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I would like to generate $n\sim 10,000$ random log-normally correlated variable $\rho$ with the following parameters:

  1. Mean of 10,000 $\rho$ is 1.
  2. Variance of 10,000 $\rho$ is 2.
  3. Correlation function $\langle\log\rho(x_1)\log\rho(x_2)\rangle=\log\left(1+\frac{1}{1+|x_2-x_1|^{1/3}}\right).$

Does any one have any idea on how to proceed?

Clarification:

  1. I said the mean of randomly generated 10,000 variables should be 1, and the square root of the mean of the squares of those randomly generated variables should be 2.
  2. $x_1$ and $x_2$ are some discrete values, in this case $x_1=1,..., 10,000$ and $x_2=1,....,10,000$.
  3. I have re-written the question in such a way that the one does not need to know what correlation length is.
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  • $\begingroup$ Some questions for clarification: Do $x_1$ and $x_2$ represent indices between 1 and 10,000? Is "correlation length" a physics term? So you're wanting to generate a time series of length 10,000? $\endgroup$ – JimB Aug 25 '16 at 14:42
  • $\begingroup$ More clarifications: The question refers to a correlation variable $\rho$, and then states that the parameter $\rho = 1$. This leaves me very confused. The next line states that the variance is 2, and defines a parameter $\sigma_\rho$ = 2. But even if $\sigma$ denotes one of the Lognormal parameters (and you fail to specify which parameterisation of several you are using for the Lognormal), in any event, the variance of a Lognormal is not equal to that parameter. Which leaves me even more confused. Then $x_1$ and $x_2$ are not defined. and the term correlation length is not defined ... etc $\endgroup$ – wolfies Aug 25 '16 at 14:50
  • $\begingroup$ Ok, I have clarified the question. $\endgroup$ – titanium Aug 25 '16 at 15:34
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    $\begingroup$ Thanks. It appears that because you've defined the desired correlation between the log of the lognormal random variables, you can equivalently state the problem as generating a multivariate normal with a specified correlation structure. $\endgroup$ – JimB Aug 25 '16 at 15:38

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