Covariance kernel representations of multidimensional second-order stochastic processes

2006 ◽  
Vol 217 (1) ◽  
pp. 82-99 ◽  
Author(s):  
C.H. Su ◽  
Didier Lucor
Keyword(s):  
Stochastic Processes ◽  
Second Order ◽  
Covariance Kernel

Basic Fourier series

1979 ◽  
Vol 369 (1736) ◽  
pp. 115-136 ◽  
Keyword(s):  
Time Series ◽  
Fourier Series ◽  
Stochastic Processes ◽  
Numerical Approximation ◽  
Basic Number ◽  
Second Order ◽  
Orthogonal Functions ◽  
Sturm Liouville

The concept of basic number is applied to the development of a simple analogue of the Sturm–Liouville system of the second order. This is then employed to deduce a family of q -orthogonal functions, which leads to a generalization of the Fourier and Fourier–Bessel expansions. The numerical approximation of basic integrals is discussed and some aspects of the evaluation of C a (q; x) are mentioned. A few of the zeros of this function are listed, and, in conclusion, an indication is given of the possibility of applying the analysis presented in this paper to thé study of stochastic processes and time-series.

Banach space valued weak second order stochastic processes

2021 ◽  
pp. 71-96
Author(s):  
Yûichirô Kakihara
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Abelian Group ◽  
Stochastic Processes ◽  
Compact Abelian Group ◽  
Covariance Function ◽  
Second Order ◽  
Locally Compact Abelian Group ◽  
Locally Compact ◽  
Content Type

Banach space valued stochastic processes of weak second order on a locally compact abelian group G G is considered. These processes are recognized as operator valued processes on G G . More fully, letting U \mathfrak {U} be a Banach space and H \mathfrak {H} a Hilbert space, we study B ( U , H ) B(\mathfrak {U},\mathfrak {H}) -valued processes. Since B ( U , H ) B(\mathfrak {U},\mathfrak {H}) has a B ( U , U ∗ ) B(\mathfrak {U},\mathfrak {U}^*) -valued gramian, every B ( U , H ) B(\mathfrak {U},\mathfrak {H}) -valued process has a B ( U , U ∗ ) B(\mathfrak {U},\mathfrak {U}^*) -valued covariance function. Using this property we can define operator stationarity, operator harmonizability and operator V V -boundedness for B ( U , H ) B(\mathfrak {U},\mathfrak {H}) -valued processes, in addition to scalar ones. Interrelations among these processes are obtained together with the operator stationary dilation.

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