WebThe Chernoff bound is like a genericized trademark: it refers not to a particular inequality, but rather a technique for obtaining exponentially decreasing bounds on tail probabilities. … 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 or exponential moments. The minimum of all such exponential bounds forms the Chernoff or Chernoff-Cramér bound, which may decay … See more The generic Chernoff bound for a random variable $${\displaystyle X}$$ is attained by applying Markov's inequality to $${\displaystyle e^{tX}}$$ (which is why it sometimes called the exponential Markov or exponential … See more The bounds in the following sections for Bernoulli random variables are derived by using that, for a Bernoulli random variable $${\displaystyle X_{i}}$$ with probability p of being equal to 1, One can encounter … See more Rudolf Ahlswede and Andreas Winter introduced a Chernoff bound for matrix-valued random variables. The following version of the … See more The following variant of Chernoff's bound can be used to bound the probability that a majority in a population will become a minority in a sample, or vice versa. Suppose there is a general population A and a sub-population B ⊆ A. Mark the relative size of the … See more When X is the sum of n independent random variables X1, ..., Xn, the moment generating function of X is the product of the individual moment generating functions, giving that: See more Chernoff bounds may also be applied to general sums of independent, bounded random variables, regardless of their distribution; this is known as Hoeffding's inequality. The proof follows a similar approach to the other Chernoff bounds, but applying See more Chernoff bounds have very useful applications in set balancing and packet routing in sparse networks. The set balancing problem arises while designing statistical … See more
Cherno bounds, and some applications 1 Preliminaries
WebNov 23, 2024 · Siegel, A.: Towards a usable theory of Chernoff–Hoeffding bounds for heterogeneous and partially dependent random variables (manuscript) Van de Geer, … Webn be independent random variables with values in the interval r0;1s. If X X 1 X 2 X n and ErXs , then for every a ¡0 we get the bounds 1 PrrX ¥ as⁄e a 2{2n, 2 PrrX ⁄ as⁄e a 2{2n. … line the design firm uphar chibber
Chernoff bound - Wikipedia
WebSince the application of the Chernoff-Hoeffding bound above does not change if the subset defined by R q does not change, to prove Theorem 2.8.1 we need to show (2.3) holds … WebApr 22, 2024 · Then holding the lower and upper bounds of the numerator constant, try to get concentration bounds for the (lower(or upper) bound/denominator random variable) … WebJun 7, 2016 · How to apply Chernoff's bound when variables are not independent. Let X = ∑ i = 1 n X i, for Bernoulli random variables X i which are not necessarily independent. … line the coffers