Suppose, there are $n$ vectors $\mathbf{X}_1$, $\mathbf{X}_2 \ldots \mathbf{X}_n$ of unequal lengths which can be combined to a new vector as $$ \mathbf{X} = \begin{bmatrix} \mathbf{X}_1 & \mathbf{X}_2 & \cdots & \mathbf{X}_n \end{bmatrix} $$ For each $\mathbf{X}_i$, there is a corresponding vector $\mathbf{A}_i$ such that $$ \mathbf{A} = \begin{bmatrix} \mathbf{A}_1 & \mathbf{A}_2 & \cdots & \mathbf{A}_n \end{bmatrix} $$ It follows that one can write, $$ \mathbf{A}^\mathrm{T}\mathbf{X} = \sum_{i=1}^n \mathbf{A}_i^\mathrm{T}\mathbf{X}_i $$ Also, let $\mathbf{\Lambda}$ be a corresponding block diagonal matrix, $$\mathbf{\Lambda} = \begin{bmatrix} \mathbf{\Lambda}_1 & 0 & \cdots & 0 \\ 0 & \mathbf{\Lambda}_2 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & \mathbf{\Lambda}_n \\ \end{bmatrix}$$ It again follows that one can write, $$ \mathbf{X}^\mathrm{T}\mathbf{\Lambda}^{-1}\mathbf{X} = \sum_{i=1}^n \mathbf{X}_i^\mathrm{T}\mathbf{\Lambda}_i^{-1}\mathbf{X}_i $$ For an application, we are required to calculate the cumulative distribution function of the normal $\mathbf{\Phi} \left( \mathbf{A}^{\mathrm{T}} \mathbf{X} \left[k + \mathbf{X}^\mathrm{T}\mathbf{\Lambda}^{-1}\mathbf{X}\right]^{-1/2} \right)$ where $k$ is a constant. Due to constraints (not mentioned here), we have to approximate the cdf as product of function of $\mathbf{X}_i$'s, that is, we want to find a function $f$ such that $$ \mathbf{\Phi} \left( \mathbf{A}^{\mathrm{T}} \mathbf{X} \left[k + \mathbf{X}^\mathrm{T}\mathbf{\Lambda}^{-1}\mathbf{X}\right]^{-1/2} \right) \approx \prod_{i=1}^{n}f\left( \mathbf{X}_i, \mathbf{A}_i, \mathbf{\Lambda}_i \right) $$ Any suggestions or thoughts would be greatly helpful.
Thanks.