# Correlated random variables

Assume we have a random variable $$X(t)$$ that changes as a function of time satisfying a correlation $$\left\langle X(0) X(\tau) \right\rangle=e^{-\tau/\tau_c}$$. Is Mathematica able to generate random numbers obeying the correlation above? Is this feature already implemented in Mathematica?

Thanks!

• Do you mean a single sample at X(t) is correlated with X(0)? Or do you mean all samples X(t) are correlated with some sample at time X(t-s) ? And what is tc ? It sounds like you want an ARProcess. – flinty Jul 4 at 16:05
• There are many different types of processes that obey this relation. This is like asking if you can generate random numbers with a given variance. It's not difficult to do, but there are many different ways to do it that are not equivalent. You could generate numbers from a normal distribution, an exponential distribution, a Laplace distribution, etc. All with the same variance, but very different results. – Sjoerd Smit Jul 4 at 16:08
• You might also be interested in OrnsteinUhlenbeckProcess because it has a covariance containing an exponential: (E^(-θ Abs[s - t]) σ^2)/(2 θ) – flinty Jul 4 at 16:17