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| Format: | Preprint |
| Published: |
2024
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| Online Access: | https://arxiv.org/abs/2402.01563 |
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| _version_ | 1866913668967432192 |
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| author | Shklyar, Sergiy |
| author_facet | Shklyar, Sergiy |
| contents | This paper establishes the conditions of existence of a stationary solution to the first order autoregressive equation on a plane as well as properties of the stationarity solution. The first-order autoregressive model on a plane is defined by the equation
$X_{i,j} = a X_{i-1,j} + b X_{i,j-1} + c X_{i-1,j-1} + ε_{i,j}.$
A stationary solution $X$ to the equation exists if and only if $(1-a-b-c) (1-a+b+c) (1+a-b+c) (1+a+b-c) > 0$. The stationary solution $X$ satisfies the causality condition with respect to the white noise $ε$ if and only if $1-a-b-c>0$, $1-a+b+c>0$, $1+a-b+c>0$ and $1+a+b-c>0$. A sufficient condition for X to be purely nondeterministic is provided.
An explicit expression for the autocovariance function of $X$ at some points is provided. With Yule-Walker equations, this allows to compute the autocovariance function everywhere. In addition, all situations are described where different parameters determine the same autocovariance function of $X$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_01563 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | First-order planar autoregressive model Shklyar, Sergiy Probability 60G60 (Primary) 62M10 (Secondary) This paper establishes the conditions of existence of a stationary solution to the first order autoregressive equation on a plane as well as properties of the stationarity solution. The first-order autoregressive model on a plane is defined by the equation $X_{i,j} = a X_{i-1,j} + b X_{i,j-1} + c X_{i-1,j-1} + ε_{i,j}.$ A stationary solution $X$ to the equation exists if and only if $(1-a-b-c) (1-a+b+c) (1+a-b+c) (1+a+b-c) > 0$. The stationary solution $X$ satisfies the causality condition with respect to the white noise $ε$ if and only if $1-a-b-c>0$, $1-a+b+c>0$, $1+a-b+c>0$ and $1+a+b-c>0$. A sufficient condition for X to be purely nondeterministic is provided. An explicit expression for the autocovariance function of $X$ at some points is provided. With Yule-Walker equations, this allows to compute the autocovariance function everywhere. In addition, all situations are described where different parameters determine the same autocovariance function of $X$. |
| title | First-order planar autoregressive model |
| topic | Probability 60G60 (Primary) 62M10 (Secondary) |
| url | https://arxiv.org/abs/2402.01563 |