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In probability theory, there exist several different notions of convergence of sequences of random variables, including convergence in probability, convergence in distribution, and almost sure convergence. The different notions of convergence capture different properties about the sequence, with some notions of convergence being stronger than ...
Each of the probabilities on the right-hand side converge to zero as n → ∞ by definition of the convergence of {X n} and {Y n} in probability to X and Y respectively. Taking the limit we conclude that the left-hand side also converges to zero, and therefore the sequence {( X n , Y n )} converges in probability to {( X , Y )}.
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 .
Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations , probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms .
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 ...
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.
A more rigorous definition takes into account the fact that θ is actually unknown, and thus, the convergence in probability must take place for every possible value of this parameter. Suppose { p θ : θ ∈ Θ } is a family of distributions (the parametric model ), and X θ = { X 1 , X 2 , …
In probability terms, a sequence of random variables converging in probability also converge in the mean if and only if they are uniformly integrable. [17] This is a generalization of Lebesgue's dominated convergence theorem , see Vitali convergence theorem .