UDC 519.95:519.211:511

MATHEMATICS

V. N. AGAFONOV

NORMAL SEQUENCES AND FINITE AUTOMATA

(Presented by Academician P. S. Novikov on 15 V 1967)

In a number of works \((^{1-6})\) infinite binary sequences that are “random” in one sense or another are studied. The notion of a random sequence considered in \((^{1-3})\) is Wald’s \((^2)\) refined notion of a collective, introduced by von Mises \((^1)\). In \((^{5,6})\) normal sequences are studied. In the present note the connection between these notions is clarified in terms of finite automata.

1. Collective. We shall call a strategy a predicate defined on the set of all finite binary sequences (including the empty sequence \(\Lambda\)). A strategy \(f\) defines the following procedure for selecting from an infinite binary sequence \(x_1x_2x_3\ldots\) a subsequence (possibly finite) \(y_1y_2y_3\ldots\). If \(z_1z_2z_3\ldots\) is the sequence of values of the strategy \(f\) corresponding to the sequence \(x_1x_2x_3\ldots\): \(z_1=f(\Lambda)\), \(z_2=f(x_1)\), \(z_3=f(x_1x_2),\ldots\), and \(i_1<i_2<i_3<\cdots\) are the numbers of those places for which \(z_{i_k}=1\), then \(y_1=x_{i_1}\), \(y_2=x_{i_2}\), \(y_3=x_{i_3},\ldots\). The sequence \(x_1x_2x_3\ldots\) is called a collective with respect to a set of strategies \(F\), containing the strategy identically equal to one, if for all infinite subsequences selected from \(x_1x_2x_3\ldots\) by strategies of the set \(F\), there exists one and the same limit of the relative frequency of ones

\[ p=\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^{n}x_i \quad (0<p<1). \]

2. Normal sequence. An infinite binary sequence \(x_1x_2x_3\ldots\) is called normal if there exists a number \(p\) \((0<p<1)\) such that for every positive integer \(n\) and every word \(a_1a_2\ldots a_n\) of length \(n\) in the alphabet \(\{0,1\}\), the following condition is satisfied: in the sequence of words
\(x_1x_2\ldots x_n,\ x_{n+1}x_{n+2}\ldots x_{2n},\ x_{2n+1}x_{2n+2}\ldots x_{3n},\ldots\)
the relative frequency of occurrence of the word \(a_1a_2\ldots a_n\) has a limit equal to \(p^k(1-p)^{n-k}\), where \(k=\sum_{i=1}^{n}a_i\) is the number of ones in the word \(a_1a_2\ldots a_n\).

3. Normal sequence as a collective. In \((^7)\) it is established that a sequence is normal if and only if it is a collective with respect to the set of strategies possessing the following property: the strategy selects or does not select the next symbol of the sequence depending on what the preceding \(l\) symbols are, where for the given strategy the number \(l\) is fixed.

It is clear that for every strategy \(f\) of this kind there exists a finite automaton \(\langle \{0,1\}, Q, Q^*, \varphi, q\rangle\) with input alphabet \(\{0,1\}\), set of states \(Q\), subset of selected states \(Q^*\subseteq Q\), transition function \(\varphi: Q\times\{0,1\}\to Q\), and initial state \(q\in Q\), which computes the strategy \(f\) in the following way:
\[ f(x_1x_2\ldots x_n)=1 \Longleftrightarrow \varphi(\varphi(\ldots \varphi(\varphi(q,x_1),x_2)\ldots),x_n)\in Q^* . \]
Hence it follows that a collective with respect to the set of strategies computable by finite automata,

is a normal sequence. The natural question arises whether the converse assertion is true. An affirmative answer to it is given by the following

Theorem. A normal sequence is a collective with respect to the set of strategies computable by finite automata.

The proof of the theorem uses the ergodic property of a finite automaton to whose input a sequence of independent random variables is fed, and the following “nonincrease of measure” property possessed by the transformation defined by a strategy on words. Let the \(p\)-weight of a word \(a_1 \ldots a_n\) be the number \(p^l(1-p)^{n-l}\), where \(l\) is the number of ones in the word \(a_1 \ldots a_n\) and \(0 < p < 1\), and let the \(p\)-weight of a subset of words of fixed length be the sum of the \(p\)-weights of all words in this subset. Then for any strategy \(f\) and any word \(a_1 \ldots a_k\), the set of words of length \(n \geq k\) from which \(f\) selects words beginning with the word \(a_1 \ldots a_k\) has \(p\)-weight not exceeding the \(p\)-weight of the word \(a_1 \ldots a_k\).

In conclusion I express my gratitude to Prof. B. A. Trakhtenbrot, who proposed the problem and made a number of valuable comments.

Institute of Mathematics
Siberian Branch of the Academy of Sciences of the USSR

Received
14 V 1967

REFERENCES

1. P. Mises, Probability and Statistics, Moscow, Gosizdat, 1930.
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