Let X be a d-dimensional random vector with unknown density function f(z) = f (z1, ..., z(d)), and let f(n) be teh nearest neighbor estimator of f proposed by Loftsgaarden and Quesenberry (1965). In this paper, we est...Let X be a d-dimensional random vector with unknown density function f(z) = f (z1, ..., z(d)), and let f(n) be teh nearest neighbor estimator of f proposed by Loftsgaarden and Quesenberry (1965). In this paper, we established the law of the iterated logarithm of f(n) for general case of d greater-than-or-equal-to 1, which gives the exact pointwise strong convergence rate of f(n).展开更多
We consider the convolution transforms of measures on R<sup>d</sup> defined by some approximate identity. We shall establish some relations between the irregular boundary properties of the convolution func...We consider the convolution transforms of measures on R<sup>d</sup> defined by some approximate identity. We shall establish some relations between the irregular boundary properties of the convolution function and the local Lipschitz exponent of the measure. In particular, the results can be applied to the Poisson and Gauss-Weierstrass kernels. We can then obtain some singular boundary behavior of positive harmonic or parabolic functions on R<sub>+</sub><sup>d+1</sup> by multifractal analysis of measures.展开更多
基金Research supported by National Natural Science Foundation of China.
文摘Let X be a d-dimensional random vector with unknown density function f(z) = f (z1, ..., z(d)), and let f(n) be teh nearest neighbor estimator of f proposed by Loftsgaarden and Quesenberry (1965). In this paper, we established the law of the iterated logarithm of f(n) for general case of d greater-than-or-equal-to 1, which gives the exact pointwise strong convergence rate of f(n).
文摘We consider the convolution transforms of measures on R<sup>d</sup> defined by some approximate identity. We shall establish some relations between the irregular boundary properties of the convolution function and the local Lipschitz exponent of the measure. In particular, the results can be applied to the Poisson and Gauss-Weierstrass kernels. We can then obtain some singular boundary behavior of positive harmonic or parabolic functions on R<sub>+</sub><sup>d+1</sup> by multifractal analysis of measures.
