148 RANDOM NUMBERS 3.5 We can also show (Web hosting control panel)
148 RANDOM NUMBERS 3.5 We can also show that the serial correlation test is satisfied: Corollary S. If a [ 0,l) sequence is (k + 1)-distributed, the serial correlation coefficient between U, and Un+k tends to zero: lim k c UjUj+k -($ c uj)(k c Uj+k) n oo d(~cU32-(~~uj)2)(~CUjZ+k-(~Cuj+k)2) = * (All summations here are for 0 5 j < n.) Proof. By Theorem B, the quantities tend to the respective limits 4, 6, 6, 4, 4 as 72 + 00. 1 Let us now consider some slightly more general distribution properties of sequences. We have defined the notion of /c-distribution by considering all of the adjacent k-tuples; for example, a sequence is a-distributed if and only if the points @Jo, w, (Ul, u2>, 672, U3), 073, U4), P4, u51, . . . are equidistributed in the unit square. It is quite possible, however, that this can happen while alternate pairs of points (UI, Uz), (Us, U4), (Us, UG), . . . are not equidistributed; if the density of points (U2n–1, Uzn) is deficient in some area, the other points (Uz,, U2n+l) might compensate. For example, the periodic binary sequence (Xn) = o,o,o, 1, o,o,o, 1, l,l,O, 1, l,l,O, 1, o,o,o, 1, . . . , (11) with a period of length 16, is seen to be 3-distributed; yet the sequence of even- numbered elements (Xsn) = 0, 0, 0, 0, 1, 0, 1, 0, . . . has three times as many zeros as ones, while the subsequence of odd-numbered elements (Xan+i) = 0, 1, 0, 1, 1, 1, 1, 1, . . . has three times as many ones as zeros. If a sequence (Un) is oo-distributed, example (11) shows that it is not at all obvious that the subsequence of alternate terms (Uzn) = UO, U2, U4, Us, . . . will be co-distributed or even l-distributed. But we shall see that (Uzn) is, in fact, m-distributed, and much more is true. Definition E. A [ 0,l) sequence (Un) is said to be (m, k)-distributed if Pr( l 5 Umn+j < %r u2 2 Umn+j+l < 212, . . . , uk 5 Umn+j+k-1 < vk) = (211 -u1) . . . ( ?& -t k) for all choices of red numbers ur, V~ with 0 2 u7 < v, 5 1 for 1 < r 5 k, and for a1J integers j with 0 5 j < m.