Standardizing a multivariate normal random vector

Problem

\(X=\left( X_1, \ldots ,X_k \right)\) is a \(k×1\) random vector following a multivariate normal, \(X \sim N\left( μ, Σ \right)\) where \(μ=E\left( X \right)\) is \(k×1\) and \(Σ\) is the \(k×k\) variance (or variance-covariance) matrix of \(X\) . Each random variable \(X_i\) follows a normal distribution with expected value \(μ_i\) and variance \(Σ_{i,i}\) and the covariances \(Cov\left( X_i,X_j \right)\) are equal to \(Σ_{i,j}\) .

We define \(Σ^{-1/2}\) as the inverse of \(Σ^{1/2}\) , which in turn is defined by \(Σ^{1/2}Σ^{1/2}=Σ\) . \(Σ^{-1/2}\) is \(k×k\) and symmetric and always exist if \(Σ\) is positive definite. The standard power rules apply:

• \(Σ^{-1/2}Σ^{-1/2}=Σ^{-1}\)
• \(Σ^{-1/2}Σ^{1/2}=I\)

Show that

\[Σ^{-1/2}(X-μ) \sim N(0,I)\]

Learning point:

• For a random variable \(X\) following normal distribution, \(X \sim N\left( μ,σ^2 \right)\) , we have the important result that we can normalize it:

\[ \frac{X-μ}{σ} \sim N\left( 0,1 \right)\]

• The multivariate result corresponding to this is

\[Σ^{-1/2}(X-μ) \sim N(0,I)\]

• We can write the single variable result as

\[{\left( σ^2 \right)}^{-1/2}(X-μ) \sim N\left( 0,1 \right)\]

• which makes the multivariate result a more natural extension.