Convergence rates of the LIL for random fields in Hilbert spaces

Rok, strany: 2011, 275 - 288

Considering the positive d$-dimensional lattice point $Z₊^d$ ($d≥2$) with partial ordering $≤$, let $\{X_{\mathbf{k}}: {\mathbf{k}}\in Z₊^d\}$ be i.i.d. random variables taking values in a real separable Hilbert space $({\mathbf{H}},|·|)$ with mean zero and covariance operator $Σ$, and set $S_{\mathbf{n}}=∑_{{\mathbf{k}} ≤{\mathbf{n}}}X_{\mathbf{k}}$, ${\mathbf{n}}\in Z₊^d$. Let $σ_i²$, $i≥1$, be the eigenvalues of $Σ$ arranged in the non-increasing order and taking into account the multiplicities. Let $l$ be the dimension of the corresponding eigenspace, and denote the largest eigenvalue of $Σ$ by $σ²$. Let $ log x=\ln(x\vee e)$, $x≥0$. This paper studies the convergence rates for $∑_{\mathbf{n}} \frac{( log log|\mathbf{n}|)^b}{|\mathbf{n}| log|\mathbf{n}|} \mathsf{P}\b(|S_\mathbf{n}|≥σε \sqrt{2|\mathbf{n}| log log|\mathbf{n}|} \bb)$. We show that when $l≥2$ and $b>-l/2$, $\mathsf{E}[|X|²( log|X|)^d-2 ( log log|X|)^b+4]<∞$ implies \begin{align*} &\lim_{ε\searrow\sqrt{d-1}}(ε²-d+1)^b+l/2 ∑_{\mathbf{n}} \frac{( log log|\mathbf{n}|)^b}{|\mathbf{n}| log|\mathbf{n}|} \mathsf{P}\b(|S_\mathbf{n}|≥σε \sqrt{2|\mathbf{n}| log log|\mathbf{n}|} \bb) {}={}&\frac{K(Σ) (d-1)^{((l-2) / (2))}Γ(b+l/2)} {Γ(l/2)(d-1)!}, \end{align*} where $Γ(·)$ is the Gamma function and $K(Σ)=\prod_i=l+1^∞((σ²-σ_i²)/σ²)^-1/2$.

ISO 690:

Fu, K., Yang, X. 2011. Convergence rates of the LIL for random fields in Hilbert spaces. In Mathematica Slovaca, vol. 61, no.2, pp. 275-288. 0139-9918. DOI: https://doi.org/10.2478/s12175-011-0011-8

APA:

Fu, K., Yang, X. (2011). Convergence rates of the LIL for random fields in Hilbert spaces. Mathematica Slovaca, 61(2), 275-288. 0139-9918. DOI: https://doi.org/10.2478/s12175-011-0011-8