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This theorem follows from the fact that if X n converges in distribution to X and Y n converges in probability to a constant c, then the joint vector (X n, Y n) converges in distribution to (X, c) . Next we apply the continuous mapping theorem , recognizing the functions g ( x , y ) = x + y , g ( x , y ) = xy , and g ( x , y ) = x y −1 are ...
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 ...
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 result in the article is not known as Slutsky's Theorem (that is a different result), but rather Slutsky's Lemma. The two results are cited often enough that the distinction should be made. — Preceding unsigned comment added by 98.223.197.174 16:34, 2 January 2013 (UTC) The claim is wrong for general X_n, Y_n.
Suppose one has a sequence of statistically independent observations {X 1, X 2, ...} from a normal N(μ, σ 2) distribution. To estimate μ based on the first n observations, one can use the sample mean: T n = (X 1 + ... + X n)/n. This defines a sequence of estimators, indexed by the sample size n.
In probability theory, the continuous mapping theorem states that continuous functions preserve limits even if their arguments are sequences of random variables. A continuous function, in Heine's definition , is such a function that maps convergent sequences into convergent sequences: if x n → x then g ( x n ) → g ( x ).
Demonstration of this result is fairly straightforward under the assumption that () is differentiable near the neighborhood of and ′ is continuous at with ′ ().To begin, we use the mean value theorem (i.e.: the first order approximation of a Taylor series using Taylor's theorem):
Thin lenses produce focal points on either side that can be modeled using the lensmaker's equation. [5] In general, two types of lenses exist: convex lenses, which cause parallel light rays to converge, and concave lenses, which cause parallel light rays to diverge. The detailed prediction of how images are produced by these lenses can be made ...