(This article was first published on

**Xi'an's Og » R**, and kindly contributed to R-bloggers)**T**he 24 questions asked by John Halton in the conclusion of his 1970 survey are

- Can we obtain a theory of convergence for random variables taking values in Fréchet spaces?
- Can the study of Monte Carlo estimates in separable Fréchet spaces give a theory of global approximation?
- When sampling functions, what constitutes a representative sample of function values?
- Can one apply Monte Carlo to pattern recognition?
- Relate Monte Carlo theory to the theory of random equations.
- What can be said about quasi-Monte Carlo estimates for finite-dimensional and infinite-dimensional integrals?
- Obtain expression, asymptotic forms or upper bounds for L² and L
^{∞}discrepancies of quasirandom sequences. - How should one improve quasirandom sequences?
- How to interpret the results of statistical tests applied to pseudo- or quasirandom sequences?
- Can we develop a meaningful statistical theory of quasi-Monte Carlo estimates?
- Can existing Monte Carlo techniques be improved and applied to new classes of problems?
- Can the design of Monte Carlo estimators be made more systematic?
- How can the idea of sequential Monte Carlo be extended?
- Can sampling with signed probabilities be made practical?
- What is the best allocation effort in obtaining zeroth- and first-level estimators in algebraic problems?
- Examine the Monte Carlo analogues of the various matrix iterative schemes.
- Develop the schemes of grid refinement in continuous problems.
- Develop new Monte Carlo eigenvectors and eigenvalue techniques.
- Develop fast, reliable true canonical random generators.
- How is the output of a true random generator to be tested?
- Develop fast, efficient methods for generating arbitrary random generators.
- Can we really have useful general purpose pseudorandom sequences.
- What is the effect of the discreteness of digital computers on Monte Carlo calculations?
- Is there a way to estimate the accuracy of Monte Carlo estimates?

Filed under: Books, R, Statistics, University life Tagged: Fréchet, low discrepancy sequences, Monte Carlo methods, pseudo-random generator, quasi-random sequences, random generator

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