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Suppose a source produces an indefinite sequence of positive integers. How can you check whether the numbers are generated truly randomly?

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What does that mean? –  Qiaochu Yuan Mar 12 '11 at 16:15
    
Do you mean to ask what I mean by 'random'? –  dbrane Mar 12 '11 at 16:22
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I can't comment, so... I assume you've seen this [previous question][1]? [1]: math.stackexchange.com/questions/6196 –  Sergio Leone Mar 12 '11 at 16:28
    
That's actually precisely the kind of thing I was looking for. Thanks! –  dbrane Mar 12 '11 at 16:30
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See: kodyaz.com/forums/22308/ShowThread.aspx –  Lie Ryan Mar 12 '11 at 16:41
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marked as duplicate by Oliver, How About a Nice Big Cup of, RecklessReckoner, voldemort, Umberto P. Apr 12 at 3:23

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2 Answers

There are two answers.

In classical probability theory, the question doesn't even make sense. From the usual perspective of probability theory, if I roll a fair die, I get a "random number" from 1 to 6, but none of those numbers is "random" on its own. "Randomness" here corresponds to the process of obtaining a measurement; it's a property of a random variable, not the property of a particular value I measure from the random variable. So I roll the die over and over and get "1,1,1,1,1,...", that's still the outcome of a random variable, and in this sense that sequence was still "generated randomly". Individual measurements are not random on their own, and so any sequence of numbers from 1-6 could be generated randomly by rolling a fair die.

There is a separate theory, called "Kolmogorov complexity" or "algorithmic randomness", which can be used to measure "how random" certain objects are, but the meaning of "random" here is not the same. Instead, a sequence of numbers is called "algorithmically random" if it satisfies a large collection of randomness properties (more formally, the sequence is random if it does not lie in any effective $G_\delta$ set of measure 0). This area is also well studied, and a lot is known about the sequences that are random in this sense. But a "random process", such as repeatedly rolling a fair die, is perfectly capable of generating a sequence that is not algorithmically random.

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Given some time dependent data-set, autocorrelation can be used to detect randomness and lack thereof. Take a look at for example http://en.wikipedia.org/wiki/Correlogram

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