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Sequences and Limit Theorems

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Sequences  ::  Convergence  ::  Limit  ::  Central Limit

Laws of Large Numbers

Suppose is a sequence of r.v.s. We are often interested in sums , as becomes large. What can we say about such sums?

Suppose all have the same means , , and are uncorrelated. We would expect the average to ''approach'' in some way as . If , consider

Let us look at m.s. convergence;

Summarizing: If and are mutually uncorrelated and have finite variance, , so that

This is an example of a weak law of large numbers.

In the example we just gave, and .

Kolmogorov's Strong Law

Theorem 1 (Kolmogorov's Strong Law)   Suppose is a sequence of independent r.v.s with finite means for each . If

then

where

Note that in the case that all the variances are bounded, e.g.

then

So, if the variances grow sublinearly , the theorem can apply.

We can get an even stronger conclusion:

Theorem 2   Kinchine's Strong Law of Large Numbers.

Suppose is an i.i.d. sequence (i.e., a sequence of i.i.d. r.v.s) with finite mean

Then the sample mean converges almost surely to the ensemble mean:

Proving these types of theorems

The proofs follow from more general limit theorems.

So is in infinitely many of the sets . (It keeps coming back.)

Another notation is: i.o. (infinitely often).

We observe that if or then .

Lemma 1   The Borel Cantelli lemma . [This is frequently a good problem for math qualifiers.]
1. If then . That is, .
2. (Conversely) If are independent events and then .

Kolmogorov's Inequality

Suppose are independent with zero means and finite variances. Define to be the running sum

Then for each ,

This is a lot like the Chebyshev inequality, but instead of looking at the variance of all of the terms, we simply look at the variance of the last one.

Copyright 2008, by the Contributing Authors. Cite/attribute Resource . admin. (2006, May 31). Sequences and Limit Theorems. Retrieved January 07, 2011, from Free Online Course Materials — USU OpenCourseWare Web site: http://ocw.usu.edu/Electrical_and_Computer_Engineering/Stochastic_Processes/lec5_3.html. This work is licensed under a Creative Commons License