autocorrelation for truncated normal distribution

Mathematics Asked by bearcub on August 17, 2020

I have a (symmetric) truncated normal distribution from which I’m sampling. There is also auto-correlation of the values, so I know:
$mu$, $sigma$, $text {autocorr} = a$.

All three of these values are single-valued. E.g., $mu_R = 1$, $sigma_R = 1$, $a_R = 0.5$.

My understanding of the application of autocorrelation to a series, $R sim N (mu_R=1,sigma_R=1)$, with the normal distribution results in the new series $S$ as:

$ S(1)=R(1) $
$S(i)=a_Rcdot R(i-1)+w(i)$,
where $w(i)sim N(0,sigma_w)$.

(taken from these notes –

However, I have also seen this formulation:

$R sim N (mu_R=1,sigma_R=1)$,
$ b = sqrt{1-a_R^2} $,
$ S(1)=R(1) $,
$ S(i)=a_Rcdot S(i-1)+bcdot R(i) $.

Where does the $b$ come from?

Why are we scaling the second term instead of drawing from $w(i)$?

Why is the autocorrelation coefficient recursively applied via the first term? (e.g., $S(2)approx a_Rcdot S(1)$, $S(3)approx a_Rcdot S(2)approx a_Rcdotbig(a_Rcdot S(1)big)$)

The only example I’ve found is this: How to transform/shift the mean and standard deviation of a normal distribution?

But it doesn’t fully explain these questions.

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