Michael Kleber's comment << . . . (Pretending that pi is a string of random digits and other such falsities...)
got me thinking. Suppose we have a probability distribution D on the set S of all countable sequences of decimal digits. And suppose that for all *finite* sets of indices {i_1 < ... < i_n}, the corresponding marginal distribution is uniform on {0,...,9}^n. 1. Does it then follow that D is uniform on S ??? 1a. Is there a way countably many variables can be correlated even if no finite subset of them are? Related question: I like considering numbers' "digits" in "factorial" base. For x in [0,1], I mean by this writing x = a_1/2! + a_2/3! + ... + a_n/(n+1)! + ... where each a_n is an integer such that 0 <= a_n <= n. Luckily, 1/2! + 2/3! + 3/4! ... = 1/2 + 1/3 + 1/8 + 1/30 + 1/144 +... = 1. 2. Is pi normal to the factorial base? Here I mean only the fractional part of pi. I.e., for any finite set of indices {i_1 < i_2 < ... < i_n}, does the joint distribution on a_(k + i_1)/(k + i_1+1)! , , ... , a_(k + i_n)/(k+i_n+1)! approach a uniform cdf on [0,1]^n as k -> oo ? N.B. It's easy to check that e would definitely NOT be normal to the factorial base. --Dan