MATHEMATICS

V. V. PETROV

ONE-SIDED INEQUALITIES OF CHEBYSHEV TYPE

(Presented by Academician V. I. Smirnov on 21 X 1963)

1. Numerous generalizations and refinements of Chebyshev’s inequality are known, according to which

\[ \mathbf P\{|X|\geqslant \varepsilon\}\leqslant \frac{E(X^2)}{\varepsilon^2} \tag{1} \]

for any random variable \(X\) and any \(\varepsilon>0\). (Surveys of these results are contained in \((^{1,2})\).) Here it is usually assumed that the random variables under consideration have finite moments of some order. For example, inequality (1) is nontrivial only when \(E(X^2)<\infty\). Such conditions impose restrictions on the behavior of the distribution function \(F(x)\) of the random variable \(X\) on the whole real line. At the same time, one can obtain inequalities for the probabilities of events of the form \(X\geqslant \varepsilon\) or \(X\leqslant -\varepsilon\) under conditions concerning the behavior of the distribution function \(F(x)\) only on one of the half-lines. These inequalities may be nontrivial also in the case when \(E|X|^r=\infty\) for every \(r\geqslant 1\).

1. We shall give the following direct generalization of Chebyshev’s inequality.

Theorem 1. Let \(X\) be a random variable with arbitrary distribution function \(F(x)=\mathbf P(X<x)\), and let \(c\) be any nonnegative constant. Let \(g(x)\) be a nonnegative even function, defined in the domain \(|x|\geqslant c\) and nondecreasing for \(x\geqslant c\). Further, let \(t\) be any constant satisfying the conditions \(t>c\), \(g(t-0)>0\). Then

\[ \mathbf P(X\geqslant t)\leqslant \frac{1}{g(t-0)} \int_{c+0}^{+\infty} g(x)\,dF(x), \tag{2} \]

\[ \mathbf P(X\leqslant -t)\leqslant \frac{1}{g(t-0)} \int_{-\infty}^{-c-0} g(x)\,dF(x). \tag{3} \]

Proof. The proof of Theorem 1 is as simple as the proof of Chebyshev’s inequality. Namely,

\[ \int_{c+0}^{+\infty} g(x)\,dF(x)\geqslant \int_{t-0}^{+\infty} g(x)\,dF(x)\geqslant g(t-0)\int_{t-0}^{+\infty} dF(x) = g(t-0)\mathbf P(X\geqslant t). \]

This implies (2). Inequality (3) is proved in the same way.

In order for inequality (2) to hold, it is sufficient to require that \(g(x)\) be a nonnegative function nondecreasing for \(x\geqslant c\).

From (2) and (3), for \(c=0\), it follows that for any \(t>0\)

\[ \mathbf P(|X|\geqslant t)\leqslant \frac{1}{g(t-0)} \int_{-\infty}^{+\infty} g(x)\,dF(x), \tag{4} \]

where the prime means that the point \(x=0\) is excluded from the domain of integration. It is easy to indicate such distribution laws for which, for a given function \(g\), inequality (4) is satisfied trivially (i.e., the right-hand side of (4) is not less than one), while one of the inequalities (2) and (3) is nontrivial.

Putting in Theorem 1 \(g(x)=|x|^r\) \((r>0)\) and \(c=0\), we obtain the following result.

Theorem 2. Let \(X\) be a random variable with arbitrary distribution function \(F(x)\). For any \(t>0\) and \(r>0\) we have
\[ \mathbf P(X\geq t)\leq \frac{1}{t^r}\int_{+0}^{+\infty} x^r\,dF(x), \tag{5} \]
\[ \mathbf P(X\leq -t)\leq \frac{1}{t^r}\int_{-\infty}^{-0} |x|^r\,dF(x). \tag{6} \]

Inequalities (2)—(6) become equalities for some special distributions. For example, if \(X\) has two possible values \(-1\) and \(+1\), to each of which corresponds probability \(1/2\), then \(\mathbf P(X\geq 1)=\mathbf P(X\leq -1)=1/2\), and the right-hand sides of inequalities (5) and (6) for \(t=1\) are also equal to \(1/2\). More interesting are extremal distributions with infinite moments. Let the random variable \(X\) have distribution function \(F(x)\), which for \(x\leq 0\) has derivative \(F'(x)=C(1+|x|^{1+\delta})^{-1}\), where \(\delta\) is an arbitrarily small positive constant, and the other constant \(C\) is determined from the condition \(F(0)=1/2\). Let, further, \(F(1+0)-F(1-0)=\mathbf P(X=1)=1/2\). Obviously, \(E|X|^r=\infty\) for any \(r\geq \delta\). Inequality (5) for \(t=1\) takes the form \(\mathbf P(X\geq 1)\leq 1/2\).

