Webfirst estimate of this kind can be traced back to Cramer’s paper [´ 6], which deals with variables possessing a density and exponential moments. In [5] Chernoff relaxed the first assumption. Bahadur [2] finally gave a proof without any assumption on the law of X1. Coming from statistical mechanics, Lanford imported the subadditivity argument 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, 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 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 … 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. … See more Rudolf Ahlswede and Andreas Winter introduced a Chernoff bound for matrix-valued random variables. The following version of the inequality can be found in the work of Tropp. See more

切尔诺夫限 - 百度百科

WebApr 29, 2024 · 3. Let X denote a standard normal random variable. As the comments and Chaconne's answer have noted, the question here is to bound P { X > x } = 2 Q ( x) where Q ( x) = 1 − Φ ( x) is the complementary normal distribution function. Now, a well-known bound is. (1) Q ( x) < 1 2 e − x 2 / 2 for x > 0. which immediately gives. WebLet X,Xn,n ≥ 1 X, X n, n ≥ 1 be a sequence of independent and identically distributed random variables. The classical Cramér-Chernoff large deviation states that limn→∞n−1lnP ((∑n i=1Xi)/n ≥x) =lnρ(x) lim n → ∞ n − 1 ln P ( ( ∑ i = 1 n X i) / n ≥ x) = ln ρ ( x) if and only if the moment generating function of X X is ... remember no anime

An Extended Perron–Frobenius Theorem and Large Deviation

Web摘要. laksa是2024新加坡科技设计大学的博后提出的一种基于链的权益证明协议,laksa通过设计支持大量节点，并提供概率安全保证，客户端通过基于其区块链视图计算事务恢复的概率来做出提交决策,轻量级委员会投票将节点之间的交互降至最低，从而产生比竞争算力要更简单、更健壮、更可扩展的协议。 WebJul 1, 2009 · In this paper a Chernoff type theorem for the L1 distance between kernel estimators from two independent and identically distributed random samples is developed. The harmonic mean is used to correct the distance for inequal sample sizes case. Moreover, the proved result is used to compute the Bahadur slope of a test based on L1 distance … WebIn this lecture, two methods of finding sub-Gaussian tail bounds are highlighted. In particular, the concepts introduced are the use of Talagrand’s concentration function and the Cramér-Chernoff bound. In order to sketch a brief heuristic, if we intend to summarize the behavior of a random variable by a few estimates, then broadly speaking, the mean or median … professor ian galton