuous time models using the reweighted Nadaraya-Watson estimator to the true function.
Given a sequence of mixing random variables {X1,Xn;n≥1} taking values in a separable Banach space B,and Sn denoting the partial sum,a general law of the iterated logarithm is established,that is,we have with probability one,lim supn→∞‖Sn‖/cn = α0 < ∞ for a regular normalizing sequence {cn}1,where α 0 is a precise value.
We study tile local linear estimator for tile drift coefficient of stochastic differential equations driven by α-stable Levy motions observed at discrete instants. Under regular conditions, we derive the weak consis- tency and central limit theorem of the estimator. Compared with Nadaraya-Watson estimator, the local linear estimator has a bias reduction whether the kernel function is symmetric or not under different schemes. A silnu- lation study demonstrates that the local linear estimator performs better than Nadaraya-Watson estimator, especially on the boundary.
