Abstract
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MATHEMATICS
V. I. LADOHIN
CRAMÉR’S, LINDEBERG’S, AND CHEBYSHEV’S THEOREMS FOR COMPLEX DISTRIBUTIONS
(Presented by Academician A. N. Kolmogorov on 19 XI 1967)
A complex continuous function of bounded variation satisfying the conditions \(F(-\infty)=0,\ F(+\infty)=1\) is called the distribution function \(F(x)\) of a random variable \(\xi\) taking values in \(R=(-\infty,+\infty)\). Denote by \(V(F,x)\) the variation of \(F(x)\) on \((-\infty,x)\). Many properties of classical (real, nondecreasing) distribution functions and of the corresponding characteristic functions
\[ f(t)=\int (\exp itx)\,dF(x) \]
are preserved, in particular the theorem on the one-to-one correspondence between distribution functions and characteristic functions. A sequence of characteristic functions \(f_n(t)\) is called convergent to the characteristic function \(f(t)\) \((f_n\Rightarrow f)\), if \(f_n\) converges to \(f\) in the sense of weak convergence over the space \(K\). A sequence of distribution functions \(F_n\) converges to the distribution function \(F\) in the sense of weak convergence over the space \(Z\) if and only if the sequence of the corresponding characteristic functions \(f_n\Rightarrow f\). For the definition of the spaces \(K\) and \(Z\), see \((^1)\).
A random variable \(\xi\) is called normally distributed, or normal with parameter \(a_{2q}^{r}=(a_1,\ldots,a_{2q},\ldots,a_r)\) \((q\) and \(r\) are integers, \(2q\le r)\), if its characteristic function has the form
\[ \varphi(t)=\exp \sum_{p=1}^{r} a_p t^p . \]
The function \(\varphi(t)\) will be characteristic if and only if the equation
\[ \frac{\partial u}{\partial \tau} = \sum_{p=1}^{r} a_p \left(i\frac{\partial}{\partial x}\right)^p u \]
is parabolic in the sense of G. E. Shilov \((^2)\), or hyperbolic (when \(r=1\)).
Addition theorem. The sum of independent normal variables is normal. The parameter of the sum is the sum of the parameters.
The proof is obvious.
Converse theorem (Cramér). Let the random variables \(\xi_1\) and \(\xi_2\) be independent, and let their distribution functions \(F_1\) and \(F_2\) satisfy the conditions \((i=1,2)\):
\[ \text{K1. }\quad V(F_i,R)-V(F_i,x)<A\exp(-\varepsilon x^{\gamma_i}). \]
\[ \text{K2. }\quad V(F_i,-x)<A\exp(-\varepsilon x^{\gamma_i}) \]
for \(x>0\) and some \(\varepsilon>0,\ \gamma_i>1\). If the sum \(\xi=\xi_1+\xi_2\) is normal with parameter \(a_{2q}^{r}\), then \(\xi_1\) and \(\xi_2\) are also normal with parameters \(a_{2q_1}^{r_1},\ a_{2q_2}^{r_2}\), respectively, where \(r_1\le \gamma_1',\ r_2\le \gamma_2',\ \max(q_1,q_2)=q\), where \(1/\gamma_i+1/\gamma_i'=1\).
Proof. Elementary estimates show that \(f_i(t)\) are entire analytic functions of the complex variable \(t\) of growth order
\(\leqslant \gamma_i'\). Obviously, the \(f_i(t)\) have no zeros. By Hadamard’s theorem,
\[ f_i(t)=\exp P_i(t), \]
where \(P_i(t)\) is a polynomial of degree \(r_i \leqslant \gamma_i'\).
Remark. For real symmetric distributions \(F_1, F_2\) and the ordinary normal distribution with parameter \(a_2^2=(0,1)\), the corresponding converse theorem was proved in \((^3)\). There, too, an example is constructed showing that the converse theorem becomes false even if, in conditions K1, K2, \(x^{\gamma_i}\) is replaced by \(x\ln x\).
