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I was reading this article related to autoregressive processes of order $1$. According to wiki it is given by

$$ x_t = \phi{x_{t-1}} + \epsilon \\ |\phi| < 1 \\ x_t|x_1,\ldots,x_{t-1} \sim N(\phi{x_{t-1},1}) $$

I didn't get what this function $$\phi$$ is and how come the conditional distribution has mean equal to $$\phi{x_{t-1}}$$

Also I was referring to this book Gaussian Markov Random Fields Theory and Applications and they have assumed that the marginal distribution of $x_1$ is normal with mean zero and variance $$\frac1{1-\phi^2}$$ and modeled the joint density of $x$,

$$\pi(x)=\pi(x_1)\pi({x_2|x_1})\cdots\pi(x_n|x_{n-1})$$ to be gaussian distribution with precision matrix given by

$$Q = ( (1) (-\phi) \cdots\cdots)$$ $$( (-\phi) ({1 + \phi^2}) (-\phi)\cdots\cdots)$$ and so on I didn't get how this precision matrix was obtained. Any clarifications?

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