How do you check if a sequence of numbers is truly random? [duplicate]

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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marked as duplicate by bwv869, user127096, RecklessReckoner, voldemort, Umberto P.Apr 12 '14 at 3:23

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
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
– Lie Ryan Mar 12 '11 at 16:41

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.