Central limit theorem (Lindeberg). Let \(\xi_1,\ldots,\xi_k,\ldots\) be a sequence of independent random variables; \(F_1,\ldots,F_k,\ldots\) and \(f_1,\ldots,f_k,\ldots\) the sequence of the corresponding distribution functions and characteristic functions. Suppose that the following conditions are satisfied:
L1. For every \(k=1,2,\ldots\) and some integer \(q\geqslant 1\),
\[ \int x\,dF_k(x)=\cdots=\int x^{2q-1}\,dF_k(x)=0;\qquad \int x^{2q}\,dF_k(x)=(-1)^{q+1}(a_k+ib_k). \]
L2. There exists a constant \(C<\infty\) such that, for every \(n\),
\[ \frac{1}{n}\sum_{k=1}^{n}\int x^{2q}\,dV(F_k,x)\leqslant C. \]
L3. Uniformly in \(t\) on every finite interval,
\[ \max_{1\leqslant k\leqslant n}\left|\,f_k\left(\frac{t}{n^{1/2q}}\right)-1\,\right|\to 0 \qquad (n\to\infty). \]
L4. For every \(\lambda>0\), the limit
\[ \lim_{n\to\infty}\frac{1}{n}\sum_{k=1}^{n} \int_{|x|>\lambda n^{1/2q}} x^{2q}\,dV(F_k,x)=0. \]
L5. There exists the limit
\[ \lim_{n\to\infty}\frac{1}{n}\sum_{k=1}^{n}(a_k+ib_k)=a+ib. \]
Then the distributions of the sums
\[ \eta_n=\frac{1}{n^{1/2q}}\sum_{k=1}^{n}\xi_k \]
have a limit if and only if \(a>0\). This limit is the normal distribution with parameter
\[ a_{2q}^{2q}=(0,\ldots,0,-(a+ib)/(2q)!). \]
Corollary 1. The preceding theorem remains true if condition L3 is replaced by the condition
L6. As \(n\to\infty\),
\[ \max_{1\leqslant k\leqslant n}\frac{1}{n}\int x^{2q}\,dV(F_k,x)\to 0 \]
or by the condition
L7. The limit
\[ \lim_{\varepsilon\to 0}\max_{\substack{1\leqslant k\leqslant n\\ 1\leqslant n<\infty}} \frac{1}{n} \int_{|x|<\varepsilon n^{1/2q}} x^{2q}\,dV(F_k,x)=0. \]
The theorem is true if, instead of condition L4, the following condition is satisfied:
L8. There is a \(\delta>0\) such that, as \(n\to\infty\),
\[ \frac{1}{n^{2q+\delta}}\sum_{k=1}^{n}\int |x|^{2q+\delta}\,dV(F_k,x)\to 0. \]
Corollary 2. Suppose the following condition of uniform boundedness of the variations is satisfied:
L9. There exists a constant \(B<\infty\) such that, for every \(k=1,2,\ldots\),
\[ \int dV(F_k,x)\leqslant B. \]
Then the preceding theorem remains true even without condition L3.
Remark. For the case when \(\xi_1,\ldots,\xi_k,\ldots\) have real densities and conditions L1, L2, L7, L4 are satisfied, the central limit theorem for normalized sums was proved in [4].
