I understand, How Cholesky decomposition can be used to generate correlated random numbers. But unable to understand, why and how does it allow for correlation?
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An excellent exposition of this topic is in Brian Ripley: "Stochastic Simulation". There are two main methods for generating multivariate normal random numbers: conditioning and the Cholesky method. To understand how they works, it helps to understand that in a time series setting, they correspond to an autoregressive (AR) representation (conditioning) and an moving average (MA) representaion (cholesky), respectively. Write out the details of that and you will understand how they generate correations!
answered Jul 11, 2013 at 10:23