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Bennett's inequality, an upper bound on the probability that the sum of independent random variables deviates from its expected value by more than any specified amount; Bhatia–Davis inequality, an upper bound on the variance of any bounded probability distribution; Bernstein inequalities (probability theory) Boole's inequality; Borell–TIS ...
Such articles are marked here by a code of the form (X:Y), which refers to number of random variables involved and the type of the distribution. For example (2:DC) indicates a distribution with two random variables, discrete or continuous. Other codes are just abbreviations for topics. The list of codes can be found in the table of contents.
Doob's inequality for discrete-time martingales implies Kolmogorov's inequality: if X 1, X 2, ... is a sequence of real-valued independent random variables, each with mean zero, it is clear that
An unbiased random walk, in any number of dimensions, is an example of a martingale. For example, consider a 1-dimensional random walk where at each time step a move to the right or left is equally likely. A gambler's fortune (capital) is a martingale if all the betting games which the gambler plays are fair.
In mathematics — specifically, in stochastic analysis — the infinitesimal generator of a Feller process (i.e. a continuous-time Markov process satisfying certain regularity conditions) is a Fourier multiplier operator [1] that encodes a great deal of information about the process.
Jensen's inequality provides a simple lower bound on the moment-generating function: (), where is the mean of X. The moment-generating function can be used in conjunction with Markov's inequality to bound the upper tail of a real random variable X.
A special case of MDS, denoted as {X t, t} 0 is known as innovative sequence of S n; where S n and are corresponding to random walk and filtration of the random processes {}. In probability theory innovation series is used to emphasize the generality of Doob representation .
Another example of a complex random variable is the uniform distribution over the filled unit circle, i.e. the set {| |}. This random variable is an example of a complex random variable for which the probability density function is defined. The density function is shown as the yellow disk and dark blue base in the following figure.