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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 ...
Let in the theorem denote a random variable that takes the values / and / with equal probabilities. With = the summands of the first two series are identically zero and var(Y n)=. The conditions of the theorem are then satisfied, so it follows that the harmonic series with random signs converges almost surely.
This article is supplemental for “Convergence of random variables” and provides proofs for selected results. Several results will be established using the portmanteau lemma: A sequence {X n} converges in distribution to X if and only if any of the following conditions are met:
Download as PDF; Printable version; ... Probability theory is used extensively in statistics, ... Convergence of random variables (Related topics: ...
In the theory of probability, the Glivenko–Cantelli theorem (sometimes referred to as the Fundamental Theorem of Statistics), named after Valery Ivanovich Glivenko and Francesco Paolo Cantelli, describes the asymptotic behaviour of the empirical distribution function as the number of independent and identically distributed observations grows. [1]
A sequence of random variables ,, …, converges weakly to the random variable if their respective CDF converges,, … converges to the CDF of , wherever is continuous. Weak convergence is also called convergence in distribution .
Uniform convergence in probability has applications to statistics as well as machine learning as part of statistical learning theory. The law of large numbers says that, for each single event A {\displaystyle A} , its empirical frequency in a sequence of independent trials converges (with high probability) to its theoretical probability.
Applied to probability theory, Scheffe's theorem, in the form stated here, implies that almost everywhere pointwise convergence of the probability density functions of a sequence of -absolutely continuous random variables implies convergence in distribution of those random variables.