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Moderate deviations and central limit theorem for small perturbation Wishart processes
Lei CHEN, Fuqing GAO, Shaochen WANG
Front Math Chin. 2014, 9 (1): 1-15.
https://doi.org/10.1007/s11464-013-0291-0
Let X? be a small perturbation Wishart process with values in the set of positive definite matrices of size m, i.e., the process X? is the solution of stochastic differential equation with non-Lipschitz diffusion coefficient: dXt?=?Xt?dBt+dBt'?Xt?+ρImdt, X0 = x, where B is an m × m matrix valued Brownian motion and B′denotes the transpose of the matrix B. In this paper, we prove that {Xt?-Xt0/?h2(?),?>0} satisfies a large deviation principle, and (Xt?-Xt0)/? converges to a Gaussian process, where h(?)→+∞ and ?h(?)→0 as ?→0. A moderate deviation principle and a functional central limit theorem for the eigenvalue process of X? are also obtained by the delta method.
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Optimal integrability of some system of integral equations
Yutian LEI, Chao MA
Front Math Chin. 2014, 9 (1): 81-91.
https://doi.org/10.1007/s11464-013-0290-1
We obtain the optimal integrability for positive solutions of the Euler-Lagrange system of the weighted Hardy-Littlewood-Sobolev inequality in ?n:{u(x)=1|x|α∫?nυ(y)q|y|β|x-y|λdy,u(x)=1|x|β∫?nυ(y)p|y|α|x-y|λdy.C. Jin and C. Li [Calc. Var. Partial Differential Equations, 2006, 26: 447-457] developed some very interesting method for regularity lifting and obtained the optimal integrability for p, q>1. Here, based on some new observations, we overcome the difficulty there, and derive the optimal integrability for the case of p, q≥1 and pq≠ 1. This integrability plays a key role in estimating the asymptotic behavior of positive solutions when |x| → 0 and when |x| → ∞.
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Multiplication formulas for Kubert functions
Hailong LI, Jing MA, Yuichi URAMATSU
Front Math Chin. 2014, 9 (1): 101-109.
https://doi.org/10.1007/s11464-013-0348-0
The aim of this paper is two folds. First, we shall prove a general reduction theorem to the Spannenintegral of products of (generalized) Kubert functions. Second, we apply the special case of Carlitz’s theorem to the elaboration of earlier results on the mean values of the product of Dirichlet L-functions at integer arguments. Carlitz’s theorem is a generalization of a classical result of Nielsen in 1923. Regarding the reduction theorem, we shall unify both the results of Carlitz (for sums) and Mordell (for integrals), both of which are generalizations of preceding results by Frasnel, Landau, Mikolás, and Romanoff et al. These not only generalize earlier results but also cover some recent results. For example, Beck’s lamma is the same as Carlitz’s result, while some results of Maier may be deduced from those of Romanoff. To this end, we shall consider the Stiletjes integral which incorporates both sums and integrals. Now, we have an expansion of the sum of products of Bernoulli polynomials that we may apply it to elaborate on the results of afore-mentioned papers and can supplement them by related results.
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Asymptotic behavior for bi-fractional regression models via Malliavin calculus
Guangjun SHEN, Litan YAN
Front Math Chin. 2014, 9 (1): 151-179.
https://doi.org/10.1007/s11464-013-0312-z
Let BH1,K1 and BH2,K2 be two independent bi-fractional Brownian motions. In this paper, as a natural extension to the fractional regression model, we consider the asymptotic behavior of the sequenceSn:=∑i=0n-1K(nαBiH1,K1)(Bi+1H2,K2-BiH2,K2),where K is a standard Gaussian kernel function and the bandwidth parameter αsatisfies certain hypotheses. We show that its limiting distribution is a mixed normal law involving the local time of the bi-fractional Brownian motion BH1,K1. We also give the stable convergence of the sequence Sn by using the techniques of the Malliavin calculus.
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14 articles
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