Does convergence in probability imply convergence in expectation?
Convergence in probability implies convergence in distribution. In the opposite direction, convergence in distribution implies convergence in probability when the limiting random variable X is a constant. Convergence in probability does not imply almost sure convergence.
What is the difference between convergence and uniform convergence?
I know the difference in definition, pointwise convergence tells us that for each point and each epsilon, we can find an N (which depends from x and ε)so that and the uniform convergence tells us that for each ε we can find a number N (which depends only from ε) s.t. .
How to show convergence in probability does not imply convergence in quadratic mean?
As a counterexample to show convergence in probability does not imply convergence in quadratic mean, take U ∼ U n i f ( 0, 1) and let X n = n I [ 0, 1 n] ( U). Then With the additional constraint that P ( | X n | < M) = 1 for some constant M > 0, we do have that convergence in probability implies convergence in quadratic mean or L2.
Why do we have convergence in mean square?
My first thought is that since we have E ( X n 2) → E ( X 2) and convergence in distribution, we might somehow have convergence in mean square which would imply we have convergence in mean. We may have a sequence ( X n) n ⩾ 1 which converges in probability to 0 but such that lim n → + ∞ E [ X n 2] is not 0.
Which is an example of convergence of expectation?
For example, consider the set of positive integers and the probability measure P { k } = 2 − k, k ⩾ 1. Then define the random variable X n := c n 1 { j ⩾ n }, where 1 denotes the indicator function. For any choice of c n, the sequence ( X n) converges in probability to 0.
How is expected convergence of expectation-cross validated?
Using Skorokhod’s representation theorem, we may assume X n → X almost surely without loss of generality. Note also that | X n | ≤ 1 + X n 2 and that, by assumption, E ( 1 + X n 2) → 1 + E X 2. Thus, assuming E X n 2 < ∞, we get E X n → E X using a dominated convergence theorem.