U-Statistic Limits for Distance Covariance

Why the Null Limit Is Not Usually a Normal Distribution

Distance covariance detects dependence that ordinary covariance can miss. Its null behavior also differs from the familiar pattern in which a suitably scaled statistic approaches a normal distribution. Under independence, the distance-covariance kernel is degenerate: its conditional expectation given one observation is zero. That removes the usual first-order fluctuation and makes the second-order structure, encoded by eigenvalues, determine the limit.

We will study a population-centered order-two U-statistic, which isolates that structure. It is unbiased for squared population distance covariance. Its null limit is generally a weighted sum of centered chi-squared variables. The weights are determined by the distance kernels and the marginal distributions, so the limit is not generally a single, distribution-free chi-squared law.

The Centered-Distance Kernel and Its U-Statistic

Let \((X,Y)\) be a random vector in \(\mathbb{R}^{p}\times\mathbb{R}^{q}\), and let \((X',Y')\) be an independent copy. Assume \(E\|X\|^2<\infty\) and \(E\|Y\|^2<\infty\). Define the population-centered distance kernels

and define \(b(y,y')\) by the same formula with \(Y\) in place of \(X\). The centering is with respect to each marginal distribution, not the joint distribution.

The expectation identity follows because every summand has the same distribution as \(a(X,X')b(Y,Y')\). Under independence, \(\mathcal V^2(X,Y)=0\). The important additional fact is conditional: for a fixed observation \(z=(x,y)\), independence of \(X'\) and \(Y'\) gives

Each factor is zero by population centering. This is the order-two degeneracy condition under the null. It is not the statement that each observed summand is zero; the summands fluctuate, but their conditional mean given either observation is zero.

From Degeneracy to the Spectral Limit

Let \(P\) denote the distribution of \(Z=(X,Y)\). The kernel \(g\) is symmetric and square-integrable under the stated second-moment assumptions. Its integral operator on \(L^2(P)\) is

This is a compact self-adjoint operator. Write its nonzero eigenvalues, repeated according to multiplicity, as \(\lambda_1,\lambda_2,\ldots\), with corresponding orthonormal eigenfunctions \(e_1,e_2,\ldots\). Square-integrability ensures \(\sum_r\lambda_r^2<\infty\). Degeneracy means the constant function is in the null space and the nonzero eigenfunctions can be taken to have mean zero.

Proof. First truncate the spectral expansion at a finite \(K\), writing \(g_K(z,z')=\sum_{r=1}^K\lambda_r e_r(z)e_r(z')\). Its U-statistic satisfies the exact identity

For fixed \(K\), the multivariate central limit theorem applies to the vectors \((e_1(Z_i),\ldots,e_K(Z_i))\): their means are zero and their covariance matrix is the identity by orthonormality. Thus the vector of sums divided by \(\sqrt n\) converges in distribution to \(K\) independent standard normals. Also, the law of large numbers gives \(n^{-1}\sum_i e_r(Z_i)^2\xrightarrow{p}1\) for each \(r\leq K\). Slutsky’s theorem therefore yields

It remains to remove the truncation. The residual kernel \(g-g_K\) is still canonical, meaning its conditional expectation given either argument is zero. For any square-integrable canonical symmetric kernel \(q\), two different unordered-pair summands \(q(Z_i,Z_j)\) and \(q(Z_k,Z_l)\) have zero covariance: disjoint pairs are independent, and for pairs sharing one index the covariance is zero by conditioning on that observation. Consequently,

Here \(U_n(q)\) denotes the pair average formed from \(q\). Since \(E[(g-g_K)^2]=\sum_{r>K}\lambda_r^2\), the mean-square contribution of the residual is bounded by \(\frac{2n}{n-1}\sum_{r>K}\lambda_r^2\), which tends to zero uniformly for \(n\geq2\) as \(K\) tends to infinity. The limiting series also converges in mean square, since its terms are independent, centered, and have total variance \(2\sum_r\lambda_r^2\). The finite-\(K\) limits therefore extend to the full kernel, proving the result. \(\square\)

Why Distance Covariance Has Nonnegative Weights

For Euclidean distance, the centered distance kernel defines a negative semidefinite integral operator: its eigenvalues are nonpositive. One way to see the relevant property is that Euclidean distance is conditionally negative definite: for any finite points \(x_i\) and real coefficients \(c_i\) summing to zero, \(\sum_{i,j}c_ic_j\|x_i-x_j\|\leq0\). Centering removes the constant component, giving the corresponding nonpositive quadratic form for the operator.

Under independence, the operator for \(g\) is the product of the marginal centered-distance operators. If \(\alpha_r\leq0\) and \(\beta_s\leq0\) are their respective eigenvalues, the product eigenfunctions have eigenvalues \(\lambda_{rs}=\alpha_r\beta_s\geq0\). Thus the distance-covariance null limit has the form \(\sum_{r,s}\lambda_{rs}(\xi_{rs}^2-1)\), with nonnegative weights. There may be many nonzero weights, rather than just one.

Unbiased and Biased Statistics: A Crucial Distinction

The U-statistic above uses population centering, so its definition involves unknown marginal expectations. Practical sample distance covariance typically estimates the centering from the data and includes diagonal terms; this is a V-statistic. Under the null, the corresponding scaled V-statistic has an uncentered weighted limit of the form \(\sum_r\lambda_r\xi_r^2\), whereas the canonical unbiased U-statistic has the centered form \(\sum_r\lambda_r(\xi_r^2-1)\). The subtraction of \(1\) reflects the removal of diagonal contributions. These limits are related but not interchangeable.

This distinction explains a useful diagnostic: the population squared distance covariance is nonnegative, but an unbiased finite-sample U-statistic need not be. A negative value is not by itself evidence of an error; it is compatible with unbiased estimation of a nonnegative quantity. It does, however, rule out treating the unbiased statistic as if every finite-sample value were a squared norm.

Common Mistake

• Wrong reasoning: “Squared distance covariance is nonnegative, so its unbiased estimator must be nonnegative and have a nonnegative chi-squared limit.”
• Correction: Nonnegativity applies to the population quantity (and to the usual biased squared-distance V-statistic), not necessarily to its unbiased U-statistic estimate. Under independence, degeneracy gives the centered terms \(\lambda_r(\xi_r^2-1)\), which can be negative even when every weight is nonnegative.