M. N. MARUSHIN

ON THE QUESTION OF THE APPLICABILITY OF A LIMIT THEOREM OF ORDER \(p>0\) TO AN INHOMOGENEOUS MARKOV CHAIN WITH TWO STATES

(Presented by Academician S. N. Bernstein on 23 VII 1963)

An inhomogeneous Markov chain is considered in the form of a sequence of series of random variables

\[ x_{k1}, x_{k2}, \ldots, x_{kk}\qquad (k=1,2,\ldots,n), \]

each of which assumes only the values 0 and 1.

The transition matrix is

\[ p_{ki}= \begin{vmatrix} p_{11}^{(i)}(k) & p_{10}^{(i)}(k)\\ p_{01}^{(i)}(k) & p_{00}^{(i)}(k) \end{vmatrix}, \qquad i=1,2,\ldots,k. \]

Here \(p_{\alpha\beta}^{(i)}(k)\) denotes the transition probability; the first index \(\alpha\) indicates the value of the preceding random variable \(x_{i-1}\), and the second index \(\beta\) the value of the random variable \(x_i\); \(k\) is the number of the series (in what follows the index \(k\) will be omitted). To specify the Markov chain completely it is necessary to give \(\mathbf P(x_1=1)=p_1\) and \(\mathbf P(x_1=0)=q_1\).

Following S. N. Bernstein \((^1)\), we shall say that a limit theorem of order \(p>0\) is applicable to the sum \(s_n=x_1+x_2+\cdots+x_n\) if the following two limiting relations hold simultaneously:

\[ \lim_{n\to\infty}\mathbf P\left(\frac{s_n-Ms_n}{\sqrt{B_n}}<t\right) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{t} e^{-z^2/2}\,dz, \]

\[ \lim_{n\to\infty}\mathbf M\left|\frac{s_n-Ms_n}{\sqrt{B_n}}\right|^p = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty}|t|^p e^{-t^2/2}\,dt, \]

where \(B_n=\mathbf M(s_n-Ms_n)^2\).

In the present note several theorems are formulated in which necessary and sufficient conditions are indicated for the applicability of a limit theorem of arbitrary order \(p>0\) to the sum of quantities connected in an inhomogeneous Markov chain with two states.

For brevity of exposition we introduce a number of notations and definitions.

\[ p_i=\mathbf P(x_i=1),\qquad q_i=\mathbf P(x_i=0),\qquad \delta_i=p_{11}^{(i)}-p_{01}^{(i)}, \]

\[ \Pi_{i+1,i+k}=\delta_{i+1}\delta_{i+2}\cdots\delta_{i+k} \qquad (\Pi_{i+1,i}=1/2), \]

\[ T_i=\sum_{k=0}^{n}\Pi_{i+1,i+k},\qquad T_i^{+}=\sum_{k=0}^{n}\left|\Pi_{i+1,i+k}\right|. \]

It is known that

\[ B_n=\sum_{i=1}^{n} p_i q_i T_i,\qquad B_n^{+}=\sum_{i=1}^{n} p_i q_i T_i^{+}. \]

We shall call the sequence \(x_j,x_{j+1},\ldots,x_{j+k}\) a Markov chain on the interval \([j,j+k]\). In particular, the sequence \(x_1,x_2,\ldots,x_n\) will be called a Markov chain on the whole interval. A Markov chain will be called positively stable on the interval \([j,j+k]\) if \(\delta_i\ge 0\) for all \(i\) satisfying the inequality \(j\le i\le j+k\). If \(\delta_i\le 0\) for all \(i\) satisfying the inequality \(j\le i\le j+k\), then such a Markov chain will be called negatively st-

stable on the interval \([j, j+k]\). We shall say that a Markov chain decomposes into two independent chains if there exists such an \(i\) for which \(\delta_i=0\).

Theorem 1. If

\[ \overline{\lim}\,\frac{B_n^+}{B_n}=C<\infty,\qquad \lim_{n\to\infty}\frac{\max_{1\le i\le n-1}T_i^+}{\sqrt{B_n}}=0, \tag{1} \]

then the limit theorem of arbitrary order \(p>0\) is applicable to the sum \(s_n\).

Theorem 2. If the Markov chain is positively stable on the entire interval and

\[ \lim_{n\to\infty}\frac{\max_{1\le i\le n-1}T_i}{\sqrt{B_n}}=0, \tag{2} \]

then the limit theorem of arbitrary order \(p>0\) is applicable to the sum \(s_n\).

Theorem 3. If the Markov chain is negatively stable on the entire interval and condition (1) is satisfied, then the limit theorem of arbitrary order \(p>0\) is applicable to the sum \(s_n\).

Theorem 4. If the Markov chain is positively stable and

1) \(0<p_1-q_1<1\),

2) \(p_{01}^{(i)}\ge p_{10}^{(i)}\),

then (2) is a necessary condition for applicability to the sum \(s_n\) of the limit theorem of order \(p\ge 4\).

Theorem 5. If the Markov chain is negatively stable and

1) \(0<q_1-p_1<1\),

2) \(p_{01}^{(i)}\ge p_{10}^{(i)}\),

then (2) is a necessary condition for applicability to the sum \(s_n\) of the limit theorem of order \(p\ge 4\).

Theorem 6. If the Markov chain decomposes into two independent chains, each of which is either positively stable or negatively stable, then (1) is a sufficient condition for applicability to \(s_n\) of the limit theorem of order \(p\ge 4\) and a necessary condition if:

1) \(B_n\to\infty\) as \(n\to\infty\),

2) \(p_{01}^{(i)}=p_{10}^{(i)}\),

3) \(p_1=q_1\).

To illustrate these theorems we give two examples.

Example 1. Let \(p_{01}^{(n)}\sim \dfrac{b}{n^\beta}\), \(p_{10}^{(n)}\sim \dfrac{a}{n^\alpha}\), where \(a>0\), \(b>1\), \(\alpha\le \beta\). Then

\[ 1-\frac{a+b}{n^\alpha}\le \delta_n\le 1-\frac{a}{n^\alpha} \]

for sufficiently large \(n\). As shown in [2], \(B_n\ge l n^{1+2\alpha-\beta}\), where \(l>0\). Therefore

\[ \frac{2}{(a+b)n^{1-\alpha}}\le \frac{1}{\sqrt{B_n}}\max_{1\le i\le n-1}T_i \le \frac{1}{a\sqrt{l}}\frac{1}{n^{(1-\beta)/2}}, \]

since \(B_n\le \frac14 n^2\). Hence it follows that for \(\beta<1\) the limit theorem of arbitrary order \(p>0\) is applicable, whereas for \(\alpha\ge 1\) it is not applicable.

Example 2 (S. N. Bernstein [1]). Let

\[ p_{01}^{(i)}=p_{10}^{(i)}=\frac{1}{a n^{1/3}},\qquad i\le n^{1/3}; \]

\[ p_{11}^{(1)}=p_{01}^{(i)}=\frac12,\qquad i=n^{1/3}+1;\qquad p_{11}^{(i)}=p_{00}^{(i)}=\frac{1}{n^{1/3}},\quad i>n^{1/3}+1. \]

In this case \(B_n\sim c^2 n^{2/3}\). Applying Theorem 6, it is not difficult to show that condition (1) is not fulfilled. Therefore the limit theorem is not applicable.

Kyiv Institute
of the Civil Air Fleet

Received
15 VII 1963

References

1. S. N. Bernstein, UMN, 10, 65 (1944).
2. B. V. Shirokorad, Izv. AN SSSR, ser. matem., 18, No. 1 (1954).