Following the (Vasicek 2002) paper, on page 2, it is specified that if the variables $X^i$ are jointly standard normal with equal pairwise correlations $rho$ they can be represented as:

$X^i = Ysqrt{rho} + Z^isqrt{1 – rho}$

where $Y$ and $Z^1…Z^n$ are mutually independent standard normal variables. The paper states that

**"This is a property of the equicorrelated normal distribution."**

How can I derive this property?

I tried finding the correlation of two variables $X^i$ and $X^j$ constructed in that way, but I don't get $rho$ as the result.

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#### Best Answer

*Which* property is being called out as a property enjoyed by jointly normal standard normal random variables? That they have a representation as $$X_i = sqrt rho, Y + sqrt{1-rho},Z_i tag{1}$$ where $Y$ and the $Z_i$ are independent standard normal random variables? The joint normality of the $X_i$ follows from the fact that they are obtained as linear combinations of jointly normal random variables. That the $X_i$ have expected value $0$ follows straightforwardly from the *linearity of expectation* applied to $(1)$ while bilinearity of covariance and independence gives begin{align} operatorname{cov}(X_i,X_j) &= operatorname{cov}left(sqrt rho, Y + sqrt{1-rho},Z_i,sqrt rho, Y + sqrt{1-rho},Z_jright)\ &= rhooperatorname{cov}(Y,Y) + 0 + 0 + (1-rho)operatorname{cov}(Z_i,Z_j)\ &= rho + begin{cases}1-rho, & i=j,\ 0, &i neq j, end{cases}\ &= begin{cases}1, & i=j,\ rho, &i neq j, end{cases} end{align} showing that the $X_i$ have unit variance and correlation coefficient as desired.

Note that all this works only when $rho geq 0$. The common correlation $rho$ satisfies $-frac{1}{n-1}leq rho leq 1$ but the construction above does not work for $-frac{1}{n-1}leq rho < 0$. For a construction of unit random variables with common correlation coefficient $-frac{1}{n-1}$, see this answer of mine.

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