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71.
72.
Stamatis Cambanis 《Journal of Mathematical Analysis and Applications》1973,41(3):603-620
Series representations are obtained for the entire class of measurable, second order stochastic processes defined on any interval of the real line. They include as particular cases all earlier representations; they suggest a notion of “smoothness” that generalizes well-known continuity notions; and they decompose the stochastic process into two orthogonal parts, the smooth part and a strongly discontinuous part. Also linear operations on measurable, second order processes are studied; it is shown that all “smooth” linear operations on a process, and, in particular, all linear operations on a “smooth” process, can be approximated arbitrarily closely by linear operations on the sample paths of the process. 相似文献
73.
Let $s_n(f,z):=\sum_{k=0}^{n}a_kz^k$ be the $n$th partial sum of
$f(z)=\sum_{k=0}^{\infty{}}a_kz^k$. We show that $\RE s_n(f/z,z)>0$ holds for all $z\in\D,\ n\in\N$, and all starlike functions $f$ of order
$\lambda$ iff $\lambda_0\leq\lambda<1$ where
$\lambda_0=0.654222...$ is the unique solution
$\lambda\in(\frac{1}{2},1)$ of the equation
$\int_{0}^{3\pi/2}t^{1-2\lambda}\cos t \,dt=0$. Here $\D$ denotes
the unit disk in the complex plane $\C$. This result is the best
possible with respect to $\lambda_0$. In particular, it
shows that for the Gegenbauer polynomials $C_{n}^{\mu}(x)$ we
have $\sum_{k=0}^n C_{k}^{\mu}(x)\cos k \theta>0$ for all
$n\in\N,\ x\in[-1,1]$, and
$0<\mu\leq\mu_0:=1-\lambda_0=0.345778...$. This result complements
an inequality of Brown, Wang, and Wilson (1993) and extends a
result of Ruscheweyh and Salinas (2000). 相似文献
74.
A random vector is said to have a 1-symmetric distribution if its characteristic function is of the form φ(|t1| + … + |tn|). 1-Symmetric distributions are characterized through representations of the admissible functions φ and through stochastic representations of the radom vectors, and some of their properties are studied. 相似文献
75.
The theory of elliptically contoured distributions is presented in an unrestricted setting, with no moment restrictions or assumptions of absolute continuity. These distributions are defined parametrically through their characteristic functions and then studied primarily through the use of stochastic representations which naturally follow from the work of Schoenberg [5] on spherically symmetric distributions. It is shown that the conditional distributions of elliptically contoured distributions are elliptically contoured, and the conditional distributions are precisely identified. In addition, a number of the properties of normal distributions (which constitute a type of elliptically contoured distributions) are shown, in fact, to characterize normality. 相似文献
76.