UDC 519.210

MATHEMATICS

Academician of the Academy of Sciences of the Uzbek SSR T. A. SARYMSAKOV, Ya. Kh. KUCHKAROV

ON THE ERGODIC PRINCIPLE FOR A PARTIALLY SEMIGROUP FAMILY OF OPERATORS ON TOPOLOGICAL SEMIFIELDS

In this article we shall adhere to the definitions and notation of works \((^1,^2)\).

1. Let a family \(\{E_k,\ 1 \leq k \leq \infty\}\) be given, where \(E_k\) is a complete topological semifield; \(K_k\) is the cone of nonnegative elements in \(E_k\), and \(\nabla_k\) is the topological Boolean algebra of all idempotents, on which a certain measure is defined. Elements of \(E_k\) for which the integral with respect to this measure exists are called summable; denote their set by \(L_k\). Obviously, \(L_k\) is a linear topological space in the topology induced from \(E_k\). An element \(x \in L_k \cap K_k\) will be called a distribution if \(\mu(x)=1\), where \(\mu\) denotes the integral sign.

Introduce the notation

\[ P_k=\{x:x\in L_k\cap K_k,\quad \mu(x)=1\}. \]

Consider, generally speaking, a noncommutative family \(\{T_k^{[k+1]},\ 1\leq k\leq \infty\}\) of linear operators, where \(T_k^{[k+1]}:L_k\to L_{k+1}\) and

\[ T_k^{[m]}=T_{m-1}^{[m]}\times T_{m-2}^{[m-1]}\cdots T_k^{[k+1]}\quad (k<m). \]

Definition 1. A linear operator \(T_k^{[k+1]}\) \((1\leq k\leq \infty)\) will be called stochastic if it satisfies the following conditions:

1) \(T_k^{[k+1]}(L_k)\subset L_{k+1}\);

2) \(\mu(T_k^{[k+1]}x)=\mu(x)\) for \(x\in L_k\);

3) \(T_k^{[k+1]}x^{(n)}\to T_k^{[k+1]}x^{(0)}\), if \(x^{(n)}\to x^{(0)}\) \((x^{(0)},x^{(n)}\in L_k)\).

Here convergence is to be understood in the sense of the topology in \(E_k\).

Definition 2. A family \(\{T_k^{[k+1]},\ 1\leq k\leq \infty\}\) of stochastic operators will be called ergodic if, for any \(k\) and any \(x,y\) from \(P_k\) and \(g\in\nabla_2\), the relation

\[ \lim_{r\to\infty}\mu(gT_k^{[r]}(x-y))=0 \]

holds.

We shall say that a family \(\{\widetilde T_k^{[k+1]},\ 1\leq k\leq \infty\}\) of operators includes the family \(\{T_k^{[k+1]},\ 1\leq k\leq \infty\}\) if, for some \(i_1,i_2,\ldots,i_h,\ldots\) \((i_h=i_j\ \text{when }h=j)\),

\[ \widetilde T_{i_h}^{[i_{h+1}]}=T_h^{[h+1]}. \]

Definition 3. A family \(\{\widetilde T_k^{[k+1]},\ 1\leq k\leq \infty\}\) of stochastic operators will be called strongly ergodic if any family \(\{\widetilde T_k^{[k+1]},\ 1\leq k\leq \infty\}\) of stochastic operators that includes the family \(\{T_k^{[k+1]},\ 1\leq k\leq \infty\}\) is ergodic.

Condition A. Whatever \(x,y\) from \(P_k\) may be, there exist a natural number \(m\), elements \(z\in L_m\), \(u\) and \(v\) from \(P_m\), depending on \(x,y\), and a number \(\lambda(T_k^{[m]})\) \((0<\lambda(T_k^{[m]})<1)\) such that

\[ T_k^{[m]}x-z=(1-\lambda(T_k^{[m]}))u, \]

\[ T_k^{[m]}y-z=(1-\lambda(T_k^{[m]}))v, \]

where \(z<T_k^{[m]}x,\ z<T_k^{[m]}y,\ k=1,2,\ldots\)

1.1. If, for the family \(\{T_k^{[k+1]},\ 1\leq k\leq\infty\}\) of stochastic operators, condition (A) is satisfied, then the equality holds

\[ T_1^{[n]}(x-y)=(1-\lambda(T_1^{[m_1]}))(1-\lambda(T_{m_1}^{[m_2]}))\cdots(1-\lambda(T_{m_{h-1}}^{[m_h]}))T_{m_h}^{[m_h+r]}(u-v), \]

\[ x,y\in P_1,\qquad u,v\in P_{m_h}\qquad \text{for } n=m_1+m_2+\cdots+m_h+r,\quad r<m_{h+1}-m_h. \]

