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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Kolmogorov extension theorem ： ウィキペディア英語版 Kolmogorov extension theorem In mathematics, the Kolmogorov extension theorem (also known as Kolmogorov existence theorem or Kolmogorov consistency theorem) is a theorem that guarantees that a suitably "consistent" collection of finite-dimensional distributions will define a stochastic process. It is credited to the Soviet mathematician Andrey Nikolaevich Kolmogorov. ==Statement of the theorem== Let $T$ denote some interval (thought of as "time"), and let $n\; \backslash in\; \backslash mathbb$. For each $k\; \backslash in\; \backslash mathbb$ and finite sequence of times $t\_,\; \backslash dots,\; t\_\; \backslash in\; T$, let $\backslash nu\_\}$ be a probability measure on $(\backslash mathbb^)^$. Suppose that these measures satisfy two consistency conditions: 1. for all permutations $\backslash pi$ of $\backslash$ and measurable sets $F\_\; \backslash subseteq\; \backslash mathbb^$, :$\backslash nu\_\}\; \backslash left(\; F\_\; \backslash times\; \backslash dots\; \backslash times\; F\_\; \backslash right)\; =\; \backslash nu\_\}\; \backslash left(\; F\_\; \backslash times\; \backslash dots\; \backslash times\; F\_\; \backslash right);$ 2. for all measurable sets $F\_\; \backslash subseteq\; \backslash mathbb^$,$m\; \backslash in\; \backslash mathbb$ :$\backslash nu\_\}\; \backslash left(\; F\_\; \backslash times\; \backslash dots\; \backslash times\; F\_\; \backslash right)\; =\; \backslash nu\_\; t\_,\; \backslash dots\; ,\; t\_\}\; \backslash left(\; F\_\; \backslash times\; \backslash dots\; \backslash times\; F\_\; \backslash times\; \backslash mathbb^\; \backslash times\; \backslash dots\; \backslash times\; \backslash mathbb^\; \backslash right).$ Then there exists a probability space $(\backslash Omega,\; \backslash mathcal,\; \backslash mathbb)$ and a stochastic process $X\; :\; T\; \backslash times\; \backslash Omega\; \backslash to\; \backslash mathbb^$ such that :$\backslash nu\_\}\; \backslash left(\; F\_\; \backslash times\; \backslash dots\; \backslash times\; F\_\; \backslash right)\; =\; \backslash mathbb\; \backslash left(\; X\_,\; \backslash dots,\; X\_\; \backslash right)$ for all $t\_\; \backslash in\; T$, $k\; \backslash in\; \backslash mathbb$ and measurable sets $F\_\; \backslash subseteq\; \backslash mathbb^$, i.e. $X$ has $\backslash nu\_\}$ as its finite-dimensional distributions relative to times $t\_\; \backslash dots\; t\_$. In fact, it is always possible to take as the underlying probability space $\backslash Omega\; =\; (\backslash mathbb^n)^T$ and to take for $X$ the canonical process $X\backslash colon\; (t,Y)\; \backslash mapsto\; Y\_t$. Therefore, an alternative way of stating Kolomogorov's extension theorem is that, provided that the above consistency conditions hold, there exists a (unique) measure $\backslash nu$ on $(\backslash mathbb^n)^T$ with marginals $\backslash nu\_\}$ for any finite collection of times $t\_\; \backslash dots\; t\_$. Kolmogorov's extension theorem applies when $T$ is uncountable, but the price to pay for this level of generality is that the measure $\backslash nu$ is only defined on the product σ-algebra of $(\backslash mathbb^n)^T$, which is not very rich. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Kolmogorov extension theorem」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.