## On some singular integrals and their applications to problems of statistical estimation.(English. Ukrainian original)Zbl 0939.62099

Theory Probab. Math. Stat. 53, 19-31 (1996); translation from Teor. Jmovirn. Mat. Stat. 53, 17-28 (1995).
Singular integrals $$I(\phi)$$ with linked arguments of the form $\int_{R^d}\dots\int_{R^d} K_1(x_1-x_2)K_2(x_2-x_3)\dots K_{m-1}(x_{m-1}-x_m)K_m(x_m-x_1) \phi(x_1,\dots,x_m)dx_1\dots dx_m,$ are investigated, where $$K_i: R^d\to R$$ are $$\delta$$-like kernels, and $$\phi$$ is a function which satisfies $$|\phi(x_1,\dots,x_m)|\leq|\phi_0(x_1,\dots,x_m)|\prod_{j=1}^m |\phi_j(x_j)|.$$ The following estimates for $$|I(\phi)|$$ are obtained in terms of $$L_p$$-norms of functions $$K_i(x)$$ and $$\phi_i(x):$$ $|I(\phi)|\leq \prod_{j=1}^m \|K_j\|_{L_q} \prod_{j=1}^m \|\phi_j\|_{L_p} \|\phi_0\|_\infty,\;\;\forall p\in[1,2], \;\;p^{-1}+q^{-1}=1.$ Conditions for the convergence $$I(\phi)\to 0$$ as $$K_j$$ tends to the Dirac $$\delta$$-function are obtained. These results are used to prove the asymptotic normality of the empirical correlograms of homogeneous random fields as well as the asymptotic normality of some estimates of the correlation functions of homogeneous isotropic random fields.

### MSC:

 62M40 Random fields; image analysis 62G20 Asymptotic properties of nonparametric inference 60F17 Functional limit theorems; invariance principles