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Informally, Doob's inequality states that the expected value of the process at some final time controls the probability that a sample path will reach above any particular value beforehand. As the proof uses very direct reasoning, it does not require any restrictive assumptions on the underlying filtration or on the process itself, unlike for ...
Matiyasevich's theorem, also called the Matiyasevich–Robinson–Davis–Putnam or MRDP theorem, says: . Every computably enumerable set is Diophantine, and the converse.. A set S of integers is computably enumerable if there is an algorithm such that: For each integer input n, if n is a member of S, then the algorithm eventually halts; otherwise it runs forever.
In mathematics, the QM-AM-GM-HM inequalities, also known as the mean inequality chain, state the relationship between the harmonic mean, geometric mean, arithmetic mean, and quadratic mean (also known as root mean square). Suppose that ,, …, are positive real numbers. Then
Grönwall's inequality is an important tool to obtain various estimates in the theory of ordinary and stochastic differential equations. In particular, it provides a comparison theorem that can be used to prove uniqueness of a solution to the initial value problem; see the Picard–Lindelöf theorem. It is named for Thomas Hakon Grönwall (1877 ...
For instance, to solve the inequality 4x < 2x + 1 ≤ 3x + 2, it is not possible to isolate x in any one part of the inequality through addition or subtraction. Instead, the inequalities must be solved independently, yielding x < 1 / 2 and x ≥ −1 respectively, which can be combined into the final solution −1 ≤ x < 1 / 2 .
The uses of this inequality are not limited to applications in probability theory.One example of its use in analysis is the following: if we let be a linear operator between two L p spaces (,) and (,), < <, with bounded norm ‖ ‖ <, then one can use Khintchine's inequality to show that
Times Square New Year's Eve celebration is in its 120th year. The tradition of thousands of people crowding to watch a shimmering ball be lowered in Times Square began in 1907, though the first ...
Note that the expression of Pinsker inequality depends on what basis of logarithm is used in the definition of KL-divergence. () = .Given the above comments, there is an alternative statement of Pinsker's inequality in some literature that relates information divergence to variation distance: