Search results
Results from the WOW.Com Content Network
Proofs of convergence of random variables; Convergence of measures; Convergence in measure; Continuous stochastic process: the question of continuity of a stochastic process is essentially a question of convergence, and many of the same concepts and relationships used above apply to the continuity question. Asymptotic distribution
The continuous mapping theorem states that this will also be true if we replace the deterministic sequence {x n} with a sequence of random variables {X n}, and replace the standard notion of convergence of real numbers “→” with one of the types of convergence of random variables.
So let f be such arbitrary bounded continuous function. Now consider the function of a single variable g(x) := f(x, c). This will obviously be also bounded and continuous, and therefore by the portmanteau lemma for sequence {X n} converging in distribution to X, we will have that E[g(X n)] → E[g(X)].
In probability theory, Lévy’s continuity theorem, or Lévy's convergence theorem, [1] named after the French mathematician Paul Lévy, connects convergence in distribution of the sequence of random variables with pointwise convergence of their characteristic functions.
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 .
For (,) a measurable space, a sequence μ n is said to converge setwise to a limit μ if = ()for every set .. Typical arrow notations are and .. For example, as a consequence of the Riemann–Lebesgue lemma, the sequence μ n of measures on the interval [−1, 1] given by μ n (dx) = (1 + sin(nx))dx converges setwise to Lebesgue measure, but it does not converge in total variation.
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.
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.