UDC 519.217

MATHEMATICS

M. I. GORDIN

ON RANDOM PROCESSES GENERATED BY NUMBER-THEORETIC ENDOMORPHISMS

(Presented by Academician Yu. V. Linnik on 19 II 1968)

1. A. Rényi \((^{1})\) studied transformations of the interval \((0,1)\) defined by the formula \(Tx=\{\varphi(x)\}\), where \(\{a\}\) denotes the fractional part of the number \(a\), and \(\varphi=f^{-1}\), with the function \(f\) satisfying one of the conditions A, B and condition C. These conditions are as follows:

A. \(f\) is defined and decreases on \([1,+\infty)\), \(f(1)=1\), \(f\) is strictly positive, continuous and strictly decreasing on \([1,T)\), and \(f=0\) on \([T,+\infty)\), where \(T\) is either a natural number or \(+\infty\) (in this case it is meant that \(\lim\limits_{t\to\infty} f(t)=0\)). In addition, \(|f(t_2)-f(t_1)|\le |t_2-t_1|\), and there exists \(\lambda,\ 0<\lambda<1\), such that \(|f(t_2)-f(t_1)|\le \lambda |t_2-t_1|\) if \(1+f(2)<t_1<t_2\).

B. \(f\) is defined and increasing on \([0,+\infty)\), \(f(0)=0\), \(f\) is continuous and strictly increasing on \([0,T]\), and \(f=1\) on \([T,+\infty)\), where \(T\) is either a natural number or \(+\infty\) (in this case \(\lim\limits_{t\to\infty} f(t)=1\)). In addition, \(f(t_2)-f(t_1)\le t_2-t_1\), if \(0\le t_2<t_1\).

C.
\[ \operatorname*{ess\,sup}_{0<x<1} f_{E_n}'(x)\le C\,\operatorname*{ess\,inf}_{0<x<1} f_{E_n}'(x). \]
Here \(E_n=(\varepsilon_1,\varepsilon_2,\ldots,\varepsilon_n)\), where \(\varepsilon_i\) are numbers from the range of values of the function \([\varphi(x)]\), \(0<x<1\) (\([a]\) is the integer part of \(a\)), \(f_{E_n}(x)=f(\varepsilon_1+f(\varepsilon_2+\cdots+f(\varepsilon_n+x)\cdots))\), and the constant \(C\) depends neither on \(E_n\) nor on \(n\).

It follows from Rényi’s results that if \(f\) satisfies one of the conditions A, B and condition C, then the sequence of functions \(a(T^n x)\), \(n\ge 0\), \(a(x)=[\varphi(x)]\), generates on \((0,1)\) the entire Borel \(\sigma\)-algebra, and for the transformation T on \((0,1)\) there is an invariant measure \(\mathbf P\), having density \(p(x)\) with respect to Lebesgue measure, with
\[ 1/C\le p(x)\le C \]
almost everywhere on \((0,1)\).

V. A. Rokhlin \((^{2})\) proved that under these conditions the transformation T of the space \(X=(0,1)\) with measure \(\mathbf P\) is an exact endomorphism, i.e. that
\[ \bigcap_{k=0}^{\infty} M_k=N. \]
Here \(M_0=M\) is the \(\sigma\)-algebra of Lebesgue subsets of \((0,1)\);
\[ M_k=\mathrm T^{-k}(M) \]
is the \(\sigma\)-algebra of sets differing only by a set of measure 0 from the complete inverse image of some Lebesgue set with respect to \(T^k\); \(N\) is the \(\sigma\)-algebra containing only sets of measure 0 or 1. Hence follows the ergodicity of T, proved in \((^{1})\).

The equality
\[ \int_A Vg(x)\,\mathbf P(dx) = \int_{\mathrm T^{-1}(A)} g(x)\,\mathbf P(dx), \qquad A\in M, \]
defines a linear operator \(V\), acting in each of the spaces \(L_p(\mathbf P)\) \((1\le p\le \infty)\) with norm 1. The operator \(V\), which can be defined for any measure-preserving transformation, has a number of interesting properties. In particular, if T is an exact endomorphism of a space—

space \(X\) with measure \(\mathbf P\), then for \(g\in L_p(\mathbf P)\), \(1\le p<\infty\),

\[ V^n g \xrightarrow[L_p(\mathbf P)]{} \int_X g(x)\mathbf P(dx). \]

This relation, in the situation considered by us, can be sharpened.

Let us formulate conditions D and E:

D. All the functions \(f'_{E_n}\) have bounded variation and

\[ \sum_{E_n}\operatorname{var}(f'_{E_n})\le K, \]

where \(K\) does not depend on \(n\).

E.

\[ \operatorname{var}(f'_{E_n})\le K\int_0^1 |f'_{E_n}(x)|\,dx, \]

where \(K\) does not depend on \(E_n\) or \(n\).

Condition E is stronger than condition D, since

\[ \sum_{E_n}\int_0^1 |f'_{E_n}(x)|\,dx=1. \]

In Theorems 1, 2, 3 (see below) it is assumed that \(f\) satisfies one of conditions A, B and condition C. By \(\alpha(n)\) is denoted a function of the form \(Ae^{-\lambda n}\), where the constants \(A>0\), \(\lambda>0\) depend only on \(f\).

