M. Rosenblatt-Roth

ON THE RELATIVE STABILITY OF SUMS OF NONNEGATIVE RANDOM VARIABLES LINKED IN A MARKOV CHAIN

(Presented by Academician A. N. Kolmogorov, 11 I 1964)

Let \(\xi_k\) \((k=1,2,\ldots)\) be a sequence of random variables linked in a Markov chain and capable of taking only nonnegative values; let \(a_i\) be the coefficient of ergodicity \((^{1,2})\) of the \(i\)-th transition probability function of the chain; let
\[ \alpha^{(n)}=\min_{1\le i<n}\alpha_i>0,\qquad S_n=\sum_{i=1}^{n}\xi_i. \]
We shall call the sums \(S_n\) relatively stable \((^{3,4})\) if there exists some sequence of numbers \(A_n>0\) \((n=1,2,\ldots)\) such that
\[ \mathbf P\left(\left|A_n^{-1}S_n-1\right|\ge \delta\right)\to 0\qquad (n\to\infty) \tag{1} \]
for every \(\delta>0\).

Theorem 1. In order that the sums \(S_n\) of nonnegative random variables linked in a Markov chain be relatively stable, it is sufficient that there exist some sequence of numbers \(A_n>0\) \((n=1,2,\ldots)\) such that, for every \(\varepsilon>0\),
\[ \sum_{k=1}^{n}\int_{|x-a_k|\ge \varepsilon A_n\alpha^{(n)}} dF_k(x)\to 0\qquad (n\to\infty), \tag{2} \]
\[ \frac{1}{A_n}\sum_{k=1}^{n}\int_{|x-a_k|<\varepsilon A_n\alpha^{(n)}} x\,dF_k(x)\to 1\qquad (n\to\infty), \tag{3} \]
where \(F_k(x)\) is the unconditional distribution function of the random variable \(\xi_k\), and \(a_k=A_k-A_{k-1}\). In this case the numbers \(A_n\) may be taken as normalizing divisors.

Theorem 2. The relative stability of sums \(S_n\) of nonnegative random variables linked in a Markov chain with normalizing divisors \(A_n\to A\) \((n\to\infty)\) entails the following:

a) if \(A=\infty\), the summands \(\xi_k\) \((k=1,2,\ldots,n)\) are asymptotically constant, i.e., there exist certain numbers \(b_k\) \((k=1,2,\ldots,n)\) such that, however small \(\delta>0\) may be,
\[ \mathbf P\left(A_n^{-1}|\xi_k-b_k|\ge \delta\right)\to 0\qquad (n\to\infty) \tag{4} \]
uniformly with respect to \(k\le n\);

b) if \(A<\infty\), the summands \(\xi_k\) \((k=1,2,\ldots)\) are, with probability 1, equal to certain numbers \(a'_k\).

Theorem 3. In order that the sums \(S_n\) of nonnegative random variables linked in a Markov chain have relative stability with a prescribed normalizing divisor \(A_n\to\infty\) \((n\to 0)\), it is sufficient that, for every \(\varepsilon>0\), conditions (2) and (3) hold, where \(a_k=A_k-A_{k-1}\). For \(A_n\to A<\infty\) this is false.

Theorem 4. In order that the sums \(S_n\) of nonnegative random variables \(\xi_k\) \((k=1,2,\ldots)\), linked in a Markov chain and possessing finite-

with finite mathematical expectations \(\mathbf{M}\xi_k\), are relatively stable with normalizing divisor \(A_n=\mathbf{M}S_n\), it is sufficient that for every \(\varepsilon>0\) conditions (2) and (3) be fulfilled, where \(a_k=\mathbf{M}\xi_k\). In this case the possibility \(A_n\to A<+\infty\) is not excluded.

