. . . (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