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The concept of convergence in probability is used very often in statistics. For example, an estimator is called consistent if it converges in probability to the quantity being estimated. Convergence in probability is also the type of convergence established by the weak law of large numbers.
provided c is a constant. Proof: Fix ε > 0. Let Bε ( c) be the open ball of radius ε around point c, and Bε ( c) c its complement. Then. By the portmanteau lemma (part C), if Xn converges in distribution to c, then the limsup of the latter probability must be less than or equal to Pr ( c ∈ Bε ( c) c ), which is obviously equal to zero.
In statistics, a consistent estimator or asymptotically consistent estimator is an estimator —a rule for computing estimates of a parameter θ0 —having the property that as the number of data points used increases indefinitely, the resulting sequence of estimates converges in probability to θ0. This means that the distributions of the ...
Convergence in total variation is stronger than weak convergence. An important example of a log-concave density is a function constant inside a given convex body and vanishing outside; it corresponds to the uniform distribution on the convex body, which explains the term "central limit theorem for convex bodies".
Uniform convergence in probability is a form of convergence in probability in statistical asymptotic theory and probability theory. It means that, under certain conditions, the empirical frequencies of all events in a certain event-family converge to their theoretical probabilities. Uniform convergence in probability has applications to ...
The order in probability notation is used in probability theory and statistical theory in direct parallel to the big-O notation that is standard in mathematics.Where the big-O notation deals with the convergence of sequences or sets of ordinary numbers, the order in probability notation deals with convergence of sets of random variables, where convergence is in the sense of convergence in ...
Slutsky's theorem. In probability theory, Slutsky's theorem extends some properties of algebraic operations on convergent sequences of real numbers to sequences of random variables. [ 1] The theorem was named after Eugen Slutsky. [ 2] Slutsky's theorem is also attributed to Harald Cramér.
On the right-hand side, the first term converges to zero as n → ∞ for any fixed δ, by the definition of convergence in probability of the sequence {X n}. The second term converges to zero as δ → 0, since the set B δ shrinks to an empty set. And the last term is identically equal to zero by assumption of the theorem.
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