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As a motivation for what we are going to discuss in this Chapter, let us recall something that we noted towards the end of Section 9.3. Let \((E, \mathcal {E})\) be a measurable space and suppose we are given a probability \(\mu _1\) on \((E,\mathcal{E})\) and, for each \(2\le j\le n\) , a probability kernel \(\mu _j\) on \(E^{j-1}\times \mathcal{E}\) . Theorem 9.3.12 (and Remark 9.3.13) then guarantees that there exists an n-dimensional random vector \(\textbf{X}=(X_1,\ldots , X_n)\) on a probability space, where each \(X_j\) is an E-valued random variable and \(\mu _1\) is the distribution of \(X_1\) , while \(\mu _j\) , for \(2\le j\le n\) , is the Regular Conditional Distribution of \(X_j\) , given \((X_1,\ldots , X_{j-1})\) .

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Infinite Products

  • 章节

  • 发表日期：

    2025年

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  • Alok Goswami,
  • B. V. Rao

摘要

As a motivation for what we are going to discuss in this Chapter, let us recall something that we noted towards the end of Section 9.3. Let \((E, \mathcal {E})\) be a measurable space and suppose we are given a probability \(\mu _1\) on \((E,\mathcal{E})\) and, for each \(2\le j\le n\) , a probability kernel \(\mu _j\) on \(E^{j-1}\times \mathcal{E}\) . Theorem 9.3.12 (and Remark 9.3.13) then guarantees that there exists an n-dimensional random vector \(\textbf{X}=(X_1,\ldots , X_n)\) on a probability space, where each \(X_j\) is an E-valued random variable and \(\mu _1\) is the distribution of \(X_1\) , while \(\mu _j\) , for \(2\le j\le n\) , is the Regular Conditional Distribution of \(X_j\) , given \((X_1,\ldots , X_{j-1})\) .