【摘要】 本文主要讨论实平稳正态序列谱函数估计的a.s.(一致)收敛速度。首先,对实平稳正态序列的观察值的二次型建立指数不等式和概率1的界;在此基础上,得到了协方差函数和谱函数估计的收敛速度及一致收敛速度。更多还原

【Abstract】 Suppose X={x（n）, n=0, ±1, …} is a real strictly stationary process, Ex（n）=0, Ex（n）x （m） =B （n-m）. Let f （λ） and F（λ）=integral from n=0 to λ f（u）du be the spectral density and spectral function of X respectively. One way to estimate B（k） and F（λ） from N consecutive observations x（j）, j= 1, 2, …, N is by means of B_N（k）=1/N sum from n=1 to n-k x（n）x（n+k） 0≤k≤N-1 0 k≥N and Fx（λ） =integral form n=0 to λ 1/2πN sum form k=1 to N x （k）e^iku|²du respectively. This paper is mainly concerned with the a. s. convergence rates of |F_N（λ）-F（λ）| and sup |F_N（λ）-F（λ）| for a real stationary Gaussian process. First, we establish the a. s. convergence rates of the quadratic forms of the observations. Then we obtain the convergence rates of estimates of the covariances and spectral function. The main results are as follows. Lemma 1. Let X= {x（n）} be a real stationary Gaussian process with zero mean and B_N= （B （n-m）） the N×N covariance matrix of X. Y_N= X’_NA_NX_N-EX’_NA_NX_N where X’_N=（x（1）, …, x（N））, A_N is a N×N real symmetric matrix. If λ_μ is the maximum of the absolute value of the eigenvalues of B_N^1/2A_NB_N^1/2, then for any fixed δ>0 and 0≤α≤δ/2（1+δ）λ_μ^-1, we have E[e^aYN]≤exp{1+δ/2 α² Var（Y_N）} and E[e^a|Y_N|]≤2 exp {1+δ/2 a² Var（Y_N）}. Theorem 1. Let X={x （n）} be a real stationary Gaussian process with zero mean, and f² log⁺ f∈ L[0, π]. Y_N= X’_N A_N X_N-EX’_NA_NX_N where X’_N=（x（1）, …, x（N））, A_N is a N×N real symmetric matrix. If ‖A_N‖（?）sup ‖Y‖₂=1 |Y’A_NY|≤c N≥1 and lim n→∞ 1/N Var（Y_N）=a²>0 Then lim n→∞ （1/2N log N）^1/2|Y_N|≤a a.s. Corollary 1. Let X={x（n）} satisfy the conditions of Theorem 1. Then （1） （?）k≥0 lim n→∞ （N/2log N）^1/2|B_N（k）-B（k）|≤d_k a. s. where d_k²=4π integral from n=-π to π cos² kuf² （u）du. （2） （?）λ∈[0, π] lim N→∞ （N/2logN）^1/2|F_N（λ）-（λ）|≤c_λ a. s. where Theorem 2. Let X={x （n）} satisfy the conditions ef Theorem 1. Then for P（N）=O（N log N）^1/2 lim N→∞ （N/2logN）^1/2 sup 0≤k≤P（N）|B_N（k）-B（k）|≤3/2^1/2 do a. s. where d₀²-4π integral from n=π to -π f² （u） du. Moreover, if f∈ Lip-1/2 in [0, π], then lim N→∞ （N/2logN）^1/2 sup 0≤k≤∞|B_N（k）-B（k）|≤2^1/2 do a. s. Theorem 3. Let X={x（n）} satisfy the conditions of Theorem 1. Then lim N→∞ （N/2logN）^1/2 sup 0≤λ≤x |F_n（λ）-F（λ）|≤(2^1/2+2) c_π a. s. where c_π²=2π integral from n=0 to π f²（u）du. Finally, using a result of [5], we obtain the convergence rates of estimate of spectral function for a real linear process. Proposition. Let X={x （n）} be a real linear process, x（n）=sum from j=0 to ∞ α（j）ε（n-j） sum from j=0 to ∞|α（j）|<∞ α（0）=1 where {ε （n）} is a strictly stationary martingale difference and E(ε²（n）|（?）_n-1) =σ², Eε⁴（n）<∞; （?）_n（?）σ{ε（m）; m≤n}. If lim n→∞N^1/2 sum from k=N to ∞|α（k）|=0 Then sup 0≤λ≤x|F_N（λ） -F （λ）|=O(N^-1/2（log N）^3/2) a. s.更多还原