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  2. Convergence of random variables - Wikipedia

    en.wikipedia.org/wiki/Convergence_of_random...

    The definition of convergence in distribution may be extended from random vectors to more general random elements in arbitrary metric spaces, and even to the “random variables” which are not measurable — a situation which occurs for example in the study of empirical processes. This is the “weak convergence of laws without laws being ...

  3. Proofs of convergence of random variables - Wikipedia

    en.wikipedia.org/wiki/Proofs_of_convergence_of...

    Proof of the theorem: Recall that in order to prove convergence in distribution, one must show that the sequence of cumulative distribution functions converges to the F X at every point where F X is continuous. Let a be such a point. For every ε > 0, due to the preceding lemma, we have:

  4. Convergence proof techniques - Wikipedia

    en.wikipedia.org/wiki/Convergence_proof_techniques

    Convergence proof techniques are canonical patterns of mathematical proofs that sequences or functions converge to a finite limit when the argument tends to infinity.. There are many types of sequences and modes of convergence, and different proof techniques may be more appropriate than others for proving each type of convergence of each type of sequence.

  5. Real analysis - Wikipedia

    en.wikipedia.org/wiki/Real_analysis

    In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. [1] Some particular properties of real-valued sequences and functions that real analysis studies include convergence, limits, continuity, smoothness, differentiability and integrability.

  6. Monotone convergence theorem - Wikipedia

    en.wikipedia.org/wiki/Monotone_convergence_theorem

    In more advanced mathematics the monotone convergence theorem usually refers to a fundamental result in measure theory due to Lebesgue and Beppo Levi that says that for sequences of non-negative pointwise-increasing measurable functions (), taking the integral and the supremum can be interchanged with the result being finite if either one is ...

  7. Rate of convergence - Wikipedia

    en.wikipedia.org/wiki/Rate_of_convergence

    In asymptotic analysis in general, one sequence () that converges to a limit is said to asymptotically converge to with a faster order of convergence than another sequence () that converges to in a shared metric space with distance metric | |, such as the real numbers or complex numbers with the ordinary absolute difference metrics, if

  8. Uniform convergence - Wikipedia

    en.wikipedia.org/wiki/Uniform_convergence

    In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions converges uniformly to a limiting function on a set as the function domain if, given any arbitrarily small positive number , a number can be found such that each of the functions ...

  9. Convergence tests - Wikipedia

    en.wikipedia.org/wiki/Convergence_tests

    While most of the tests deal with the convergence of infinite series, they can also be used to show the convergence or divergence of infinite products. This can be achieved using following theorem: Let { a n } n = 1 ∞ {\displaystyle \left\{a_{n}\right\}_{n=1}^{\infty }} be a sequence of positive numbers.