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In probability theory, a Chernoff bound is an exponentially decreasing upper bound on the tail of a random variable based on its moment generating function. The minimum of all such exponential bounds forms the Chernoff or Chernoff-Cramér bound , which may decay faster than exponential (e.g. sub-Gaussian ).
In probability theory, a subgaussian distribution, the distribution of a subgaussian random variable, is a probability distribution with strong tail decay. More specifically, the tails of a subgaussian distribution are dominated by (i.e. decay at least as fast as) the tails of a Gaussian.
In probability theory, Bernstein inequalities give bounds on the probability that the sum of random variables deviates from its mean. In the simplest case, let X 1, ..., X n be independent Bernoulli random variables taking values +1 and −1 with probability 1/2 (this distribution is also known as the Rademacher distribution), then for every positive ,
In probability theory and statistics, the moment-generating function of a real-valued random variable is an alternative specification of its probability distribution.Thus, it provides the basis of an alternative route to analytical results compared with working directly with probability density functions or cumulative distribution functions.
Therefore, the theorem above gives a tighter bound than the Ahlswede–Winter result. The chief contribution of (Ahlswede & Winter 2003) was the extension of the Laplace-transform method used to prove the scalar Chernoff bound (see Chernoff bound#Additive form (absolute error)) to the case of self-adjoint
This is a generalization of Hoeffding's since it can handle random variables with not only almost-sure bound but both almost-sure bound and variance bound. 6. Chernoff bounds have a particularly simple form in the case of sum of independent variables, since [] = = [].
The Chernoff bound of the Q-function is () ... As in the one dimensional case, there is no simple analytical formula for the Q-function. Nevertheless, ...
This bound is rather sharp, in the sense that () cannot be replaced with a larger number which would yield a strict inequality for all positive . [3] However, the exponential bound can still be reduced by a subexponential factor on the order of 1 / N {\displaystyle 1/{\sqrt {N}}} ; this follows from the Stirling approximation applied to the ...