Scatter Versus Covariance in Heavy-Tailed Elliptical Laws

Contour Shape Is Not the Same as Covariance

An elliptical distribution can have well-defined, informative contours even when its variance is infinite. The key distinction is between scatter, which describes the shape and orientation of those contours, and covariance, which also depends on how much probability lies far from the center. A heavy tail can leave the contour geometry intact while making second moments diverge.

This distinction matters in multivariate Student models. Their matrix parameter determines contour shape, but it equals covariance only after multiplication by a degrees-of-freedom factor—and only when the degrees of freedom are large enough for that covariance to exist.

The eigenvectors of \(S\) give the principal directions of the contours. If the contour is defined by \((x-\mu)^{\mathsf T}S^{-1}(x-\mu)=c\), then its semiaxis lengths are \(\sqrt{c}\) times the square roots of the eigenvalues of \(S\). Thus eigenvalue ratios are ratios of squared semiaxis lengths, not ratios of semiaxis lengths.

Radial Moments Determine Covariance

A useful representation of a spherical vector in \(d\) dimensions is \(Z=RU\), where \(R\geq 0\) is a random radius and \(U\) is independent of \(R\) and uniform on the unit sphere. In an elliptical representation, a linear transformation maps this spherical vector into the desired contour shape.

Proof. Rotational symmetry implies \(E[U]=0\). The matrix \(E[UU^{\mathsf T}]\) is unchanged by rotations, so it must be a scalar multiple of the identity matrix \(I_d\). Its trace is \(E[\operatorname{tr}(UU^{\mathsf T})]=E[U^{\mathsf T}U]=1\), so that scalar is \(1/d\). Therefore \(E[UU^{\mathsf T}]=I_d/d\).

Because \(E[R^2]<\infty\), \(R\) is integrable as well: \(E[R]\leq \sqrt{E[R^2]}\). Independence and \(E[U]=0\) give \(E[RU]=0\), hence \(E[X]=\mu\). Independence also gives \(E[(RU)(RU)^{\mathsf T}]=E[R^2]I_d/d\). Applying the linear transformation \(S^{1/2}\) on both sides yields \(\operatorname{Cov}(X)=S^{1/2}(E[R^2]I_d/d)S^{1/2}=(E[R^2]/d)S\). \(\square\)

This result specifies exactly when scatter is proportional to covariance under this radial normalization. If \(E[R^2]=d\), then covariance equals \(S\). If \(E[R^2]\) is finite but differs from \(d\), covariance is a scalar multiple of \(S\). If \(E[R^2]=\infty\), the covariance does not exist as a finite matrix: for each coordinate \(U_i\), independence and \(E[U_i^2]=1/d\) give \(E[R^2U_i^2]=\infty\). In particular, the second moment in each nonzero direction of the whitened vector is infinite.

A scale convention is necessary when using a radial representation: replacing \(S\) by \(aS\) and \(R\) by \(R/\sqrt{a}\), for \(a>0\), leaves \(X\) unchanged. In common elliptical density parameterizations, the named scatter matrix is fixed by the density and its contour expression. The radial result then tells us how covariance relates to that parameter; it does not make the parameter a covariance matrix by definition.

Multivariate Student Distributions

For degrees of freedom \(\nu>0\), a multivariate Student distribution with location \(\mu\) and positive-definite scatter parameter \(S\) has density

The density makes the role of \(S\) in the contours explicit. To find the covariance, whiten the vector: \(Y=S^{-1/2}(X-\mu)\). Its density is proportional to \((1+\|y\|^2/\nu)^{-(\nu+d)/2}\), so its radial density is proportional to \(r^{d-1}(1+r^2/\nu)^{-(\nu+d)/2}\). In polar coordinates, the ratio giving \(E[\|Y\|^2]\) is a ratio of radial integrals. Substitution \(s=r^2/\nu\) gives, for \(k=0,2\) with \(\nu>k\),

Consequently, when \(\nu>2\), the ratio of the integral for \(k=2\) to that for \(k=0\) is

The final equality follows from \(B(a,b)=\Gamma(a)\Gamma(b)/\Gamma(a+b)\), \(\Gamma(a+1)=a\Gamma(a)\), and \(\Gamma(b+1)=b\Gamma(b)\). Rotational symmetry gives equal second moments in the \(d\) coordinates, so each has second moment \(\nu/(\nu-2)\).

To check the conditions, the radial integrand for the first absolute moment of \(Y\) behaves like \(r^{-\nu}\) as \(r\) tends to infinity. Its integral converges exactly when \(\nu>1\). The integrand for the radial second moment behaves like \(r^{1-\nu}\), which is integrable exactly when \(\nu>2\). When \(\nu>2\), the radial-angular result applies to \(Y=RU\), with \(E[R^2]=d\nu/(\nu-2)\); transforming back from \(Y\) gives the stated covariance. At \(\nu=2\), the second-moment divergence is logarithmic; for smaller \(\nu\), it also diverges.

Common Mistakes and Practical Interpretation

• Calling every scatter parameter a covariance. A matrix in an elliptical density describes its contours. It equals covariance only if the relevant second moments exist and the radial normalization gives the appropriate factor.
• Assuming a center guarantees a mean. Symmetric contours identify a location parameter, but a distribution can have too-heavy tails for its expectation to exist. For a Student distribution, the mean requires \(\nu>1\).
• Assuming a mean guarantees finite variance. The Pareto radial example has a finite mean and infinite second moment. For a Student distribution, the corresponding threshold for finite covariance is \(\nu>2\).
• Reading eigenvalues as axis lengths. For a quadratic contour based on \(S\), semiaxis lengths are proportional to square roots of its eigenvalues. Eigenvalue ratios are ratios of squared axis lengths.

Scatter remains useful when covariance is unavailable: it describes contour geometry and can be used to parameterize an elliptical model. But conclusions requiring covariance, such as variance-based summaries, need finite second moments. In a heavy-tailed model, reporting the scatter matrix without checking the tail condition can suggest that a covariance has been calculated when none exists.