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As an example one may consider random variables with densities f n (x) = (1 + cos(2πnx))1 (0,1). These random variables converge in distribution to a uniform U(0, 1), whereas their densities do not converge at all. [3] However, according to Scheffé’s theorem, convergence of the probability density functions implies convergence in ...
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.
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:
The uniform convergence of more general empirical measures becomes an important property of the Glivenko–Cantelli classes of functions or sets. [2] The Glivenko–Cantelli classes arise in Vapnik–Chervonenkis theory, with applications to machine learning. Applications can be found in econometrics making use of M-estimators.
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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. [3]
If X n: Ω → X is a sequence of random variables then X n is said to converge weakly (or in distribution or in law) to the random variable X: Ω → X as n → ∞ if the sequence of pushforward measures (X n) ∗ (P) converges weakly to X ∗ (P) in the sense of weak convergence of measures on X, as defined above.
It is important to note that the convergence in Doob's first martingale convergence theorem is pointwise, not uniform, and is unrelated to convergence in mean square, or indeed in any L p space. In order to obtain convergence in L 1 (i.e., convergence in mean), one requires uniform integrability of the random variables .