1. We now give a one-sided analogue of the inequality of S. N. Bernstein [3].

Theorem 3. Let \(X_1,X_2,\ldots,X_n\) be mutually independent random variables. Put
\[ m_{kj}=\int_c^{+\infty} x^k\,dF_j(x), \]
where \(F_j(x)\) is the distribution function of the random variable \(X_j\). Suppose that for some value \(c\) in the range \(-\infty\leq c\leq 0\) there exists a positive constant \(H\) such that
\[ |m_{kj}|\leq \frac{k!}{2}H^{k-2}m_{2j} \tag{7} \]
for all integers \(k\geq 2\) and \(j=1,\ldots,n\). Suppose, further, that \(m_{2j}<\infty\) for the same \(c\) and \(j=1,\ldots,n\). Then
\[ \mathbf P\left(\sum_{j=1}^n X_j-\sum_{j=1}^n m_{1j}\geq xM_n\right)\leq e^{-x^2/4} \tag{8} \]
for \(0<x\leq \dfrac{M_n}{H}\), where \(M_n^2=\sum_{j=1}^n m_{2j}\).

Proof. The proof of this theorem is a modification of the proof of Bernstein’s inequality. Choose the constant \(\varepsilon\) subject to the condition \(0<\varepsilon\leq \dfrac{1}{2H}\). Put
\[ R_j=Ee^{\varepsilon X_j},\qquad r_j=\int_c^\infty e^{\varepsilon x}\,dF_j(x)\quad (j=1,\ldots,n). \]

Expanding \(e^{\varepsilon x}\) in a power series, we obtain

\[ r_j=\sum_{p=0}^{\infty}\frac{\varepsilon^p}{p!}\int_c^{\infty}x^p\,dF_j(x) =\sum_{p=0}^{\infty}\frac{\varepsilon^p}{p!}\,m_{pj}. \]

The series \(\sum_{p=2}^{\infty}\frac{\varepsilon^p}{p!}m_{pj}\) is majorized, by virtue of (7), by the series
\(\sum_{p=0}^{\infty}(\varepsilon H)^p\frac{\varepsilon^2}{2}m_{2j}\), whose sum is equal to
\(\frac{\varepsilon^2 m_{2j}}{2(1-\varepsilon H)}\leqslant \varepsilon^2 m_{2j}\). Therefore

\[ \begin{aligned} R_j &=\int_{-\infty}^{c} e^{\varepsilon x}\,dF_j(x)+r_j \leqslant \int_{-\infty}^{c} dF_j(x)+r_j \leqslant 1+\varepsilon m_{1j}+\varepsilon^2 m_{2j} \\ &\leqslant \exp\{\varepsilon m_{1j}+\varepsilon^2 m_{2j}\}. \end{aligned} \tag{9} \]

Applying Chebyshev’s inequality, we obtain

\[ \mathbf P\left(\exp\left\{\varepsilon\sum_{j=1}^{n}X_j\right\}\geqslant e^{x^2/4}R_1\cdots R_n\right) \leqslant e^{-x^2/4}. \]

Taking (9) into account, we obtain from this

\[ \mathbf P\left(\varepsilon\sum_{j=1}^{n}X_j\geqslant \frac{x^2}{4}+\varepsilon\sum_{j=1}^{n}m_{1j} +\varepsilon^2 M_n^2\right) \leqslant e^{-x^2/4}. \tag{10} \]

If we take here \(\varepsilon=x/2M_n\), then for \(0<x\leqslant M_n/H\) we shall have
\(0<\varepsilon\leqslant 1/2H\). From (10), after small computations, we obtain (8).

For \(c=-\infty\) and \(EX_j=0\) \((j=1,\ldots,n)\), Theorem 3 reduces to Bernstein’s theorem. A result close to Theorem 3 was obtained by V. M. Zolotarev\({}^{4}\) under the additional condition \(E|X_j|<\infty\) \((j=1,\ldots,n)\).

It is of interest to seek one-sided analogues of other probability inequalities of Chebyshev type.

Leningrad State University
named after A. A. Zhdanov

Received
2 X 1963

CITED LITERATURE

\({}^{1}\) H. J. Godwin, J. Am. Statist. Assoc., 50, 923 (1955).
\({}^{2}\) I. R. Savage, J. Res. Nat. Bur. Stand., 65B, No. 3, 211 (1962); Collected Translations: Mathematics, 6, No. 4, 71 (1962).
\({}^{3}\) S. N. Bernstein, Theory of Probability, 1946.
\({}^{4}\) V. M. Zolotarev, Proceedings of the Fourth All-Union Conference on Probability Theory and Mathematical Statistics, 1962, p. 43.