There are uncountably many binary sequences ...

There are uncountably many binary sequences whose limiting relative frequency is x Phil Pollett and Victor Scharaschkin March 21, 2015 Definition. PLet b = (b1 , b2 , . . . ) be a sequence of binary digits (bi ∈ {0, 1}). The limit limn→∞ (1/n) ni=1 bi (if it exists) is called the limiting relative frequency of b, which we denote by lrf(b). When we write lrf(b) = x we mean that lrf(b) exists and that its value is x. Remark 1. Notice that there is a connection between the notion of limiting relative frequency and natural density, for we may construct a corresponding subset A of the natural numbers by including the number i if bi = 1. Then, the limiting relative frequency of b is precisely the natural density of A. It is well known that there are sets A which do not have a natural density (for example the numbers whose binary expansions contain an odd number of digits). So, the limiting relative frequency does not exist for all binary sequences. Theorem. For each x ∈ [0, 1], there are uncountably many binary sequences whose limiting relative frequency exists and equals x. The theorem follows immediately from the following simple result. Here ⊕ denotes both addition modulo 2 (1 ⊕ 1 = 0), as well as the operation extended to addition of binary sequences: (b1 , b2 , . . .) ⊕ (c1 , c2 , . . .) = (b1 ⊕ c1 , b2 ⊕ c2 , . . .). Lemma. (a) For every x ∈ [0, 1] there is a binary sequence b with lrf(b) = x. (b) There are uncountably many binary sequences z with lrf(z) = 0. (c) If lrf(b) exists and lrf(z) = 0 then lrf(b ⊕ z) = lrf(b). Proof . (a) Let s = (s1 , s2 , . . . ) be any equidistributed sequence on [0, 1], and set bi = 1 if si ≤ x; bi = 0 otherwise. (For example, if α is any irrational number, the fractional parts of members of the sequence α, 2α, 3α, . . . are equidistributed on [0, 1].) (b) Let b be any binary sequence. Define a new sequence b∗ by ( 0 if n is not the square of an integer b∗n = bk if n = k 2 for some integer k. Pn ∗ Pb√nc √ n, and so lrf(b∗ ) = 0. If b and c are any two distinct biThen b = k=1 bk ≤ j=1 j nary sequences, the corresponding sequences b∗ and c∗ will be distinct. It follows that there 1

are uncountably many binary sequences b∗ with lrf(b∗ ) = 0. (Note that any suitably sparse subsequence for the indices, rather than the squares, will do here.) (c) Given ε > 0, we have, for all n sufficiently large, that P lrf(b) − 2ε < n1 nj=1 bj < lrf(b) + 2ε P − 2ε < n1 nj=1 zj < 2ε . Now, for any u, v ∈ {0, 1} it is clear that u − v ≤ u ⊕ v ≤ u + v. Therefore, ! ! n n n n n 1X 1X 1X 1X 1X lrf(b) − ε < bj − zj ≤ (bj ⊕ zj ) ≤ bj + zj < lrf(b) + ε, n j=1 n j=1 n j=1 n j=1 n j=1 and so lrf(b ⊕ z) = lrf(b). Remark 2. There is a natural way to define a “uniform” probability measure for binary sequences. Every u ∈ [0, 1) has a binary expansion u = 0 · b1 b2 b3 . . . . For example, 1/3 = 0 · 01010101 . . . (which, incidentally, has lrf(b) = 1/2). The binary expansion is not unique, but only because numbers like 0.01 (= 1/4), with a “finite” expansion, also admit the expansion 0.0011111 . . . : thus two binary sequences for the same real number u (one with lrf(b) = 0, and the other with lrf(b) = 1). Clearly there are only countably many of these. It is customary to take the latter expansion, the one with infinitely many trailing 1s, thereby enabling a one-to-one mapping of [0, 1) to the collection S of infinite binary sequences. The Borel subsets of [0, 1) thus provide a natural σ-algebra B for the collection of infinite binary sequences, and we may use uniform probability measure P (Lebesgue measure) on [0, 1) to measure any member of B. The probability space (S, B, P) thus provides natural probability model for an infinite sequence of coin tosses. ´ Of particular interest in this connection is a 1904 result1 of Emile Borel, which states that almost all u ∈ [0, 1) have binary expansions b with lrf(b) = 1/2. So, the Borel subsets of [0, 1) whose members have binary expansions b with (i) no lrf or (ii) a lrf not equal to 1/2, have measure 0: if you like, the chance of observing these binary sequences as the result of an infinite coin tossing experiment is 0. Finally, we note there are non-measurable subsets of [0, 1]—the Vitali sets for example— and, to be sure, their members have binary expansions. One can only speculate on whether the lrf exists for sequences which are not members of B.

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See, for example, Nillsen, Rodney (2000) “Normal Numbers without Measure Theory”, The American Mathematical Monthly, 107:7, 639–644 (and the references therein).

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