hw6 - EE 562a Homework Set 6 Due Wednesday 4 April 2007...

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Unformatted text preview: EE 562a Homework Set 6 Due Wednesday 4 April 2007 1 (The following well-defined problems come from different sources, and the notation used may vary. Don’t let that bother you!) 1. State each of the following results from your probability theory class as y n ( u ) → y ( u ) as n → ∞ ; identifying in each case, { y n ( u ) } , y ( u ), and the mode of stochastic convergence (i.e. the sense of the limit). (a) The Weak Law of Large Numbers (b) The Strong Law of Large Numbers (c) The Central Limit Theorem (d) In what sense is the Strong Law of Large Numbers “stronger” than the Weak Law of Large Numbers? 2. Consider the space of all square-integrable complex functions on [- 1 / 2 , 1 / 2): that is, all functions f ( t ) such that R 1 / 2- 1 / 2 | f ( t ) | 2 dt < ∞ . We can define an inner product ( f, g ) on this space: ( f, g ) = Z 1 / 2- 1 / 2 f * ( t ) g ( t ) dt. We can also define a linear transformation on this space: f ( t ) → g ( t ) = Z 1 / 2- 1 / 2 K ( t, s ) f ( s ) ds ≡ K f. (a) Verify that this space is a vector space, and that ( f, g ) is an inner product. (b) An eigenfunction of a linear operator K is a function f ( t ) such that K f ≡ Z 1 / 2- 1 / 2 K ( t, s ) f ( s ) ds = λf ( t ) ∀ t, where λ is the eigenvalue of f ( t ). Suppose that K ( t, s ) is Hermitian: K ( t, s ) = K * ( s, t )....
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hw6 - EE 562a Homework Set 6 Due Wednesday 4 April 2007...

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