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soln6

# soln6 - EE 562a Homework Solutions 6 Wednesday 4 April 2007...

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EE 562a Homework Solutions 6 Wednesday 4 April 2007 1 1. Each of the convergence results deals with the following term S n ( u ) = n i =1 x i ( u ) , where { x i ( u ) } i =1 is a sequence of independent, identically distributed (iid) random variables, each with mean m and variance σ 2 . It follows that m n = E { S n ( u ) } = nm σ 2 n = V ar { S n ( u ) } = 2 . All three of these results are found in any good text on probability theory. (a) The Weak Law of Large Numbers (WLLN) says the that the sample mean converges to the mean in probability = lim n →∞ P 1 n S n ( u ) - m > = 0 , or y n ( u ) = 1 n S n ( u ) m = y ( u ) in probability. (b) The Strong Law of Large Numbers (SLLN) says the that the sample mean converges to the mean almost surely = P lim n →∞ 1 n S n ( u ) = m = 1 , or y n ( u ) = 1 n S n ( u ) m = y ( u ) almost surely. (c) The Central Limit Theorem is a statement about convergence in distribution. It says that the cumulative effect of many iid disturbances is Gaussian. Mathematically, we have that the sequence of random variables y n ( u ) = S n ( u ) - nm σ n converge in distribution to a mean zero unit variance Gaussian. In other words lim n →∞ F y n ( u ) ( z ) = z -∞ 1 2 π e - v 2 2 dv. Another way of saying this is y n ( u ) = S n ( u ) - nm σ n y ( u ) in distribution, where y ( u ) is any Gaussian random variable zero mean and unit variance.

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2 EE 562a Homework Solutions 6 Wednesday 4 April 2007 (d) We know that almost sure convergence always implies convergence in probability. Thus, if we knew that the Strong Law of Large Numbers is true, we could conclude that the Weak Law of Large Numbers is also true. If we only knew that the WLLN was true, we could not conclude that the SLLN holds. In this sense, the SLLN is “stronger” than the WLLN. You have probably all seen a proof of the WLLN, but probably none of you saw the proof of the SLLN in you probability class. In class we also proved another “law of large numbers,” namely y n ( u ) = 1 n S n ( u ) m = y ( u ) in the mss. It is easy to show that this law implies the WLLN (see problem 3). In general almost sure convergence and convergence in the mss do not imply one another. So an appropriate name for this law would be the “Equally Strong Law of Large Numbers.” A final comment about these rules is that many of the hypotheses can be relaxed; for ex- ample we can prove the “Equally Strong Law of Large Numbers” and thus the WLLN with the assumption that the x n ( u )’s are only uncorrelated (not independent). Depending on how advanced the probability text that you used was, you may have statement with weaker hypotheses. For example, see Stark and Woods version of the Central Limit Theorem. 2. Solution: Consider the space of all square-integrable complex functions on [ - 1 / 2 , 1 / 2): that is, all functions f ( t ) such that 1 / 2 - 1 / 2 | f ( t ) | 2 dt < . We can define an inner product ( f, g ) on this space: ( f, g ) = 1 / 2 - 1 / 2 f * ( t ) g ( t ) dt.
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