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The law of the iterated logarithm over a stationary Gaussian sequence of random vectors. (English) Zbl 0941.60041

Let \(\{X_j\}_{j \geq 1}\) be a stationary Gaussian sequence of \({\mathbb R}^d\)-valued vectors with mean zero. Set \[ L_n = (2n \log \log n)^{1/2} \sum_{j=1}^n (G(X_j) - E[G(X_j)]) \] and \[ \sigma^2 = \lim_{n \to \infty} n^{-1} E\Biggl[\biggl(\sum_{j=1}^n (G(X_j) - E[G(X_j)])\biggl)^2\Biggr], \] where \(G: {\mathbb R}^d \to {\mathbb R} \) is a certain function. Theorem 1 establishes that \(\limsup_{n \to \infty} L_n \leq \sigma\) a.s. under the condition \[ \sum_{k = 0}^{\infty} |E[X_1^{(p)}X_{k+1}^{(q)}]|< \infty \text{ for } 1 \leq p,q \leq d,\tag{1} \] and the requirement on \(G\) in terms of coefficients of its Fourier-Hermite expansion. Theorem 2 weakens (1) when the known Hermitean rank of \(G\) is not less than 1. These theorems improve the result by H.-C. Ho [J. Theor. Probab. 8, No. 2, 347-360 (1995; Zbl 0832.60041)] given for \(d=1\). Their counterparts, Theorems 3 and 4, refer to sequences of vectors having a moving average representation \(X_j^{(p)} = \sum_{k=-\infty}^{\infty} \sum_{q=1}^d b_{j+k}(p,q)g_{k,q}\), where \(\{ (b_j(p,q)), 1 \leq p,q \leq d, -\infty < j < \infty \}\) are matrices and \(\{(g_{j,p}), -\infty < j < \infty, 1 \leq p \leq d\}\) are i.i.d. standard normal r.v.’s. Conditions on \(b_j\) and function \(G\) are given implying that with probability one \(\{L_n\}_{n \geq 1}\) is compact and its limit set is \([-\sigma,\sigma]\).

MSC:

60F05 Central limit and other weak theorems

Citations:

Zbl 0832.60041
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