Large deviations for random matricial moment problems

Fabrice Gamboa, Jan Nagel, Alain Rouault, Jens Wagener

Abstract: We consider the moment space MnK<math><msubsup is="true"><mrow is="true"><mi mathvariant="script" is="true">M</mi></mrow><mrow is="true"><mi is="true">n</mi></mrow><mrow is="true"><mi is="true">K</mi></mrow></msubsup></math> corresponding to p×p<math><mi is="true">p</mi><mo is="true">×</mo><mi is="true">p</mi></math> complex matrix measures defined on K<math><mi is="true">K</mi></math> (K=[0,1]<math><mi is="true">K</mi><mo is="true">=</mo><mrow is="true"><mo is="true">[</mo><mn is="true">0</mn><mo is="true">,</mo><mn is="true">1</mn><mo is="true">]</mo></mrow></math> or K=T<math><mi is="true">K</mi><mo is="true">=</mo><mi mathvariant="double-struck" is="true">T</mi></math>). We endow this set with the uniform distribution. We are mainly interested in large deviation principles (LDPs) when n→∞<math><mi is="true">n</mi><mo is="true">→</mo><mi is="true">∞</mi></math>. First we fix an integer k<math><mi is="true">k</mi></math> and study the vector of the first k<math><mi is="true">k</mi></math> components of a random element of MnK<math><msubsup is="true"><mrow is="true"><mi mathvariant="script" is="true">M</mi></mrow><mrow is="true"><mi is="true">n</mi></mrow><mrow is="true"><mi is="true">K</mi></mrow></msubsup></math>. We obtain an LDP in the set of k<math><mi is="true">k</mi></math>-arrays of p×p<math><mi is="true">p</mi><mo is="true">×</mo><mi is="true">p</mi></math> matrices. Then we lift a random element of MnK<math><msubsup is="true"><mrow is="true"><mi mathvariant="script" is="true">M</mi></mrow><mrow is="true"><mi is="true">n</mi></mrow><mrow is="true"><mi is="true">K</mi></mrow></msubsup></math> into a random measure and prove an LDP at the level of random measures. We end with an LDP on Carathéodory and Schur random functions. These last functions are well connected to the above random measure. In all these problems, we take advantage of the so-called canonical moments technique by introducing new (matricial) random variables that are independent and have explicit distributions.