In
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 ...
, Ville's inequality provides an
upper 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 a
supermartingale
In probability theory, a martingale is a sequence of random variables (i.e., a stochastic process) for which, at a particular time, the conditional expectation of the next value in the sequence is equal to the present value, regardless of all ...
exceeds a certain value. The
inequality is named after
Jean Ville
Jean Ville, also known under the names Jean-André Ville et André Ville, born 24 June 1910 in Marseille, died 22 January 1989 in Blois, was a French mathematician. He is known for having proved an extension of von Neumman's minimax theorem, as we ...
, who proved it in 1939.
[
]
The inequality has applications in statistical testing.
Statement
Let
be a non-negative supermartingale. Then, for any
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
:
The inequality is a
generalization
A generalization is a form of abstraction whereby common properties of specific instances are formulated as general concepts or claims. Generalizations posit the existence of a domain or set of elements, as well as one or more common characteri ...
of
Markov's inequality.
References
Probabilistic inequalities
Martingale theory
{{Probability-stub