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Consider $X=X_r+jX_i$ to be a circularly symmetric Gaussian random variable, and $$X\sim\mathcal{N}_c(0,\sigma^2)$$ where $\mathcal{N_c}$ denotes the Gaussian distribution for complex random variable. Here $X_r$ and $X_i$ are the real and the imaginary parts of $X$. Due to circularly symmetry assumption, zero mean random variables $X_r$ and $X_i$ are independent and identically distributed (i.i.d) with each $$X_r\sim\mathcal{N_r}(0,\sigma^2/2)$$ $$X_i\sim\mathcal{N_r}(0,\sigma^2/2)$$ where $\mathcal{N_r}$ denotes the Gaussian distribution for real random variable. Now, I want to calculate Variance of $$\mathbb{Var}[\min(|X|,\eta)]$$

This Question I already asked but for Real random variable. Moments of min of a random variable and a constant.. Let $z_{\eta}\equiv(\eta-m)/\sigma$. Then \begin{align} \mathbb{E}[X\wedge \eta]=&E[X1\{X\le \eta\}]+\eta \mathbb{E}[1\{X>\eta\}] \\ =&m\Phi(z_{\eta})-\sigma\phi(z_{\eta})+\eta(1-\Phi(z_{\eta})), \end{align} \begin{align} \mathbb{E}[(X\wedge \eta)^2]=&\mathbb{E}[X^21\{X\le \eta\}]+\eta^2 \mathbb{E}[1\{X>\eta\}] \\ =&(m^2+\sigma^2)\Phi(z_{\eta})-\sigma(m+\eta)\phi(z_{\eta})+\eta^2(1-\Phi(z_{\eta})), \end{align} and $$ Var(X\wedge \eta)=\mathbb{E}[(X\wedge \eta)^2]-(\mathbb{E}[X\wedge \eta])^2. $$

I want to know what will be the changes if Random variable is complex

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Marcus
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  • Naturally there is no ordering for complex number, so perhaps you need to re-define your question – BGM Dec 21 '16 at 20:31
  • Thanks. Now I have redefined my question. I hope now it make sense. – Marcus Dec 26 '16 at 09:50
  • I do not see the differences yet. As $X$ is complex, how do you define the order of complex number? You need to define the relations $>$, $<$ for the minimum function.... – BGM Dec 26 '16 at 09:57
  • I don't understand which relation you mean for minimum function. What I want to say is here $\eta$ is constant and $X$ is complex number. So in $\min(X,\eta)$ how should I define the relations $>$, $<$ for the minimum function. – Marcus Dec 26 '16 at 10:13
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    How do you define $a + bi > c + di$? If this is undefined then $\min(X, \eta)$ is also undefined – BGM Dec 26 '16 at 11:18
  • Sorry for my mistake now I changed it to $min(|X|,\eta)$. I hope now it make sense. – Marcus Dec 26 '16 at 12:21

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