Path-Transformation

A Path-Transformation is a Random Walk on a integer lattice in which the transformed walk follows the same law as the original walk.

• • …
• Counter-Example(s):
  • Transformation Function,
  • Brownian Motion.
• See: Pitman's Representation Theorem, Random Walk, Brownian Motion, Weyl Chamber, Young Tableau, Robinson-Schensted Correspondence, RSK, Markov Functions, Hermitian Brownian Motion, Random Matrices.

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References

2003

• (O'Connell) ⇒ Neil O'Connell (2003). "A Path-Transformation for Random Walks and the Robinson-Schensted Correspondence". Transactions of the American Mathematical Society, 355(9), 3669-3697.
  • QUOTE: In O'Connell and Yor (2002) a path-transformation $G$ was introduced with the property that, for $X$ belonging to a certain class of random walks on the integer lattice, the transformed walk $G(X)$ has the same law as that of the original walk conditioned never to exit a type-A Weyl chamber. In this paper, we show that $G$ is closely related to the Robinson-Schensted algorithm, and use this connection to give a new proof of the above representation theorem. The new proof is valid for a larger class of random walks and yields additional information about the joint law of $X$ and $G(X)$. The corresponding results for the Brownian model are recovered by Donsker's theorem. These are connected with Hermitian Brownian motion and the Gaussian Unitary Ensemble of random matrix theory. The connection we make between the path-transformation $G$ and the RS algorithm also provides a new formula and interpretation for the latter. This can be used to study properties of the RS algorithm and, moreover, extends easily to a continuous setting.

2002

• (O'Connell & Yor, 2002) ⇒ Neil O'Connell and Marc Yor. “A Representation for Non-colliding Random Walks". Elect. Commun. Probab. 7 (2002) 1-12.</ref>

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