A record is an event which hasn’t happened before. Climate, weather, human feats, athletics etc are all examples of phenomena where records occur. The theory of random events has been restricted to the study of independent and identically distributed random variables, this essay hopes to cover the study of non-i.i.d random variables, with some added findings of its own.
Abstract
Previous studies of record events in a time series of random variables is surprisingly limited to those which are independent and identically distributed. Few have gone beyond this setting. This research project combines two recent studies into the statistics of records for random variables that have been rounded to a specified accuracy and those that have been drawn from the same underlying probability distribution with additive linear drift.
For the former, a formula has been added, representing the probability that two adjacent random variables in a given time series are both records. Their relevant correlations have also been touched upon and interestingly, rounding down causes i.i.d random variables to lose their independence. Ad ditionally, the project extends previous work to a non-i.i.d framework.
For the latter, I have analysed the consequences of applying both drift and rounding to a time series. Ultimately, this can be characterised by which universality class of extreme value statistics the probability distribution belongs to. The additive term increases the record rate whilst rounding has the opposite effect. The interrelation between them shows interesting results and one finds the additive drift contribution is the strongest.
The full paper, I’ve written can be found here. Feel free to contact me if there is anything unclear.
