I am studying Monte-Carlo simulations using quasi random numbers and encounter the following problem: I am given a set of 1D quasi-random numbers $(X_i)$ over $[0,1)$, and would like to generate another set of 1D quasi-random numbers $(Y_i)$ over $[0,1)$, that jointly form a set $(X_i,Y_i)$ of quasi-random numbers over $[0,1)^2$. In other words, I would like Y to be "independent" of X. This is trivial with pseudo-random numbers, but for quasi numbers this is usually not the case. In fact, Y may be exactly the same as X if one generates Y using the same algorithm as X.
More generally, how do I sequentially generate sets of independent quasi-numbers? This is necessary for simulation a Brownian motion, but so far I have not seen how this can be done. Even if one could generate 2 set of independent quasi numbers, I don't see how one could generate indefinitely many sets. Any ideas?
Thank you for your inputs.
quasi-randoms have to be created with a dimension in mind. Thus you need to develop a vector of pairs of quasi-randoms. Quasi random generators generally come with an ability to specify the dimensionality.
For developing independent sets, the usual solutions are randomized quasi Monte Carlo and scrambled Quasi Monte Carlo. Here you add pseudo-random uniforms to get independence. One uniform per dimension set.