• 자연과학논문집
• /
• 제9권1호
• /
• pp.57-60
• /
• 1997

{$X_{ni}$,1$\leq$i$\leq$,n$\geq$1}은 2p차 적률을 갖는 적당한 확률변수 X에 의해서 유계된 바나하 공간상의 값을 갖는 확률변수 열이다. 상수 열 {$a_{ni}$,1$\leq$i$\leq$,n$\geq$1}에 적당한 조건을 부여할 때 $sum_{i=1}^n a_{ni}X_{ni}$가 0에 확률적으로 수렴할 조건과 완전수렴할 조건은 서로 동치이다.

• 대한수학회보
• /
• 제53권5호
• /
• pp.1531-1548
• /
• 2016

Let C[0, t] denote the space of real-valued continuous functions on [0, t] and define a random vector $Z_n:C[0,t]{\rightarrow}\mathbb{R}^n$ by $Z_n(x)=(\int_{0}^{t_1}h(s)dx(s),{\ldots},\int_{0}^{t_n}h(s)dx(s))$, where 0 < $t_1$ < ${\cdots}$ < $ t_n=t$ is a partition of [0, t] and $h{\in}L_2[0,t]$ with $h{\neq}0$ a.e. Using a simple formula for a conditional expectation on C[0, t] with $Z_n$, we evaluate a generalized analytic conditional Wiener integral of the function $G_r(x)=F(x){\Psi}(\int_{0}^{t}v_1(s)dx(s),{\ldots},\int_{0}^{t}v_r(s)dx(s))$ for F in a Banach algebra and for ${\Psi}=f+{\phi}$ which need not be bounded or continuous, where $f{\in}L_p(\mathbb{R}^r)(1{\leq}p{\leq}{\infty})$, {$v_1,{\ldots},v_r$} is an orthonormal subset of $L_2[0,t]$ and ${\phi}$ is the Fourier transform of a measure of bounded variation over $\mathbb{R}^r$. Finally we establish various change of scale transformations for the generalized analytic conditional Wiener integrals of $G_r$ with the conditioning function $Z_n$.