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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?
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.
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