Let an,n 1 be a sequence of independent standard normal random variables.Consider the random trigonometric polynomial Tn(θ)=∑nj=1 aj cos(jθ),0≤θ≤2π 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=c0,where 0
In this paper,we consider the limiting spectral distribution of the information-plus-noise type sample covariance matrices C_n=1/N(R_n+σX_n)(R_n+σX_n),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.
Let {X, Xn ; n ≥ 0} be a sequence of independent and identically distributed random variables, taking values in a separable Banach space (B,||·||) with topological dual B* . Considering the geometrically weighted series ξ(β) =∑∞n=0βnXn for 0 < β < 1, and a sequence of positive constants {h(n), n ≥ 1}, which is monotonically approaching infinity and not asymptotically equivalent to log log n, a limit result for(1-β2)1/2||ξ(β)||/(2h(1/(1-β2)))1/2 is achieved.
This paper focuses on the dilute real symmetric Wigner matrix Mn=1√n(aij)n×n,whose offdiagonal entries aij(1 i=j n)have mean zero and unit variance,Ea4ij=θnα(θ>0)and the fifth moments of aij satisfy a Lindeberg type condition.When the dilute parameter 0<α13and the test function satisfies some regular conditions,it proves that the centered linear eigenvalue statistics of Mn obey the central limit theorem.
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 the local linear estimator for the drift coefcient of stochastic diferential equations driven byα-stable L′evy motions observed at discrete instants.Under regular conditions,we derive the weak consistency 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 diferent schemes.A simulation study demonstrates that the local linear estimator performs better than Nadaraya-Watson estimator,especially on the boundary.