Theorem 5. In order that the sums \(S_n\) of nonnegative random variables \(\xi_k\) \((k=1,2,\ldots)\), linked into a Markov chain, which possess relative stability with normalizing divisor \(A_n\), consist of summands \(\xi_k\) for which \(A_n^{-1}\xi_k\) \((k=1,2,\ldots,n)\) are negligible in the limit, i.e., however small \(\delta>0\) may be,

\[ \mathbf{P}\bigl(A_n^{-1}\xi_k\ge \varepsilon\bigr)\to 0 \quad (n\to\infty) \tag{5} \]

uniformly with respect to \(k\le n\), it is necessary and sufficient that

\[ A_n\to+\infty,\qquad A_n^{-1}A_{n-1}\to 1 \quad (n\to\infty). \tag{6} \]

Theorem 6. In order that the sums \(S_n\) of nonnegative random variables \(\xi_k\) \((k=1,2,\ldots)\), linked into a Markov chain, be relatively stable and that the quantities \(B_n^{-1}\xi_k\) \((k=1,2,\ldots,n)\), where \(B_k\) is a normalizing divisor, be negligible in the limit, it is sufficient that there exist numbers \(A_n\) \((n=1,2,\ldots)\) for which the conditions

\[ \sum_{k=1}^{n}\mathbf{P}\bigl(\xi_k\ge \varepsilon A_n\alpha^{(n)}\bigr)\to 0,\qquad (n\to\infty); \tag{7} \]

\[ \frac{1}{A_n}\sum_{k=1}^{n}\int_{0}^{\varepsilon A_n\alpha^{(n)}} x\,dF_k(x)\to 1 \qquad (n\to\infty), \tag{8} \]

hold, however \(\varepsilon>0\) may be chosen. Here the numbers \(A_n\) may be taken as the normalizing divisors \(B_n\).

Theorem 7. In order that the sums \(S_n\) of nonnegative random variables \(\xi_k\) \((k=1,2,\ldots)\), linked into a Markov chain, be relatively stable with a prescribed normalizing divisor \(A_n\), and that the quantities \(A_n^{-1}\xi_k\) \((k=1,2,\ldots,n)\) be negligible in the limit, it is sufficient that, for any \(\varepsilon>0\), relations (7) and (8) be fulfilled.

Theorem 8. In order that the sums \(S_n\) of nonnegative random variables \(\xi_k\) \((k=1,2,\ldots)\), linked into a Markov chain, possessing finite mathematical expectations \(\mathbf{M}\xi_k\), be relatively stable with normalizing divisor \(A_n=\mathbf{M}S_n\), and that the quantities \(A_n^{-1}\xi_k\) \((k=1,2,\ldots,n)\) be negligible in the limit, with \(\alpha^{(n)}>\rho>0\) \((n=1,2,\ldots)\), it is sufficient that, for every \(\varepsilon>0\), the relation

\[ \frac{1}{A_n}\sum_{k=1}^{n}\int_{0}^{\varepsilon A_n\alpha^{(n)}} x\,dF_k(x)\to 1 \quad (n\to\infty). \tag{9} \]

be fulfilled.

Theorem 9. In order that the sums \(S_n\), linked into a stationary and homogeneous Markov chain with ergodicity coefficient \(\alpha>0\), of nonnegative identically distributed random variables be relatively stable, it is sufficient that there exist some numbers \(A_n>0\) \((n=1,2,\ldots)\) satisfying the conditions

\[ n\int_{\varepsilon A_n\alpha^{(n)}}^{+\infty} dF(x)\to 0 \qquad (n\to\infty); \tag{10} \]

\[ \frac{n}{A_n}\int_{0}^{\varepsilon A_n\alpha^{(n)}} x\,dF(x)\to 1 \qquad (n\to\infty). \tag{11} \]

for any \(\varepsilon > 0\), where \(F(x)\) is the unconditional distribution function of the random variables. In this case the numbers \(A_n\) may be taken as normalizing divisors.

Remark. If the random variables \(\xi_k\) \((k = 1, 2, \ldots)\) are independent, i.e. \(\alpha^{(n)} = 1\) \((n = 1, 2, \ldots)\), then from these theorems, as special cases, follow the corresponding results of \({}^{3,4}\).

Faculty of Mathematics and Mechanics
University of Bucharest
Bucharest, Romanian People’s Republic

Received
13 V 1963

References

\({}^{1}\) R. L. Dobrushin, Theory of Probability and Its Applications, 1, no. 1, 72 (1956).
\({}^{2}\) R. L. Dobrushin, Theory of Probability and Its Applications, 1, no. 4, 365 (1956).
\({}^{3}\) A. Khintchine, Giornale dell’Istituto Italiano degli Attuari, 7, 365 (1936).
\({}^{4}\) A. A. Bobrov, Scientific Notes of Moscow State University, issue 146, Mathematics, 3, 92 (1950).