Are the digits of $\pi$ statistically random? Suppose you observe the sequence:
7, 9, 0, 5, 5, 5, 4, 8, 0, 6, 9, 5, 3, 8, 7, 8, 5, 4, 0, 0, 6, 6, 4, 5
, 3, 3, 7, 5, 9, 8, 1, 8, 6, 2, 8, 4, 6, 4, 1, 9, 9, 0, 5, 2, 2, 0, 4, 5, 2, 8 ...
What statistically tests would you apply to determine if this is truly random? 
FYI these are the $n$th digits of $\pi$. Thus, are digits of $\pi$ statistically random? Does this say anything about the constant $\pi$?

 A: It's a strange question. Numbers aren't random.
As a time series of base 10 digits, $\pi$ is completely fixed.
If you are talking about randomly selecting an index for the time series, and picking that number, sure it's random. But so is the boring, rational number $0.1212121212\ldots$. In both cases, the "randomness" comes from picking things at random, like drawing names from a hat.
If what you're talking about is more nuanced, as in "If I sequentially reveal a possibly random sequence of numbers, could you tell me if it's a fixed subset from $\pi$? And where did it come from?". Well first, though $\pi$ is not repeating, different random sequences will at least locally align for a small run. That's a number theory result, not a statistical one. As soon as you break, you have to scan on to the next instance of alignment. Computationally it's not tractable to align any random sequence because $\pi$ could match up to the $2^{2^{2^2}}+1$-th place. Heck even if the sequence did align with $\pi$ somewhere, doesn't mean it's not random. For instance, I could choose 3 at random, doesn't mean it's the first digit of $\pi$.
A: The US National Institute of Standard has put together a battery of tests that a (pseudo-)random number generator must pass to be considered adequate, see http://csrc.nist.gov/groups/ST/toolkit/rng/stats_tests.html. There are also tests known as the Diehard suite of tests, which overlap somewhat with NIST tests. Developers of Stata statistical package report their Diehard results as a part of their certification process. I imagine you can take blocks of digits of $\pi$, say in groups of consecutive 15 digits, to be comparable to the double type accuracy, and run these batteries of tests on thus obtained numbers.