Proof. First of all, it is quite easily verified that
\[ \mathrm{L1},\mathrm{L6}\to\mathrm{L3};\qquad \mathrm{L1},\mathrm{L4},\mathrm{L7}\to\mathrm{L3};\qquad \mathrm{L8}\to\mathrm{L7};\qquad \mathrm{L9},\mathrm{L4}\to\mathrm{L3}. \]
These relations prove the corollaries. In proving the theorem it is convenient to introduce the random variables \(\xi_{nk}=\xi_k/n^{1/2q}\) with characteristic functions \(f_{nk}\) \((1\leqslant k\leqslant n)\). Let \(\varphi_n(t)\) be the characteristic function of the variable
\[ \eta_n=\sum_{k=1}^{n}\xi_{nk}. \]
It follows from L3 that \(\ln\varphi_n(t)\) exists and
\[ \ln\varphi_n(t)=\sum_{k=1}^{n}(f_{nk}(t)-1)+R_n(t), \]
where
\[ |R_n(t)|\leqslant \max_{1\leqslant k\leqslant n}|f_{nk}(t)-1|\cdot \sum_{k=1}^{n}|f_{nk}(t)-1|. \]
From conditions L1, L2 it follows that
\[ \sum_{k=1}^{n}|f_{nk}(t)-1|\leqslant C\frac{t^{2q}}{(2q)!}. \]
Thus, if conditions L1, L2, L3 are satisfied, then \(|R_n(t)|\to0\) uniformly in \(t\) on every finite interval. From condition L1,
\[ \sum_{k=1}^{n}(f_{nk}(t)-1) = -\frac{1}{n}\frac{t^{2q}}{(2q)!}\sum_{k=1}^{n}(a_k+ib_k)+\rho_n(t), \]
where
\[ \rho_n(t)=\sum_{k=1}^{n}\int\left(e^{itx}-1-\cdots-\frac{(itx)^{2q}}{(2q)!}\right)dF_{nk}(x). \]
From conditions L2, L4, \(|\rho_n(t)|\to0\) uniformly on every finite interval. Using also condition L5, we obtain that
\[ \varphi_n(t)\to \exp\left[-\frac{t^{2q}}{(2q)!}(a+ib)\right] \]
uniformly on every finite interval, and therefore also in the weak sense over \(K\). The theorem is proved.
A sequence of random variables \(\xi_1,\ldots,\xi_k,\ldots\) is called convergent to 0 if the sequence of the corresponding characteristic functions \(f_n(t)\Rightarrow 1\), or, equivalently, if \(F_n(x)\Rightarrow \varepsilon(x)\).
Law of large numbers. Chebyshev’s theorem. Let \(\xi_1,\ldots,\xi_k,\ldots\) be a sequence of independent random variables, and let \(F_1,\ldots,F_k,\ldots\) be the sequence of the corresponding distribution functions. If the following conditions are satisfied:
T1. For \(k=1,2,\ldots\) and some integer \(p\geqslant1\),
\[ \int x\,dF_k(x)=\cdots=\int x^{p-1}\,dF_k(x)=0;\qquad \int x^p\,dF_k(x)=c_k+id_k. \]
T2. For some \(\alpha>0\), as \(n\to\infty\),
\[ \frac{1}{n^{\alpha p}}\sum_{k=1}^{n}\int |x|^p\,dV(F_k,x)\to 0, \]
then the sequence of random variables
\[ \eta_n=\frac{1}{n^\alpha}\sum_{k=1}^{n}\xi_k \]
tends to \(0\) as \(n\to\infty\).
Proof. Let \(\varphi_n(t)\) be the characteristic function of \(\eta_n\). Introduce the quantities \(\xi_{nk}=\xi_k/n^\alpha\). From conditions T1, T2,
\[ \max_{1\le k\le n}|f_{nk}(t)-1| \le \frac{t^p}{p!}\max_{1\le k\le n}\int |x|^p\,dV(F_{nk},x)\to 0 \qquad (n\to\infty). \]
Consequently, there exists and is finite
\[ \ln\varphi_n(t)=\ln\prod_{k=1}^{n}f_{nk}(t). \]
When
\[ \max_{1\le k\le n}|f_{nk}(t)-1|<\frac12, \]
the inequality
\[ |\ln\varphi_n(t)| \le \left(1+\max_{1\le k\le n}|f_{nk}(t)-1|\right) \sum_{k=1}^{n}|f_{nk}(t)-1| \]
is valid. From conditions T1, T2,
\[ \sum_{k=1}^{n}|f_{nk}(t)-1| \le \frac{t^p}{p!}\sum_{k=1}^{n}\int |x|^p\,dV(F_{nk},x)\to 0. \]
Thus, \(\varphi_n(t)\to 1\) uniformly in \(t\) on every finite interval. The theorem is proved.
Remark. For \(p=2\), \(\alpha=1\), and real nondecreasing \(F_1,\ldots,\ldots,F_k,\ldots\), we obtain the classical Chebyshev theorem.
Kazan State University
named after V. I. Ulyanov-Lenin
Received
7 IV 1964
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