Probability theory of {nα}
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Mon, 25/10/2010 15:45 |
Istvan Berkes (Graz University of Technology) |
Stochastic Analysis Seminar  |
Eagle House |
| The sequence {nα}, where α is an irrational number and {.} denotes fractional part, plays a fundamental role in probability theory, analysis and number theory. For suitable α, this sequence provides an example for "most uniform" infinite sequences, i.e. sequences whose discrepancy has thesmallest possible order of magnitude. Such 'low discrepancy' sequences have important applications in Monte Carlo integration and other problems of numerical mathematics. For rapidly increasing nk the behaviour of {nkα} is similar to that of independent random variables, but its asymptotic properties depend strongly also on the number theoretic properties of nk, providing a simple example for pseudorandom behaviour. Finally, for periodic f the sequence f(nα) provides a generalization of the trig-onometric system with many interesting properties. In this lecture, we give a survey of the field (going back more than 100 years) and formulate new results. | |||