probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...

, Etemadi's inequality is a so-called "maximal inequality", an

inequality Inequality may refer to: Economics * Attention inequality, unequal distribution of attention across users, groups of people, issues in etc. in attention economy * Economic inequality, difference in economic well-being between population groups * ...

that gives a bound on the

probability Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...

that the

partial sum In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, math ...

s of a

finite Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb Traditionally, a finite verb (from la, fīnītus, past partici ...

collection of

independent random variables Independent or Independents may refer to: Arts, entertainment, and media Artist groups * Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s * Independen ...

exceed some specified bound. The result is due to Nasrollah Etemadi.

Statement of the inequality

Let ''X''₁, ..., ''X''_''n'' be independent real-valued random variables defined on some common

probability space In probability theory, a probability space or a probability triple (\Omega, \mathcal, P) is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models t ...

, and let ''α'' ≥ 0. Let ''S''_''k'' denote the partial sum :

S_k = X_1 + \cdots + X_k.\,

Then :

\Pr \Bigl( \max_ ,  S_k ,  \geq 3 \alpha \Bigr) \leq 3 \max_ \Pr \bigl( ,  S_k ,  \geq \alpha \bigr).

Remark

Suppose that the random variables ''X''_''k'' have common

expected value In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a ...

zero. Apply

Chebyshev's inequality In probability theory, Chebyshev's inequality (also called the Bienaymé–Chebyshev inequality) guarantees that, for a wide class of probability distributions, no more than a certain fraction of values can be more than a certain distance from th ...

to the right-hand side of Etemadi's inequality and replace ''α'' by ''α'' / 3. The result is Kolmogorov's inequality with an extra factor of 27 on the right-hand side: :

\Pr \Bigl( \max_ ,  S_k ,  \geq \alpha \Bigr) \leq \frac \operatorname (S_n).

References

* (Theorem 22.5) * {{cite journal , last=Etemadi , first=Nasrollah , title=On some classical results in probability theory , journal=Sankhyā Ser. A , volume=47 , year=1985 , pages=215–221 , mr=0844022 , jstor = 25050536 , issue=2 Probabilistic inequalities Statistical inequalities