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Notice that for the condition to be satisfied, it is not possible that for each n the random variables X and X n are independent (and thus convergence in probability is a condition on the joint cdf's, as opposed to convergence in distribution, which is a condition on the individual cdf's), unless X is deterministic like for the weak law of ...
In probability, weak dependence of random variables is a generalization of independence that is weaker than the concept of a martingale [citation needed].A (time) sequence of random variables is weakly dependent if distinct portions of the sequence have a covariance that asymptotically decreases to 0 as the blocks are further separated in time.
In probability theory, the Helly–Bray theorem relates the weak convergence of cumulative distribution functions to the convergence of expectations of certain measurable functions. It is named after Eduard Helly and Hubert Evelyn Bray. Let F and F 1, F 2, ... be cumulative distribution functions on the real line.
In measure theory Prokhorov's theorem relates tightness of measures to relative compactness (and hence weak convergence) in the space of probability measures. It is credited to the Soviet mathematician Yuri Vasilyevich Prokhorov, who considered probability measures on complete separable metric spaces. The term "Prokhorov’s theorem" is also ...
In mathematics, weak convergence may refer to: Weak convergence of random variables of a probability distribution; Weak convergence of measures, of a sequence of probability measures; Weak convergence (Hilbert space) of a sequence in a Hilbert space more generally, convergence in weak topology in a Banach space or a topological vector space
Very often, the measures in question are probability measures, so the last part can be written as μ ( K ε ) > 1 − ε . {\displaystyle \mu (K_{\varepsilon })>1-\varepsilon .\,} If a tight collection M {\displaystyle M} consists of a single measure μ {\displaystyle \mu } , then (depending upon the author) μ {\displaystyle \mu } may either ...
In probability theory, lumpability is a method for reducing the size of the state space of some continuous-time Markov chains, first published by Kemeny and Snell. [1]
In probability theory, the continuous mapping theorem states that continuous functions preserve limits even if their arguments are sequences of random variables. A continuous function, in Heine's definition, is such a function that maps convergent sequences into convergent sequences: if x n → x then g(x n) → g(x).