We derive a Karhunen-Loève expansion of the Gauss process $B_t - g(t)\int_0^1 g'(u)\,d B_u$, $t\in[0,1]$, where $(B_t)_{t\in[0,1]}$ is a standard Wiener process and $g:[0,1]\to R$ is a twice continuously differentiable function with $g(0) = 0$ and $\int_0^1 (g'(u))^2\,d u =1$. This process is an important limit process in the theory of goodness-of-fit tests. We formulate two special cases with the function $g(t)=\frac{\sqrt{2}}π\sin(πt)$, $t\in[0,1]$, and $g(t)=t$, $t\in[0,1]$, respectively. The latter one corresponds to the Wiener bridge over $[0,1]$ from $0$ to $0$.
Karhunen-Loève expansions of mean-centered Wiener processes
对于$γ>-{1/2}$，我们提供了加权均值中心维纳过程的Karhunen-Loève展开，定义为\[W _γ(t)=\frac{1}{\sqrt{1+2γ}}\Big\{W\big(t^{1+2γ}\big)- \int_0^1W\big(u^{1+2γ}\big)du\Big\},\] for $t\in(0,1]$。我们证明了这些展开式中的正交函数具有简单的贝塞尔函数表达式。进一步证明了$W_γ$的$L^2[0,1]$范数与加权布朗桥$t^γB(t)$的$L^2[0,1]$范数在分布上是一致的.
A quantum Karhunen-Loeve expansion and quadratic-exponential functionals for linear quantum stochastic systems
Research School of Electrical, Energy and Materials Engineering, College of Engineering and Computer Science, Australian National University（澳大利亚国立大学工程与计算机科学学院电气、能源与材料工程研究学院）
Brownian Motion with a Pulse: A Biostatistician's Guide to Diffusions, Bridges, Functional PCA, and First-Passage Models
Department of Biostatistics, University of California, Los Angeles, Los Angeles, CA, USA（加州大学洛杉矶分校生物统计学系）；Kuntuo, an IQVIA company, China（IQVIA旗下公司Kuntuo，中国）
