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UDC 519.21
MATHEMATICS
V. V. PETROV
ON THE ORDER OF GROWTH OF SUMS OF INDEPENDENT RANDOM VARIABLES
(Presented by Academician Yu. V. Linnik on 10 IX 1968)
Let \(\{X_n\}\) \((n=1,2,\ldots)\) be a sequence of independent random variables,
\[ S_n=\sum_{k=1}^n X_k. \]
The set of functions \(\psi(x)\) such that each \(\psi(x)\) is positive and nondecreasing in the region \(x>x_0\) for some \(x_0\), and the series \(\sum \frac{1}{n\psi(n)}\) converges (diverges), will be denoted by \(\Psi_c\) (respectively \(\Psi_d\))\(^*\).
1. Theorem 1. Let \(g(x)\) be an even continuous function, positive and strictly increasing in the region \(x>0\), with \(g(x)\to\infty\) as \(x\to\infty\). Suppose that one of the following two conditions is satisfied:
(A) \(x/g(x)\) is nondecreasing in the region \(x>0\).
(B) \(x/g(x)\) and \(g(x)/x^2\) are nonincreasing in the region \(x>0\).
Suppose, further,\(^ {**}\)
\[ E g(X_n)<\infty \qquad (n=1,2,\ldots), \tag{1} \]
\[ A_n\to\infty, \tag{2} \]
where
\[ A_n=\sum_{k=1}^n E g(X_k). \tag{3} \]
In addition, in the case when condition (B) is satisfied, assume that
\[ E X_n=0,\qquad (n=1,2,\ldots). \tag{4} \]
Then
\[ S_n=o\bigl(g^{-1}(A_n\psi(A_n))\bigr)\quad \text{a.s.} \tag{5} \]
for any function \(\psi(x)\in\Psi_c\). Here \(g^{-1}\) denotes the function inverse to \(g\).
The following theorem shows that if, instead of \(\psi(x)\in\Psi_c\), one takes a more slowly increasing function \(\psi(x)\in\Psi_d\), then relation (5) may fail.
Theorem 2. Let \(g(x)\) be a function satisfying the conditions of Theorem 1. For any function \(\psi(x)\in\Psi_d\) there exists a sequence of independent random variables \(\{X_n\}\) satisfying conditions (1), (2), (4) and the condition
\[ \limsup |S_n|/g^{-1}(A_n\psi(A_n))>0 \quad \text{a.s.} \tag{6} \]
2. We now give simply formulated consequences of these theorems, corresponding to the case \(g(x)=|x|^p,\ 0<p\le 2\).
Theorem 3. Let
\[ E|X_n|^p<\infty \qquad (n=1,2,\ldots) \tag{7} \]
\[
\text{* Here and everywhere below, } \sum f(n) \text{ denotes summation over all positive integers } n
\]
for which the values \(f(n)\) are defined and nonnegative. In the definitions of the sets \(\Psi_c\) and \(\Psi_d\), the value \(x_0\) is not assumed to be the same for different functions \(\psi\).
\[ \text{** Everywhere the indicated limits are as } n\to\infty,\text{ unless otherwise stated. The notation a.s. means almost surely.} \]
for some positive \(p \leqslant 2\). Let
\[ A_n=\sum_{k=1}^{n} E|X_k|^p \to \infty . \tag{8} \]
Then in the case \(0<p<1\) we have
\[ S_n=o\left([A_n\psi(A_n)]^{1/p}\right) \quad \text{a.s.} \tag{9} \]
for any function \(\psi(x)\in\Psi_c\), while in the case \(1\leqslant p\leqslant 2\) the same assertion is true under the additional condition (4).
Theorem 4. For any function \(\psi(x)\in\Psi_d\) and any positive number \(p\leqslant 2\), there exists a sequence of independent random variables \(\{X_n\}\) satisfying conditions (4), (7), (8) and the condition
\[ \limsup |S_n|/[A_n\psi(A_n)]^{1/p}>0 \quad \text{a.s.} \tag{10} \]
- The case \(p=2\) is of special interest. In this case, Theorems 3 and 4 immediately imply the following results.
Theorem 5. Let \(\{X_n\}\) be a sequence of independent random variables with finite variances \(DX_n=E(X_n^2)-(EX_n)^2\) \((n=1,2,\ldots)\). Put
\[ B_n=\sum_{k=1}^{n} DX_k . \]
If \(B_n\to\infty\), then
\[ S_n-ES_n=o\left(\sqrt{B_n\psi(B_n)}\right) \quad \text{a.s.} \tag{11} \]
for any function \(\psi(x)\in\Psi_c\).
Theorem 6. For any function \(\psi(x)\in\Psi_d\) there exists a sequence of independent random variables \(\{X_n\}\) with mathematical expectations equal to zero and finite variances such that
\[ B_n=\sum_{k=1}^{n} DX_k \to \infty \]
and
\[ \limsup |S_n|/\sqrt{B_n\psi(B_n)}>0 \quad \text{a.s.} \tag{12} \]
It follows from Theorem 5 that, for sums \(S_n\) of independent random variables with finite variances and unboundedly increasing variance of the sum \(B_n=DS_n\), the following estimates of the order of growth hold, each of which is stronger than the preceding one: for any \(\varepsilon>0\),
\[ S_n-ES_n=o\left(B_n^{1/2+\varepsilon}\right) \quad \text{a.s.}, \]
\[ S_n-ES_n=o\left(B_n^{1/2}(\log B_n)^{1/2+\varepsilon}\right) \quad \text{a.s.}, \]
\[ S_n-ES_n=o\left(B_n^{1/2}(\log B_n)^{1/2}(\log\log B_n)^{1/2+\varepsilon}\right) \quad \text{a.s.}, \]
and so on. By virtue of Theorem 6, in these estimates one cannot replace \(\varepsilon>0\) by zero without introducing additional assumptions.
- The results obtained can be used in studying conditions for the applicability of the strong law of large numbers with the simplest normalization, namely in the form
\[ \frac{1}{n}(S_n-ES_n)\to 0 \quad \text{a.s.} \tag{13} \]
As is known, the condition
\[ B_n=o(n^2) \tag{14} \]
is sufficient for the applicability of the weak law of large numbers, i.e. for the convergence of \(\frac{1}{n}(S_n-ES_n)\) to zero in probability. With the help of Theorem 5 one can indicate a strengthening of condition (14) that ensures the applicability of the strong law of large numbers.
Theorem 7. If
\[ B_n=O(n^2/\psi(n)) \tag{15} \]
for some function \(\psi(x)\in\Psi_c\), then relation (13) holds.
On the other hand, whatever the function \(\psi(x) \in \Psi_d\) for which \(n/\psi(n)\) does not decrease in the domain \(n > n_0\), for some \(n_0\), there exists a sequence of independent random variables \(\{X_n\}\) with finite variances for which (15) is satisfied, but
\[ P\left\{ \frac{1}{n}(S_n - ES_n) \to 0 \right\} = 0. \]
Let us give one consequence of Theorem 7. The condition \(B_n = O(n^2/(\log n)^{1+\delta})\) for some \(\delta > 0\) is sufficient for the applicability of the strengthened law of large numbers to a sequence of independent random variables \(\{X_n\}\), i.e., for relation (13). If, however, the condition \(B_n = O(n^2/\log n)\) or even the stronger condition \(B_n = O(n^2/\log n \log \log n)\) is satisfied, then relation (13) may fail to hold.
Leningrad State University
named after A. A. Zhdanov
Received
2 IX 1968