162 RANDOM NUMBERS 3.5 sequence is more random (Adelphia web hosting)
162 RANDOM NUMBERS 3.5 sequence is more random than 000000000. Although it is true that truly random sequences will exhibit locally nonrandom behavior, we would expect such behavior only in a long finite sequence, not in a short one. Several ways for defining the randomness of a finite sequence have been proposed, and only a few of the ideas will be sketched here. Let us consider only the case of bary sequences. Given a bary sequence X1, X2, . 7 X,, we can say that . if I V(N)IN -PI 5 l/JN, where v(n) is the quantity appearing in Definition A at the beginning of this section. The above sequence can be called k-distributed if Pr(X,X,+, . . .Xn+kpI = ~~22 . . . zk) z l/bk for all bary numbers z1z2 . . . xk. (Cf. Definition D; unfortunately a sequence may be k-distributed by this new definition when it is not (k -1)-distributed.) A definition of randomness may now be given analogous to Definition Rl, as follows: Definition &I. A b-ary sequence of length N is random if it is k-distributed (in the above sense) for all positive integers k 5 log, N. According to this definition, for example, there are 170 nonrandom binary sequences of length 11: 00000001111 10000000111 11000000011 11100000001 00000001110 10000000110 11000000010 11100000000 00000001101 10000000101 11000000001 10100000001 00000001011 10000000011 01000000011 01100000001 00000000111 plus 01010101010 and all sequences with nine or more zeros, plus all sequences obtained from the preceding sequences by interchanging ones and zeros. Similarly, we can formulate a definition for finite sequences analogous to Definition R6. Let A be a set of algorithms, each of which is a combination selection and choice procedure that gives a subsequence (X,,)R, as in the proof of Theorem M. Definition Q2. The b-ary sequence XI, X2, . . . , X, is (n, c)-random with respect to a set of algorithms A, if for every subsequence X,,, X,,, . . . , Xt, determined by an algorithm of A we have either m < n or for 0 < a < b. Here 21,(x1,. . . , x,) is the number of a s in the sequence x1, . . . , xm.