Representations and properties of weight functions in Tauberian theorems.(Ukrainian, English)Zbl 1199.60176

Teor. Jmovirn. Mat. Stat. 77, 64-81 (2007); translation in Theory Probab. Math. Stat. 77, 71-90 (2008).
Let $$\xi(t),t\in\mathbb R^n$$, be a real-valued mean-square continuous homogeneous and isotropic random field with mean value $$E\xi(t)=0$$ and covariance function $$\mathbb B_n(r) =\mathbb B_n(\| t\| ) = E\xi(0)\xi(t)$$. It is known that there exists a bounded nondecreasing function $$\Phi(x), x\geq0$$, called the spectral function of the field, such that $\mathbb B_n(r)=2^{(n-2)/2}\Gamma(\frac{n}{2})\int_0^{\infty}\frac{J_{(n-2)/2}(rx)}{(rx)^{(n-2)/2}}d\Phi(x),$ where $$J_{\nu}(z)$$ is the Bessel function of the first kind and of order $$\nu>-1/2$$ (see, for example, M. I. Yadrenko [Spectral theory of random fields, Springer-Verlag (1983; Zbl 0539.60048)]). In the above mentioned book some asymptotic properties of the variance of integrals of random fields $b_n(r)=D\left[\int_{v (r)}\xi(t)dt \right] =(2\pi)^nr^{2n}\int_0^{\infty}\frac{J^2_{n/2}(rx)}{(rx)^{n}}d\Phi(x),$ where $$v(r)=\{t\in\mathbb R^n: | t| \leq r\}$$ is a ball in $$\mathbb R^n$$, are studied in terms of both covariance and spectral functions. The asymptotic behaviour of the function $$\Phi_a(\lambda):=\Phi(a+\lambda)-\Phi(a)$$ as $$\lambda\to0$$ is investigated by A. Ya. Olenko [Teor. Jmovirn. Mat. Stat. 74, 81–97 (2006); translation in Theory Probab. Math. Stat. 74, 93–111 (2007; Zbl 1150.60027)]. In this paper the Tauberian theorem is formulated in terms of the functionals $\tilde{b}^a(r)=(2\pi)^{n}\int_0^{\infty}\frac{J^2_{n/2}(rx)}{(rx)^{n}}d\Phi_a(x).$ It is also shown that there exists a real-valued function $$g_{n,r,a}(| t| )$$ such that $\tilde{b}^a(r)= D\left[\int_{\mathbb R^n} g_{n,r,a}(| t| ) \xi(t)dt \right].$ The author of the paper under review continues the studies of weight functions in Tauberian theorems for random fields. He investigates the rate of convergence of function series in the representation of the function $$g_{n,r,a}(| t| )$$. Maple 9.5 is also used to check all numerical results.

MSC:

 60G60 Random fields 62E20 Asymptotic distribution theory in statistics 40E05 Tauberian theorems

Citations:

Zbl 0539.60048; Zbl 1150.60027

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