Abstract
Full Text
V. P. LEONOV
ON THE CENTRAL LIMIT THEOREM FOR ERGODIC ENDOMORPHISMS OF COMPACT COMMUTATIVE GROUPS
(Presented by Academician A. N. Kolmogorov, 22 VI 1960)
In papers ((^{1-3})) it was proved by different methods that if (g(x)) is a real periodic function with period one with Fourier coefficients
[
C_n(g)=\int_0^1 g(x)e^{-2\pi i n x}\,dx
=O\left(\frac{1}{|n|^\beta}\right),\qquad \beta>\frac12,
]
or satisfying a Hölder condition with exponent (\alpha>0),
[
m_g=\int_0^1 g(x)\,dx,
]
(a) is an arbitrary integer not equal to (\pm 1), and (\mu) is Lebesgue measure on the interval ([0,1]), then*
[
\mu\left{x:\frac{1}{\sqrt p}\left(\sum_{t=0}^{p-1} g(a^t x)-m_g p\right)<y\right}
\xrightarrow[p\to\infty]{}
\frac{1}{\sqrt{2\pi}\sigma_g}\int_{-\infty}^{y} e^{-z^2/2\sigma_g^2}\,dz
]
under the additional condition
[
\sigma_g^2
=\int_0^1 (g(x)-m_g)^2\,dx
+2\sum_{t=1}^{\infty}\int_0^1 (g(a^t x)-m_g)(g(x)-m_g)\,dx>0.
]
The present paper generalizes the indicated results. It uses the basic definitions and notation of ((^5)). Let (G) be a compact commutative group with invariant measure (\mu), and let (T) be an endomorphism of the group (G) into (G). For (g\in L^2(G)) denote
[
m_g=\int_G g(x)\,d\mu
]
and consider the question of when the quantity
[
\frac{1}{\sqrt p}\left(\sum_{t=0}^{p-1} g(T^t x)-m_g p\right)
=
\frac{1}{\sqrt p}\left(\sum_{t=0}^{p-1} U^t g(x)-m_g p\right)
]
as (p\to\infty) is asymptotically normal or degenerate, i.e., when for some (\sigma\ge 0) the limiting relation holds
[
\lim_{p\to\infty}
\mu\left{x:\frac{1}{\sqrt p}\left(\sum_{t=0}^{p-1} g(T^t x)-m_g p\right)<y\right}
=\Phi_\sigma(y)
\tag{1}
]
* In ((^4)) the same result was proved under weaker restrictions on the function (g(x)).
for all points of continuity (y) of the function (\Phi_\sigma(y)), where
[
\Phi_\sigma(y)=\frac{1}{\sqrt{2\pi}\sigma}\int_{-\infty}^{y} e^{-z^2/2\sigma^2}\,dz,\quad \text{if } \sigma>0;
]
[
\Phi_0(y)=
\begin{cases}
0, & y\leqslant 0,\
1, & y>0.
\end{cases}
\tag{2}
]
If for a function (g(x)) relation (1) holds with (\sigma\geqslant 0), then we shall say that (g(x)) satisfies the central limit theorem (c.l.t.).
By (\chi(x)) we shall denote the characters of the group (G). As is known (see, for example, ((^6))), functions (g\in L^2(G)) are expanded in a Fourier series in characters, convergent in mean square:
[
g(x)=\sum_n C_n(g)\chi_n(x);
]
in what follows it is assumed everywhere that (\chi_0(x)=1).
For convenience of notation we introduce an operator (A^*), mapping the set of indices of characters (\mathfrak N) into itself and defined by the formula
[
\chi_{A^*n}=U\chi_n,
]
so that (\chi_{A^{*t}n}=U^t\chi_n) for all nonnegative integers (t).
Theorem 1. Let (G) be a compact commutative group with invariant measure (\mu), let (T) be its ergodic endomorphism, (g(x)\in L^2(G)), real-valued, and
[
\sum_{t=0}^{\infty}\sum_{n\ne 0}\left|C_n(g)\right|\,\left|C_{A^{*t}n}\right|<\infty;
\tag{3}
]
then (g(x)) satisfies the c.l.t.; moreover,
[
\sigma^2=\sigma_g^2=|g-m_g|^2+2\sum_{t=1}^{\infty}\bigl((U^t g-m_g),(g-m_g)\bigr)=
\tag{4}
]
[
=\sum_{n\ne 0}\left|C_n(g)\right|^2+2\sum_{t=1}^{\infty}\sum_{n\ne 0} C_n(g)\overline{C}_{A^{*t}n}(g).
\tag{4'}
]
Lemma 1. Let
[
b_p(g)=\int_G\left(\sum_{t=0}^{p-1} U^t g(x)-m_g p\right)^2\,d\mu.
]
If (3) is fulfilled, then
[
\lim_{p\to\infty}\frac{b_p(g)}{p}=\sigma_g^2.
]
Proof of the theorem. For (\sigma_g=0) the theorem is trivial. Let (\sigma_g>0). Take (G) to be the space of elementary events and (\mu) to be the probability distribution on it. Then the function
[
\xi(t,x)=U^t g(x)=g(T^t x)
]
forms a real stationary random process with discrete time (t=0,1,2,\ldots). We first consider the case when
[
g(x)=\sum_{n\in \mathfrak N'} C_n(g)\chi_n(x),
]
where (\mathfrak N' \subset \mathfrak N) is finite. Then (g'(x)) is bounded, and all moments of (\xi(t,x)) exist:
[
m_{\xi}^{(k)}(t_1,\ldots,t_k)=\int_G U^{t_1}g'(x)\cdots U^{t_k}g'(x)\,d\mu =
]
[
= \sum_{\substack{n_1,\ldots,n_k\in\mathfrak N'\
U^{t_1}\chi_{n_1}\cdots U^{t_k}\chi_{n_k}=1}}
C_{n_1}(g)\cdots C_{n_k}(g).
