Later in our statistical analysis n will be the

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. Later, in our statistical analysis, n will be the number of random draws from and underlying population. But it could be, say, a number of Bernoulli trials. All that matters for definitions is that for each positive integer n , W n is a random variable. [Technically, the sequence is defined on a probability space, , F , P .] Results for convergence of random sequences will imply results for deterministic sequences. 8
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A simple example of a random sequence is W n  n 1 / n Z where, say, Z ~ Normal 0,1 . It is clear that, if we have meaningful concepts of “convergence,” then W n “converges” to Z because n 1 / n 1as n . A more relevant example is when W n is a sample average of random variables: W n n 1 i 1 n X i X ̄ n 9
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Here is a general question: If c is a nonrandom constant, what should we mean by W n “converging” to c ? Here is a first attempt: Convergence of W n to c requires P W n c 1as n . 10
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Some examples show that this concept is not useful. Suppose that W n Normal 3, n 1 . The distribution of W n is clearly collapsing to the single value 3. Any sensible definition of W n converging must have W n converging to 3. But because the normal distribution is continuous, P W n 3 0 for all n and so P W n 3 converges to zero, not one. 11
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There is a problem with this definition in discrete cases, too. Suppose W n X ̄ n where the X i are independent Bernoulli 1/2 random variables. Then the expected value of W n is 1/2 and its variance is 1/ 4 n . The distribution of W n is collapsing to 1/2. But X ̄ n can never equal 1/2 whenever n is odd. In other words, P W n 1/2 0for n odd, and so clearly P W n 1/2 does not converge to one. Instead, a definition that does prove useful is based on a sequence of probabilties concerning how close W n is to c . 12
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DEFINITION : The sequence of random variables W n : n 1,2,. .. converges in probability to c if for every 0, P | W n c | 0as n . Equivalently, P | W n c | 1as n or P W n c n . 13
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The definition has its bite when is small. It says that no matter how small we choose a fixed distance of W n from c , eventually we can go far enough along the sequence to make the probability that W n is farther from c than as small as we want. Notice that this definition just uses the usual notion of convergence of a nonrandom sequence of numbers. Let a n P | W n c | . Then we have to show a n 0as n no matter how we choose 0.
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Later in our statistical analysis n will be the number of...

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