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Nawaf Bou-Rabee
Nawaf Bou-Rabee
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Distribution and moments/moments of transformed normally distributed random vector

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YCor
YCor
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distribution Distribution and moments of normally distributed random vector

Let $\epsilon \sim N\left ( 0,I_{k} \right )$$\varepsilon \sim N\left ( 0,I_{k} \right )$, consider the following function of $\epsilon$$\varepsilon$,

$y=\left ( A+B\epsilon \epsilon {}'B{}' \right )^{^{\frac{1}{2}}}\epsilon $$y=\left ( A+B\varepsilon \varepsilon {}'B{}' \right )^{^{\frac{1}{2}}}\varepsilon $,

where $A$ is a positive semi-definite symmetric $k\times k$ matrix, $B$ is a $k\times k$ matrix. I want to find the distribution or the approximate distribution of $y$, or some moments of $y$, such as $E\left ( yy{}' \right )$, $E\left ( yy{}'\otimes yy{}' \right )$.

distribution and moments of normally distributed random vector

Let $\epsilon \sim N\left ( 0,I_{k} \right )$, consider the following function of $\epsilon$,

$y=\left ( A+B\epsilon \epsilon {}'B{}' \right )^{^{\frac{1}{2}}}\epsilon $,

where $A$ is a positive semi-definite symmetric $k\times k$ matrix, $B$ is a $k\times k$ matrix. I want to find the distribution or the approximate distribution of $y$, or some moments of $y$, such as $E\left ( yy{}' \right )$, $E\left ( yy{}'\otimes yy{}' \right )$.

Distribution and moments of normally distributed random vector

Let $\varepsilon \sim N\left ( 0,I_{k} \right )$, consider the following function of $\varepsilon$,

$y=\left ( A+B\varepsilon \varepsilon {}'B{}' \right )^{^{\frac{1}{2}}}\varepsilon $,

where $A$ is a positive semi-definite symmetric $k\times k$ matrix, $B$ is a $k\times k$ matrix. I want to find the distribution or the approximate distribution of $y$, or some moments of $y$, such as $E\left ( yy{}' \right )$, $E\left ( yy{}'\otimes yy{}' \right )$.

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Zhizhan Zen
Zhizhan Zen
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distribution and moments of normally distributed random vector

Let $\epsilon \sim N\left ( 0,I_{k} \right )$, consider the following function of $\epsilon$,

$y=\left ( A+B\epsilon \epsilon {}'B{}' \right )^{^{\frac{1}{2}}}\epsilon $,

where $A$ is a positive semi-definite symmetric $k\times k$ matrix, $B$ is a $k\times k$ matrix. I want to find the distribution or the approximate distribution of $y$, or some moments of $y$, such as $E\left ( yy{}' \right )$, $E\left ( yy{}'\otimes yy{}' \right )$.