On the Hedenmalm-Shimorin estimate of short anti diagonals

Based on analysis of pairs of jointly Gaussian Analytic functions of Dirichlet type, Hedenmalm and Shimorin obtained very recently a sharp estimate of short antidiagonals for contrative operators on Hilbert space. In this talk, I will present an elementary and direct proof of such estimates and generalize the estimates to higher order tensors. We find also an application of our work in the following problem: consider three sequences of random variables $(X_n), （Y_n), (Z_n)$, each of which is a sequence of i.i.d. Gaussian random variables， but no a priori knowledge is known on their joint distribution, what can we say about the joint moments $E(X_i Y_j Z_k)$? This talk is based on a joint work with Yong HAN and Zipeng WANG.