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From a old handwritten note without references cited, the first canonical process of a stochastic process $\Omega \times T \to S$ is defined as the identity mapping on $S^T$. I was wondering if there are concepts such as the second canonical process, the third canonical process, ...? Thanks and regards!

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I would call it "the canonical random element representation", rather that the process (which is VERY often a "sequence"/"collection" of random elements) or "the first ..." since it does not seem there are "second canonical ..." you often meet in math. –  Ilya Feb 20 '13 at 9:04

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