## Abstract

In Akahori and Ida (2014), p-conformal (or Parisian-conformal) maps on the triangular lattice were defined. The definition of p-conformality is nonstandard in comparison to ordinary discrete derivatives, but was seen to be natural in connection with a particular type of random walk, the Parisian random walk. In this note, we establish the fact that the only p-conformal polynomials in z and z̄ on the triangle lattice are linear combinations of 1, z and z
^{2}
−z̄, but that if one extends the notion of p-conformality to functions of two variables (the complex variable z and the time variable t) we obtain a rich class of polynomials which yield martingales when applied to the Parisian walk. These polynomials make use of a particular type of martingale transform, which is defined in the paper.

Original language | English |
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Pages (from-to) | 42-48 |

Number of pages | 7 |

Journal | Statistics and Probability Letters |

Volume | 151 |

DOIs | |

Publication status | Published - 1 Aug 2019 |

## Keywords

- Discrete complex analysis
- Martingale theory
- Random walk