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One-to-one correspondences between discrete multivariate stationary, self-similar and stationary increment fields

Marko Voutilainen

Abstract

In this article, we consider three important classes of $n$-variate fields indexed by the set of $N$ dimensional integers, namely stationary, stationary increment and self-similar fields. These classes are connected through bijective transformations. In addition, we introduce generalized AR$(1)$ type equations, whose unique stationary solutions are obtained via these transformations. Lastly, we apply the transformations in order to construct stationary fractional Ornstein-Uhlenbeck fields of the first and second kind.

One-to-one correspondences between discrete multivariate stationary, self-similar and stationary increment fields

Abstract

In this article, we consider three important classes of -variate fields indexed by the set of dimensional integers, namely stationary, stationary increment and self-similar fields. These classes are connected through bijective transformations. In addition, we introduce generalized AR type equations, whose unique stationary solutions are obtained via these transformations. Lastly, we apply the transformations in order to construct stationary fractional Ornstein-Uhlenbeck fields of the first and second kind.
Paper Structure (6 sections, 14 theorems, 118 equations)

This paper contains 6 sections, 14 theorems, 118 equations.

Table of Contents

  1. Introduction
  2. Results
  3. Notation
  4. Multivariate stationary, self-similar and stationary increments fields
  5. Multivariate stationary fractional Ornstein-Uhlenbeck fields
  6. Proofs and auxiliaries

Key Result

Theorem 2.7

Let $\Theta\in(S_+^n)^N_c$. If $X$ is stationary, then $\mathcal{L}_\Theta X$ is $\Theta$-self-similar. Conversely, if $Y$ is $\Theta$-self-similar, then $\mathcal{L}^{-1}_\Theta Y$ is stationary. Moreover, for all stationary $X$ and $\Theta$-self-similar $Y$.

Theorems & Definitions (47)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Theorem 2.7
  • Definition 2.8
  • Definition 2.9
  • Remark 2.10
  • ...and 37 more