Abstract Generated abstract
This note proves a local limit theorem for sums of integer-valued variables forming a nonhomogeneous Markov chain with countably many possible states. Under a uniform minorization condition, linear growth of the variance, bounded moments of order at least three, and an aperiodicity condition expressed through state occupation probabilities, the characteristic function of the normalized sum is expanded in an Edgeworth-type form and controlled away from the origin. These estimates yield an asymptotic expansion for the lattice probabilities of the sum, uniformly in the lattice point, with the normal density as the leading term and polynomial differential corrections. A final remark states that uniform convergence to the normal local limit remains valid under a weaker bounded moment condition of order \(2+\delta\).
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Reports of the Academy of Sciences of the USSR
- Volume 115, No. 5
MATHEMATICS
V. A. STATULEVIČIUS
A LOCAL LIMIT THEOREM FOR NONHOMOGENEOUS MARKOV CHAINS WITH A COUNTABLE NUMBER OF POSSIBLE STATES
(Presented by Academician A. N. Kolmogorov on 14 III 1957)
Consider a sequence
\[ X_1, X_2,\ldots, X_n,\ldots \]
of random variables taking integer values and connected into a nonhomogeneous Markov chain with transition probabilities
\(p_{ij}^{(k)}=P\{X_k=j\mid X_{k-1}=i\}\) and absolute probabilities \(p_{k\mid j}=P\{X_k=j\}\). Let
\[ S_n=X_1+X_2+\cdots+X_n. \]
We introduce the following conditions:
I. There exists a value \(j_0\) such that \(p_{ij_0}^{(k)}\geq \alpha>0\) for all \(i\) and \(k\).
II. \(B_n^2=DS_n\geq cn;\ c>0\) *.
III. There exist uniformly bounded absolute moments \(M|X_k|^s\) up to order \(s\) \((s\geq 3)\), inclusive.
IV. The greatest common divisor of all differences \(j_l-j_0\) for which
\[ \frac{1}{\ln n}\sum_{k=1}^{n} p_{k\mid j_l}\to\infty \qquad (n\to\infty), \]
is equal to one.
For the characteristic function \(f_n(t)\) of the sum \(S_n\) the following lemmas hold.
Lemma 1 **. If conditions I—III are satisfied, then for \(|t|\leq n^{1/6}\)
\[ f_n\left(\frac{t}{B_n}\right) = \exp\left[-\frac{t^2}{2}+it\frac{MS_n}{B_n}\right] \left(1+\sum_{k=1}^{s-3}\frac{1}{n^{k/2}}P_k(it)\right) + \]
\[ +\frac{1}{n^{(s-2)/2}}\left(|t|^s+|t|^{3(s-2)}\right) \exp\left[-\frac{t^2}{2}\right]O(1), \tag{1} \]
where \(P_k(it)\) is a polynomial in \(it\) of degree \(3k\). The coefficients of the polynomial are real, depend on \(n\), but are uniformly bounded in \(n\).
Here and below, the constant in the symbol \(O(1)\) depends only on \(s\).
* Condition II is satisfied if, for example, \(DX_k^\tau\geq c_1>0\) (¹).
** Lemma 1 is valid if condition I is replaced by the condition that the coefficient of regularity \(a^{(n)}\geq \delta>0\), and the quantities \(X_k\) may be arbitrary \(\mathfrak{A}_k\)-measurable functions (for the definition of \(a^{(n)}\) and \(\mathfrak{A}_k\), see (¹)).
Lemma 2. If conditions I, IV are satisfied, then
\[ \int_{n^{1/s}<|t|<\pi B_n}\left| f_n\left(\frac{t}{B_n}\right)\right|\,dt = O\left(\frac{1}{n^{(s-2)/2}}\right). \tag{2} \]
From the equality
\[ B_n P\{S_n=m\} = \frac{1}{2\pi} \int_{|t|<n^{1/s}} f_n\left(\frac{t}{B_n}\right) \exp\left[-it\frac{m}{B_n}\right]\,dt + \]
\[ + \frac{1}{2\pi} \int_{n^{1/s}<|t|<\pi B_n} f_n\left(\frac{t}{B_n}\right) \exp\left[-it\frac{m}{B_n}\right]\,dt \]
and the estimates (1), (2), the following theorem easily follows (see, for example, (2), § 51).
Theorem. Under conditions I–IV the expansion
\[ B_n P\{S_n=m\} = g(x_{nm}) + \sum_{k=1}^{s-3} \frac{1}{n^{k/2}} P_k\left(-\frac{d}{dx_{nm}}\right)g(x_{nm}) + O\left(\frac{1}{n^{(s-2)/2}}\right), \]
holds, where
\[ g(x)=\frac{1}{\sqrt{2\pi}}\exp\left[-\frac{x^2}{2}\right]; \qquad x_{nm}=\frac{m-MS_n}{B_n}; \]
\(P_k\left(-\dfrac{d}{dx}\right)g(x)\) means that in the polynomial \(P_k(it)\) the powers \((it)^\nu\) are replaced by the expressions
\[ (-1)^\nu \frac{d^\nu}{dx^\nu}g(x). \]
Remark. The relation
\[ B_nP\{S_n=m\}-g(x_{nm})\to 0 \qquad (n\to\infty) \]
holds uniformly in all \(m\) also in the case when condition III is replaced by the weaker condition:
IIIa. There exist uniformly bounded \(M|X_k|^{2+\delta}\), \(\delta>0\).
Leningrad State University
named after A. A. Zhdanov
Received
8 XII 1956
CITED LITERATURE
- R. L. Dobrushin, Probability Theory and Its Applications, No. 1 (1956).
- B. V. Gnedenko, A. N. Kolmogorov, Limit Distributions for Sums of Independent Random Variables, 1949.