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...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〈 ∞.展开更多
In this paper, the completeness and minimality properties of some random exponential system in a weighted Banach space of complex functions continuous on the real line for convex nonnegative weight are studied. The re...In this paper, the completeness and minimality properties of some random exponential system in a weighted Banach space of complex functions continuous on the real line for convex nonnegative weight are studied. The results may be viewed as a probabilistic version of Malliavin＇s classical results.展开更多
We show that the zeros of random sequences of Gaussian systems of polynomials of increasing degree almost surely converge to the expected limit distribution under very general hypotheses. In particular, the normalized...We show that the zeros of random sequences of Gaussian systems of polynomials of increasing degree almost surely converge to the expected limit distribution under very general hypotheses. In particular, the normalized distribution of zeros of systems of m polynomials of degree N, orthonormalized on a regular compact set K ? ? m , almost surely converge to the equilibrium measure on K as N → ∞.展开更多
基金supported by National Natural Science Foundation of China (GrantNos. 10671176 and 11071213)Zhejiang Provincial Natural Science Foundation of China (Grant No. R6090034)+1 种基金Doctoral Programs Foundation of Ministry of Education of China (Grant No. J20110031)Competitive Earmarked Research Grant of Research Grants Council (Grant No. 602608)
文摘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〈 ∞.
基金Project supported by the National Natural Science Foundation of China (No.10371005)the Scientific Research Foundation of the Ministry of Education of China for Returned Overseas Chinese Scholars
文摘In this paper, the completeness and minimality properties of some random exponential system in a weighted Banach space of complex functions continuous on the real line for convex nonnegative weight are studied. The results may be viewed as a probabilistic version of Malliavin＇s classical results.
基金Research partially supported by the Notional Science Foundation(Grant No.DMS-0600982)
文摘We show that the zeros of random sequences of Gaussian systems of polynomials of increasing degree almost surely converge to the expected limit distribution under very general hypotheses. In particular, the normalized distribution of zeros of systems of m polynomials of degree N, orthonormalized on a regular compact set K ? ? m , almost surely converge to the equilibrium measure on K as N → ∞.
