Stat 5101 Lecture Notes - School of Statistics

6.1. UNIVARIATE THEORY 169density of X density of X 30The density of X is shown on the left. It is extremely skewed going to infinity atzero. On the right is the density of X 30 and the normal density with the samemean and variance. The fit is not good. The density of X 30 , a rescaled chi 2 (30)density, is still rather skewed and so cannot be close to a normal density, whichof course is symmetric.density of X 100 density of X 300The fit is better at n = 100 and n = 300, but still not as good as our bimodalexample at n = 30. The moral of the story is that skewness slows convergencein the central limit theorem.If you wish to play around with this example, varying different aspects tosee what happens, go to the web pagehttp://www.stat.umn.edu/geyer/5101/clt.html#expo6.1.3 Convergence in ProbabilityA special case of convergence in distribution is convergence in distributionDto a degenerate random variable concentrated at one point, X n −→ a where ais a constant. Theorem 2 of Chapter 5 in Lindgren says that this is equivalentto the following notion.Definition 6.1.2 (Convergence in Probability to a Constant).A sequence of random variables X 1 , X 2 , ... converges in probability to a constanta if for every ɛ>0P(|X n −a|>ɛ)→0,as n →∞.We indicate X n converging in probability to a by writingX nP−→ a,as n →∞.