Take a time series $S_t = \left\{ {S_1 ,S_2 ,S_3 ,...,S_n } \right\}$, where $S_t$ are a sequence of numerical data points in successive order, occurring in uniform intervals of time "t". In this case, how can I verify analytically if $ \left\{ {S_1 ,S_2 ,S_3 ,...,S_n } \right\}$ are iid (independent and identically distributed) ?
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For discrete observations, one usually checks the i.i.d. property on long sequences through counts of motives (letters and words of length two, at least). Local counts of letters, averaged on mesoscopic blocks must remain roughly constants, while counts of 2-words must correspond to the products of probabilities of their letters. – Did Apr 02 '16 at 12:58
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If $(S_1,\ldots,S_n)$ is a single realization, you cannot infer much about the underlying distributions from which the $S_i$ came from.
parsiad
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Ok, thanks. This is exactly point that I have doubt. Thereby, in which case would make sense to say if the time series $S_t$ is iid? – user327010 Mar 31 '16 at 14:38
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It would not make sense if $(S_1,\ldots,S_n)$ is a single realization. – parsiad Mar 31 '16 at 14:45
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As an example, think of an experiment in which you flip two independent fair coins. A possible outcome is $(H,T)$. Now, think of an experiment in which you flip a fair coin, and if it is heads, you flip an unfair coin (otherwise flip a fair coin). Again, a possible outcome is $(H,T)$. Given just $(H,T)$, you cannot determine if the underlying distributions are independent. – parsiad Mar 31 '16 at 14:50