slides_7_converge

# 64 but we must be more formal than just saying

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But we must be more formal than just saying “approximate the distribution of X ̄ n .” If we simply study the distribution of X ̄ n as n we get nothing useful: X ̄ n p which means the distribution of X ̄ n collapses to the single point with probability one. (Called point mass at .) One way to see this is that Var X ̄ n 2 / n 0as n . The next graph shows 3. The limiting distribution, which of course is discrete, is a very poor approximation to the actual distribution of X ̄ n even when n is very large. 65
n = 120 n = 50 n = 25 limiting distribution 0 1 2 3 4 3 Distributions of Sample Average with Mean = 3 66

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The problem is that we are using the wrong sequence of random variables. We need to standardize X ̄ n so that the variance does not shrink to zero. If we use Z n X ̄ n E X ̄ n  SD X ̄ n X ̄ n / n n X ̄ n then E Z n 0 and Var Z n 1. The distribution of Z n can be virtually anything, but we know the distribution is not collapsing as n . In the next section we will show that the distribution of Z n settles down to a standard normal. 67
DEFINITION : The sequence of random variables W n : n 1,2,. .. converges in distribtion to the random variable W if F n w F w as n for all w where F is continuous. Here, F n w P W n w and F w P W w , where W has distribution F . We are actually talking about (pointwise) convergence of a sequence of CDFs to a given CDF, but it is convenient to refer to a random variable with distribution F . We call F the limiting distribution or asymptotic distribution of W n . 68

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As notation, we write W n d W ( W n converges in distribution to W )or W n a ~ W ( W n is asymptotically distributed as W ). When we know W has a specific distribution, such as standard normal (the leading case), we might write W n d Normal 0,1 W n a ~ Normal 69
In the cases we are interested in, F  is continuous at all w .In addition to the normal, the chi-square distribution is an important case. In many applications, the F n  are not always continuous. In fact, important cases include when F n  has some jumps for all n (but the jumps become less and less pronounced as n increases, and so F n  converges to a continuous function). 70

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Continuous Convergence Theorem Once we have established convergence in distribution, we can derive the limiting distribution of lots of functions of the original series. THEOREM : Suppose W n d W .If g  is a continuous function on then g W n d g W EXAMPLE : Suppose we know Z n d Z ~ Normal 0,1 . Then Z n 2 d Z 2 ~ 1 2 and exp Z n d exp Z ~ Lognormal .
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64 But we must be more formal than just saying approximate...

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