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UDC 519.217
MATHEMATICS
D. V. MANEVICH
ON THE EXTENSION OF A. N. KOLMOGOROV’S UNIFORM THEOREM TO HOMOGENEOUS MARKOV CHAINS
(Presented by Academician A. N. Kolmogorov, 9 XII 1966)
In the works \((^1,^2)\) A. N. Kolmogorov gave an estimate for the approximation of the distributions of sums of independent, identically distributed summands by infinitely divisible distributions. In the present note this result is extended to the case of distributions of sums of summands connected in a homogeneous Markov chain.
Let, on a probability space \(\{\Omega,\mathcal F,P\}\), a sequence of random variables \(\{X_k=X_k(\omega)\}\) be given, connected in a homogeneous Markov chain; \(\Omega\) is an arbitrary set of states \(\omega\); \(\mathcal F\) is a \(\sigma\)-algebra of its subsets \(A\); \(P\) is a probability measure of the elements of the set \(\mathcal F\). Denote by \(P_{ij}(\omega,A)\) the probabilistic transition function in \(j-i\) steps;
\[ P(A)\text{ is the probability of }A;\quad c,c_1,c_2,\ldots\text{ are absolute constants};\quad s_n=\sum_{i=1}^n X_i, \]
\[ F_n=P(s_n<x),\quad F=P(X_i<x). \]
We shall assume that the ergodic property of the chain is characterized by the condition adopted in the works of S. V. Nagaev \((^3)\):
\[ \sup_{\omega,A}\left|P_{ij}(\omega,A)-P(A)\right|\le Cq^\mu,\quad 0<q<1,\quad \mu=j-i. \]
We note that the ergodicity condition used in the works of R. L. Dobrushin \((^4)\) would not change the subsequent reasoning.
Theorem. Let \(G\) be the set of all infinitely divisible distribution laws \(D(x)\). Then there exists an absolute constant \(C_1\) such that
\[ \rho(F_n,G)=\sup_F\inf_{D_n\in G}\sup_x |F_n-D_n|\le c_1 n^{-1/3}\ln^{\lambda/3}n, \tag{1} \]
where \(\lambda>1\).
We outline the proof. Using the idea of S. N. Bernstein \((^5)\), we construct sums of independent random variables by whose distributions we shall approximate \(F_n\). Write \(s_n\) as follows:
\[ s_n=y_1+u_1+\ldots+y_l+u_l, \]
where
\[ y_i=X_{(i-1)(r+k)+1}+\ldots+X_{i(r+k)-k};\quad u_i=X_{i(r+k)-k+1}+\ldots+X_{i(r+k)}; \]
\[ i=1,\ldots,l;\quad (r+k)l\ge n;\quad \Sigma_l=y_1+\ldots+y_l;\quad \sigma_l=u_1+\ldots+u_l; \]
\[ s_n=\Sigma_l+\sigma_l. \]
Put \(k=r=\ln^\lambda n,\ \lambda>1\). Then \(l\ge n/2\ln^\lambda n\).
Denote
\[ F_\Sigma(x)=P(\Sigma_l<x),\quad F_\sigma(x)=P(\sigma_l<x), \]
and \(F_\Sigma(x/A)\), \(F_\sigma(x/A)\) the corresponding conditional distribution functions (d.f.) under some condition \(A\); \(\Phi_k'\) is some conditional d.f., and \(\Phi_k\) the unconditional d.f. of \(y_k\).
The rest of the proof rests on three lemmas.
Lemma 1. Let \(G_l\) and \(f\) be the characteristic functions (c.f.) of \(\Sigma_l\) and \(y_k\), respectively. Then
\[ G_l-f^l=\sum_{r=1}^l \delta_r L_r,\quad \text{where }\quad \delta_r=\int e^{itx}\,d[\Phi_r'-\Phi_r],\quad L_r\equiv G_{r-1}f^{\,l-r}. \]
Proof is carried out in many respects in the same way as that of Lemma 2 in the author’s paper \((^6)\).
