Classic Problems of Probability

Problem 22

Borel and A Different Kind of Normality (1909)

Problem. A real number in [0,1] is said to be normal¹ in a given base if any finite pattern of digits occurs with the same expected frequency when the number is expanded in that particular base. Thus a number in [0,1] is normal in base 10 if, in its decimal expansion, each of the digits {0,1, . . ., 9} occurs with frequency 1/10, each of the pairs {00, 01, . . ., 99} occurs with frequency 1/100, and so on. Prove that almost every real number in [0,1] is normal for all bases.

Solution. Let the random variable ( = 0, 1, . . ., 9; j = 1, 2, . . .) be the jth digit in the decimal expansion of , that is

Then, for the digit b = 0, 1, . . ., 9, we define the binary random variable

It is reasonable to assume that the s are independent (across the j 's)² and identically distributed. Applying the Strong Law of Large Numbers (SLLN),³ we ...