Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 225 / 3

Because a strictly monotone map fixing both endpoints is increasing, the change of variables formula $t=h(s)$ givesAssuming $\mathbb{E}\Vert X{\Vert }^2>0$, defineThe Hilbert-space variance identity then gives

Under the regularity needed to interchange expectation and differentiation, $\mathbb{E}{h}^{\prime }(s)=1$ because $\mathbb{E}h(s)=s$. HenceThus $A=1$ exactly when the integrated covariance in the numerator vanishes. In particular, this holds when the amplitude $X$ and the time warp $h$ are independent random variables.

For these increasing warps, the square-root velocity function is $Q(h)(t)=\sqrt{h^{\prime }(t)}$. The chain rule givesThereforewhere the last equality again uses the change of variables formula. Taking square roots proves invariance under common right composition by $\gamma$.

For positive trace-class covariance operators $C_i=L_i{L}_i^{\ast }$, their Procrustes distance between covariance operators iswhere $\mathcal{O}(H)$ is the group of unitary operators on the real Hilbert space $H=L^2[0,1]$. Expanding the square gives the equivalent formula

Convergence in the Hilbert-Schmidt norm impliesProducts of two Hilbert-Schmidt operators are trace-class operators, and the Schatten norm Hölder inequality givesBy trace duality, every unitary $R$ satisfiesTaking the supremum over $R$ shows that the supremum terms in the two Procrustes formulas converge. Combining this with convergence of the squared norms proves