Lemma 2. One can indicate such a continuous d.f. \(\Psi_\Sigma(x)\) of the sum of \(l\) terms, related to one another in the same way as the \(y_k\) in the sum \(\Sigma_l\), that
\[ |F_\Sigma(x)-\Psi_\Sigma(x)|\leq c_2 n^{-1/3}. \]
For the proof we compose \(\Phi(x/\varepsilon)\)—a certain normal distribution—with \(F_\Sigma(x)\). We show that
\[ |F_\Sigma(x)-\Psi_\Sigma(x)| \leq |\Psi_\Sigma(x-2h\varepsilon)-\Psi_\Sigma(x+2h\varepsilon)| +\frac{2c_3}{h}e^{-h^2/2},\qquad \Psi_\Sigma=F_\Sigma*\Phi . \]
The estimate
\[ \alpha \equiv |\Psi_\Sigma(x-2h\varepsilon)-\Psi_\Sigma(x+2h\varepsilon)| \]
is carried out by S. N. Bernstein’s lemma \(((7), p. 409)\):
\[ \alpha \leq \frac{2\varepsilon}{\pi}\left[A\cdot 2h\varepsilon+\frac{1}{2\sqrt{Nh\varepsilon}}\right], \]
\[ A=\frac12\left|\int_{-N}^{N}\varphi_l(t)\,dt\right|; \]
\(\varphi_l\) is the ch.f. for the d.f. \(\Psi_\Sigma(x)\). Then, by Lemma 1,
\[ A=\frac12\left|\int_{-N}^{N} f^l(t)\,dt+\sum_{k=1}^{l}\int_{-N}^{N}\delta_k L_k\,dt\right|. \]
The first integral is estimated by known methods, while the second is reduced to the Dirichlet integral. Setting \(h=\ln^2 n\), \(\varepsilon=\ln^{-2} n\cdot n^{-1/3}\), \(\sqrt[3]{n}=n^{2/3}\), and using the ergodicity condition, we verify the validity of the lemma.
Lemma 3. Let \(\bar F_\Sigma(x)\) denote the d.f. of the sum \(y_1+\cdots+y_l\) under the assumption of continuity and independence of the terms. Then
\[ |\bar F_\Sigma(x)-\Psi_\Sigma(x)|\leq c_4 n^{-1/3}. \]
Proof. For the d.f.’s \(\bar F_\Sigma(x)\) and \(\Psi_\Sigma(x)\), the corresponding ch.f.’s can be written as follows: \(f^l\varphi; G_l\varphi\), where \(\varphi\) is the ch.f. for \(\Phi(x/\varepsilon)\). Next we use Esseen’s theorem \((^8)\), for the application of which we consider
\[ \Delta= \left| \int_{-T}^{T} e^{-itx}\frac{(f^l-G_l)\varphi}{-it}\,h(t)\,dt \right| \]
instead of
\[ \int_{-T}^{T}\frac{|f^l-G_l|\,|\varphi|}{|t|}\,dt, \]
and verify the validity of the lemma.
Further,
\[ F_n(x)=M\left\{\int F_\Sigma\left(\frac{x-t}{A}\right)\,dF_\sigma\left(\frac{t}{A}\right)\right\}. \]
Since Lemma 2 remains valid also for conditional distributions, i.e.
\[ \left|F_\Sigma\left(\frac{x-t}{A}\right)-\Psi_\Sigma\left(\frac{x}{A}\right)\right|\leq c_2 n^{-1/3}, \]
the error in replacing \(F_\Sigma((x-t)/A)\) by a continuous function will not exceed \(c_2 n^{-1/3}\). We shall assume, without making elementary explanations for simplicity, that \(F_\Sigma((x-t)/A)\) is continuous. From
\[ F_n(x)=M\left[\int_{-\infty}^{\xi}+\int_{\xi}^{\infty}\right] \]
we obtain
\[ F_\sigma(\xi)-1\leq F_n(x)-F_\Sigma(x-\theta)\leq 0, \]
\(-\infty<\theta\leq \xi\). We choose \(\xi\) so that \(1-F_\sigma(\xi)\leq n^{-1/3}\). Next we replace \(F_\Sigma(x-\theta)\) by the distribution \(\bar F_\Sigma(x-\theta)\). Then, using Lemma 3, we obtain
\[ -c_5 n^{-1/3}\leq F_n(x)-\bar F_\Sigma(x-\theta)\leq c_5 n^{-1/3}. \tag{2} \]
According to A. N. Kolmogorov’s theorem \((^2)\), there exists an infinitely divisible distribution \(D_n(x)\) such that
\[ |\bar F_\Sigma(x-\theta)-D_n(x)|\leq c_6 n^{-1/3}\ln^\lambda n, \]
since the number of terms in \(\Sigma_l\) is equal to \(l\geq n/2\ln^\lambda n\). Combining this with (2), we arrive at the conclusion that (1) holds. The theorem is proved.
Tashkent State Pedagogical Institute
named after Nizami
Received
25 XI 1966
References
- A. N. Kolmogorov, Theory of Probability and Its Applications, 1, § 4, 426 (1956).
- A. N. Kolmogorov, Trudy Moskov. matem. obshch., 12, 437 (1963).
- S. V. Nagaev, Theory of Probability and Its Applications, 2, No. 4, 389 (1957).
- R. L. Dobrushin, Theory of Probability and Its Applications, 1, No. 1, 72 (1956).
- S. N. Bernstein, UMN, 10 (1944).
- D. V. Manevich, Collected Papers: Probability Theory and Mathematical Statistics, vol. 1, Academy of Sciences of the Uzbek SSR, 1964.
- S. N. Bernstein, Probability Theory, Moscow, 1946.
- C. G. Esseen, Acta Math., 77, 1 (1945).