\tag{5}
]
Since (\mathfrak N') is finite, it follows from (5) that, for fixed (k), the set of values (\mathfrak M_k) of the function (m_{\xi}^{(k)}(t_1,\ldots,t_k)) consists of a finite number of points. But the semi-invariants (s_{\xi}^{(k)}(t_1,\ldots,t_k)) of the process (\xi(t)) (see (7), § 1) are expressed in terms of the moments (m_{\xi}^{(l)}(t_{i_1},\ldots,t_{i_l})), (1\le l\le k), (1\le i_s\le k), by formula II-c from ((^8)), which contains only a finite sum of products of a finite number of moments. Therefore, for fixed (k), the set of values (\mathfrak S_k) of the function (s_{\xi}^{(k)}(t_1,\ldots,t_k)) also consists only of a finite number of points.
But, by the theorem of V. A. Rokhlin (((^5)), § 3), every ergodic endomorphism of a compact commutative group is mixing of all degrees; hence, according to Theorem 6 in ((^9)), it follows that
[
s_{\xi}^{(k)}(t_1,\ldots,t_k)\underset{\max_{i,j}|t_i-t_j|\to\infty}{\longrightarrow}0
]
for every (k\ge 2). Since (\mathfrak S_k) is finite, it follows from this that for every (k\ge 2) there exists (D_k<\infty) such that
[
s_{\xi}^{(k)}(t_1,\ldots,t_k)=0
]
when (\max_{i,j}|t_i-t_j|>D_k). Consequently,
[
\sum_{t_1,\ldots,t_k=0}^{p-1}s_{\xi}^{(k)}(t_1,\ldots,t_k)=O(p)=o(bp^{k/2})
]
for (k\ge 3), since, by Lemma 1, (b_p=\sigma_g^2p+o(p)) and (\sigma_g>0). Hence, by Theorem 7 in ((^9)), relation (1) with (\sigma=\sigma_g) follows, and for the case in which only a finite number of Fourier coefficients (C_n(g)) are different from zero, the theorem is proved.
The theorem is extended without difficulty to the general case by using Lemma 1.
Corollary 1. If
[
\sum_n |C_n(g)|<\infty,
]
then (g(x)) satisfies the central limit theorem for every ergodic endomorphism.
Let us now consider the special case where (G) is the (k)-dimensional torus:
(x=(x_1,\ldots,x_k)), (0\le x_i<1) (see ((^5)), example 1, § 4). Let
[
\omega_{x_i}^{(2)}(\delta,g)=
]
[
=\sup_{0\le h\le\delta}
\left(
\int_0^1\cdots\int_0^1
\left|g(x_1,\ldots,x_{i-1},x_i+h,x_{i+1},\ldots,x_k)-g(x_1,\ldots,x_k)\right|^2
\,dx_1\cdots dx_k
\right)^{1/2}.
]
Theorem 1 allows us to obtain the following theorem:
Theorem 2. Let (G) be the (k)-dimensional torus; (\mu) the Lebesgue measure on it; (A) an integer square nonsingular matrix of order (k), among whose characteristic roots there are no roots of unity; (g(x)) a real-
a real function from (L^2(G)), periodic with period one in each variable (x_i), (1 \leq i \leq k), and such that for all (1 \leq i \leq k)
[
\omega_{x_i}^{(2)}(\delta,g)\leq \frac{C}{\left(\ln \frac{1}{\delta}\right)^\alpha},\qquad \alpha>1.
]
Then
[
\sum_{t=1}^{\infty}\left|(g(A^t x)-m_g),(g(x)-m_g)\right|<\infty;
\tag{6}
]
[
\lim_{p\to\infty}\mu\left{x:\frac{1}{\sqrt p}\left(\sum_{t=0}^{p-1}g(A^t x)-m_g p\right)<y\right}\Phi_{\sigma_g}(y)
\tag{7}
]
for all points of continuity of the function (\Phi_{\sigma_g}(y)), where (\Phi_\sigma(y)) is defined by formula (2), and (\sigma_g) by formula (4).
Remark 1. In terms of the rate of decrease of the Fourier coefficients of the function (g(x))
[
C_{n_1\ldots n_k}(g)=\int_0^1\cdots\int_0^1 g(x)e^{-2\pi i(n,x)}\,dx_1\cdots dx_k
]
a sufficient condition for (g(x)) to satisfy the central limit theorem can be expressed as follows: if the matrix (A) satisfies the conditions of Theorem 2 and
[
|C_{n_1\ldots n_k}(g)|\leq B\prod_{i=1}^k
\frac{1}{(1+|n_i|)^{1/2}(\ln(2+|n_i|))^\beta},
]
where (B<\infty), (\beta>3/2), then relations (6) and (7) hold.
In conclusion, the author expresses gratitude to A. N. Kolmogorov for his attention to this work, and also to A. G. Postnikov for valuable advice.
Mathematical Institute named after V. A. Steklov
Academy of Sciences of the USSR
Received
14 VI 1960
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