We study certain infinite-dimensional probability measures in connection with frame analysis. Earlier work on frame-measures has so far focused on the case of finite-dimensional frames. We point out that there are good reasons for a sharp distinction between stochastic analysis involving frames in finite vs infinite dimensions. For the case of infinite-dimensional Hilbert space $\mathcal{H}$, we study three cases of measures. We first show that, for $\mathcal{H}$ infinite dimensional, 1 one must resort to infinite dimensional measure spaces which properly contain $\mathcal{H}$. The three cases we consider are: (i) Gaussian frame measures, (ii) Markov path-space measures, and (iii) determinantal measures.
Infinite Random Matrices and Ergodic decomposition of Finite or Infinite Hua-Pickrell measures
研究了无限埃尔米特矩阵空间上一类Hua-Pickrell测度族的遍历分解.首先，我们证明了Hua-Pickrell概率测度的遍历分量不含高斯因子，推广了AlexeiBorodin和GrigoriOlshanski的一个结果.其次，我们证明了Hua-Pickrell随机矩阵的渐近特征值序列在一定意义下是平衡的，并且有一个与遍历分量的$γ_1$参数一致的“主值“.这使得我们能够完成Borodin和Olshanski关于Hua-Pickrell概率测度遍历分解的描述。最后，我们将上述结果推广到了无限Hua-Pickrell测度的情形.本文利用A. I. Bufetov，我们能够在$\mathbb{R}^*$中的点配置空间上识别Hua-Pickrell无穷测度到某些显式$σ$ -有限行列式测度的遍历分解.本文解决了Borodin和Olshanski的一个问题。
A Palm hierarchy for determinantal point processes with the confluent hypergeometric kernel, the decomposing measures in the problem of harmonic analysis on the infinite-dimensional unitary group