1.2. If, for the family \(\{T_k^{[k+1]},\ 1\leq k\leq\infty\}\) of stochastic operators, condition (A) is satisfied and the series

\[ \lambda(T_1^{[m_1]})+\sum_{h=2}^{\infty}\lambda(T_{m_{h-1}}^{[m_h]})=\infty, \tag{1} \]

then for any \(x,y\) from \(P_1\) the relation holds

\[ \lim_{n\to\infty}\mu\bigl(gT_1^{[n]}(x-y)\bigr)=0. \]

1.3. In order that the family \(\{T_k^{[k+1]},\ 1\leq k\leq\infty\}\) of stochastic operators satisfying condition (A) be strongly ergodic, it is necessary and sufficient that relation (1) hold.

All known criteria of ergodicity for inhomogeneous Markov chains (see \((^3),(^4)\), p. 206, \((^5,^6)\)) are sufficient conditions for strong ergodicity and constitute special cases of condition (1). From proposition 1.3, as a consequence, follow the results of papers \((^7)\) (Theorem 11), \((^8)\) (Lemma 2), \((^9)\) (Theorem 1).

2. Consider a topological semifield \(E_k(t)\), depending on a numerical parameter \(t\), and denote by \(\mathscr E_k(t)\) the direct sum

\[ \mathscr E_k(t)=E_k(t)\oplus iE_k(t). \]

As in item 1, it is assumed that on \(\nabla_k(t)\) a certain proper measure is defined.

Let there be given, generally speaking, a noncommutative family of linear operators \(\{T_k^{[k+1]}(t),\ 1\leq k\leq\infty\}\), where \(T_k^{[k+1]}(t):\mathscr L_k(t)\to\mathscr L_{k+1}(t)\) and

\[ T_k^{[m]}(t)=T_{m-1}^{[m]}(t)\times T_{m-2}^{[m-1]}(t)\cdots T_k^{[k+1]}(t) \]

for \(k<m\). Here \(\mathscr L_k(t)\) is the set of summable elements of the complex semifield \(\mathscr E_k(t)\).

Definition 4. A family \(\{T_k^{[k+1]}(t),\ 1\leq k\leq\infty\}\) of linear operators continuously depending on the numerical parameter \(t\) will be called a family of characteristic operators if the following relations are fulfilled:

1) \(T_k^{[k+1]}(t)(\mathscr L_k(t))\subseteq \mathscr L_{k+1}(t)\);

2) \(T_k^{[k+1]}(0)=T_k^{[k+1]}\) is a stochastic operator;

3) \(T_k^{[k+1]}(t)x^{(n)}(t)\to T_k^{[k+1]}(t)x^{(0)}(t)\), if

\[ x^{(n)}(t)\to x^{(0)}(t)\quad (x^{(0)}(t),x^{(n)}(t)\in\mathscr L_k(t)). \]

If, for the family \(\{T_k^{[k+1]}(t),\ 1\leq k\leq\infty\}\) of characteristic operators, a condition analogous to (A), taking into account the parameter \(t\), is satisfied, then results 1.1 and 1.2 remain valid for this family.

From the results of item 2, as a consequence, follows the result of paper \((^{10})\) (Lemma 5).

Tashkent State University
named after V. I. Lenin

Received
19 V 1970

CITED LITERATURE

1. M. Ya. Antonovskii, V. G. Boltyanskii, T. A. Sarymsakov, Topological Boolean Algebras, Tashkent, 1963.
2. T. A. Sarymsakov, Topological Semifields and Probability Theory, Tashkent, 1969.
3. T. A. Sarymsakov, DAN, 90, No. 1, 25 (1953).
4. S. N. Bernstein, Course of Probability Theory, 4th ed., Moscow–Leningrad, 1946.
5. A. N. Kolmogorov, UMN, 5, 3 (1938).
6. S. K. Sirazhdinov, DAN, 71, No. 5, 829 (1950).
7. R. L. Dobrushin, Probability Theory and Its Applications, 1, issue 1, 4, 72 and 365 (1956).
8. T. A. Sarymsakov, ibid., 6, issue 1, 194 (1961).
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10. V. A. Statulyavichus, Lithuanian Mathematical Collection, 1, 1–2, 231 (1961).