The paper focuses on the largest eigenvalues of theβ-Hermite ensemble and theβ-Laguerre ensemble. In particular, we obtain the precise moment convergence rates of their largest eigenvalues. The results are motivated...The paper focuses on the largest eigenvalues of theβ-Hermite ensemble and theβ-Laguerre ensemble. In particular, we obtain the precise moment convergence rates of their largest eigenvalues. The results are motivated by the complete convergence for partial sums of i.i.d, random variables, and the proofs depend on the small deviations for largest eigenvalues of the β ensembles and tail inequalities of the generalβ Tracy-Widom law.展开更多
Let {Xn; n ≥ 1} be a sequence of independent and identically distributed U[0,1]-distributed random variables. Define the uniform empirical process Fn（t） = n^-1/2 ∑^ni=1 （I{xi≤t} - t）, 0 ≤ t 〈 1, ││Fn││ = ...Let {Xn; n ≥ 1} be a sequence of independent and identically distributed U[0,1]-distributed random variables. Define the uniform empirical process Fn（t） = n^-1/2 ∑^ni=1 （I{xi≤t} - t）, 0 ≤ t 〈 1, ││Fn││ = sup0≤t≤ 1 │Fn（t）│. In this paper, the exact convergence rates of a general law of weighted infinite series of E{││Fn││ -εg^s（n）}＋ are obtained.展开更多
We consider a supercritical branching process （Zn） in an independent and identically distributed random environment ξ, and present some recent results on the asymptotic properties of the limit variable W of the nat...We consider a supercritical branching process （Zn） in an independent and identically distributed random environment ξ, and present some recent results on the asymptotic properties of the limit variable W of the natural martingale Wn = Zn/E[Zn｜ξ], the convergence rates of W - Wn （by considering the convergence in law with a suitable norming, the almost sure convergence, the convergence in Lp, and the convergence in probability）, and limit theorems （such as central limit theorems, moderate and large deviations principles） on （log Zn）.展开更多
基金Supported partly by the National Natural Science Foundation of China (Grant Nos. 11071213, 11101362)Zhejiang Provincial Natural Science Foundation of China (Grant No. R6090034)Research Fund for the Doctoral Program of Higher Education (Grant No. 20100101110001)
文摘The paper focuses on the largest eigenvalues of theβ-Hermite ensemble and theβ-Laguerre ensemble. In particular, we obtain the precise moment convergence rates of their largest eigenvalues. The results are motivated by the complete convergence for partial sums of i.i.d, random variables, and the proofs depend on the small deviations for largest eigenvalues of the β ensembles and tail inequalities of the generalβ Tracy-Widom law.
基金Supported by National Natural Science Foundation of China （Grant No. 10901138）, National Science Fundation of Zhejiang Province （Grant No. R6090034） and the Young Excellent Talent Foundation of Huaiyin Normal University Thanks are due to the referees for valuable comments that have led to improvements in this work.
文摘Let {Xn; n ≥ 1} be a sequence of independent and identically distributed U[0,1]-distributed random variables. Define the uniform empirical process Fn（t） = n^-1/2 ∑^ni=1 （I{xi≤t} - t）, 0 ≤ t 〈 1, ││Fn││ = sup0≤t≤ 1 │Fn（t）│. In this paper, the exact convergence rates of a general law of weighted infinite series of E{││Fn││ -εg^s（n）}＋ are obtained.
基金Acknowledgements This work was partially supported by the National Natural Science Foundation of China （Grant Nos. 11171044, 11101039） and the Natural Science Foundation of Hunan Province （Grant No. 11JJ2001）.
文摘We consider a supercritical branching process （Zn） in an independent and identically distributed random environment ξ, and present some recent results on the asymptotic properties of the limit variable W of the natural martingale Wn = Zn/E[Zn｜ξ], the convergence rates of W - Wn （by considering the convergence in law with a suitable norming, the almost sure convergence, the convergence in Lp, and the convergence in probability）, and limit theorems （such as central limit theorems, moderate and large deviations principles） on （log Zn）.