Theorem 1. Let \(g\) be a function of bounded variation. If condition D is satisfied, then

\[ \operatorname*{ess\,sup}_{0<x<1}\left|V^n g(x)-\int_0^1 g(u)\mathbf P(du)\right| \le \alpha(n)\operatorname{var}(g). \]

Let us now put \(X_k(x)=[\varphi(T^k x)]\) and denote by \(M_a^b\) the \(\sigma\)-algebra generated by all the functions \(X_k\), \(0\le a\le k\le b\le \infty\).

Theorem 2. Let condition D be satisfied. Then

\[ |\mathbf P(A\cap B)-\mathbf P(A)\mathbf P(B)| \le \alpha(n)\mathbf P(B), \]

if \(A\in M_0^k\), \(B\in M_{k+n}^{\infty}\), \(n\ge 1\).

Theorem 3. Let condition E be satisfied. Then

\[ |\mathbf P(A\cap B)-\mathbf P(A)\mathbf P(B)| \le \alpha(n)\mathbf P(A)\mathbf P(B), \]

if \(A\in M_0^k\), \(B\in M_{k+n}^{\infty}\), \(n\ge 1\).

Remark 1. Condition E is satisfied if, for example, \(f''\) exists and for some \(p>1\)

\[ \sup_l \int_0^1 \left|\frac{f''(x+l)}{f'(x+l)}\right|^p dx < +\infty. \]

The numbers \(l\) run through the range of values of the function \([\varphi(x)]\), \(x\in(0,1)\).

Remark 2. If \(\theta>1\), \(\theta\) nonintegral, then \(f(x)=x/\theta\) does not satisfy condition B. However, as shown in \((^1)\) and \((^3)\), the sequence \(X_k\) generates the Borel \(\sigma\)-algebra, and the transformation \(T\) has an invariant measure. For the case \(\theta>2\) one can prove analogues of Theorems 1 and 2 with \(\alpha(n)=A(2/\theta)^n\).

Remark 3. The results of Theorems 1, 3 for \(f(x)=1/x\) give estimates known in the metric theory of continued fractions \((^8)\). Under more stringent restrictions on \(f\), an estimate of order \(Ae^{-\lambda\sqrt n}\) was obtained in \((^4)\).

2. Here probabilistic notation will be used. All mathematical expectations are taken with respect to the invariant measure \(\mathbf P\). Let \(g\in L_2(\mathbf P)\), \(\mathbf E g=0\). The sequence \(g(T^k x)\), \(k\ge 0\), forms a stationary random process. A number of results are known which make it possible to apply limit theorems to similar processes

probability theory, provided only that these processes are asymptotically independent in a certain sense, as, for example, the process \(X_k\) in Theorem 2 (for the central limit theorem see, for example, (5), Ch. XVIII). However, one of the conditions for the applicability of such theorems is the relation

\[ B_n(g)=\mathbf E\left(\sum_{k=0}^{n-1} g(T^k x)\right)^2 \xrightarrow[n\to\infty]{} \infty, \]

whose verification is not always simple.

Theorem 4. If \(f\) satisfies the conditions of Theorem 1 and \(f'\) is continuous, \(g(x)=G([\varphi(x)])\), \(\mathbf E g=0\), \(\mathbf E g^2<\infty\), then

\[ B_n(g)\xrightarrow[n\to\infty]{}\infty. \]

Theorem 5. If \(f\) satisfies the conditions of Theorem 1, \(g\) is an unbounded function, \(Vg\) has bounded variation, \(\mathbf E g=0\), \(\mathbf E g^2<\infty\), then

\[ B_n(g)\xrightarrow[n\to\infty]{}\infty. \]

Let \(q_n(x)\) be the denominator of the \(n\)-th convergent of the expansion of the number \(x\) into an ordinary continued fraction. With the aid of Theorem 5, for \(f(x)=1/x\),

\[ g(x)=\ln \frac1x-\int_0^1 \ln \frac1x\,p(x)\,dx,\quad \text{where } p(x)=\frac{1}{(1+x)\ln 2}, \]

one can prove that

\[ \int_0^1\left(\ln q_n(x)-\int_0^1 \ln q_n(u)\,p(u)\,du\right)^2 p(x)\,dx \xrightarrow[n\to\infty]{} \infty. \]

Relying on this relation, one can prove the central limit theorem and the law of the iterated logarithm for \(\ln q_n\) (see (7)).

Remark 4. The result of Theorem 4 for \(f(x)=1/x\) and the above relation for \(\ln q_n\) were stated without proof by W. Doeblin (6).

In conclusion, the author expresses his gratitude to I. A. Ibragimov for posing the problem and for his attention to the work.

Leningrad State University
named after A. A. Zhdanov

Received
5 II 1968

REFERENCES

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4. Chan Ving Kh’ev, Vestn. Moskovsk. univ., ser. 1, No. 5, 28 (1963).
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6. W. Doeblin, Compositio Math., 3, 353 (1940).
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8. A. Ya. Khinchin, Continued Fractions, 1949.