Introduction-to-Asymptotics

Introduction-to-Asymptotics - Introduction to Asymptotic...

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Introduction to Asymptotic Theory ARE 210 page 1 Technical Notes on Characteristic Functions The characteristic function of a random variable X is defined as () tX x tE e ι ϕ= , , t −∞< <∞ 1 ι =− . The function ( ) x t ϕ is finite for all random variables X and all real numbers t . The distri- bution function of X , and the density function when it exists, can be obtained from the characteristic function by means of an inversion formula . Facts involving complex variables 1. We can write any complex number z as zx y = , where x and y are real numbers. 2. The modulus (absolute value) || z of a complex number is 22 zx y = + . 3. The distance between any two complex numbers 1 z and 2 z is 12 1 2 zz xx yy −= +− . 4. When a function of a real variable has a power series expansion with a positive radius of convergence, we can use that power series to define a corresponding function of a complex variable. Thus we define 0 ! n z n z e n = = for any complex number z. A direct calculation verifies that the relation ee e + = is valid for all complex numbers, 1 z and 2 z . Letting zt = ι , where t is a real number, 0 ! n t n t e n ι = ι =
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Introduction to Asymptotic Theory ARE 210 page 2 23 45 1 ... 2! 3! 4! 5! tt t  ιι =+ ι −− ++ −   24 35 1 ... ... 3! t =−+− + ι−+− Recall the McLauren series expansion for the sine and cosine functions, cos( ) 1 ... t = −+− and sin( ) ... 3! 5! = From these, we apply deMoivre’s theorem to write cos( ) sin( ) t ett ι ι . This is very helpful because it allows us to make use of some trigonometric identities and the relationships between exponential functions. We have cos( ) cos( ) = , and sin( ) sin( ) =− , so that by simply changing the signs of the above series expansions, we get cos( ) sin( ) t et t −ι ι . As a result, if we add t e ι and t e −ι , we obtain the identity ( ) cos( ) ½ te e ι , while if we subtract t e from t e ι , we obtain the identity () sin( ) ½ ½ ee e ι− ι ι == ι ι .
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Introduction to Asymptotic Theory ARE 210 page 3 The second equality uses ( ) 2 111 1 1 ι= − =− − =−ι . Finally, we obtain the modulus of t e ι as 22 cos ( ) sin ( ) 1 t et t t ι =+ \ , so that, as a function of t , t e ι is smooth and absolutely bounded on [-1, +1]. 5. If ( ) f t and ( ) gt are real-valued functions of t , then () ht f t = defines a complex-valued function of t . 6. We can differentiate ( ) by differentiating ( ) f t and ( ) separately, ′′ = so long as ( ) f t and ( ) exist. 7. Similarly, we define bb b aa a h t dt f t dt g t dt ι ∫∫ so long as the integrals for f and g exist.
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This note was uploaded on 08/01/2008 for the course ARE 210 taught by Professor Lafrance during the Fall '07 term at University of California, Berkeley.

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Introduction-to-Asymptotics - Introduction to Asymptotic...

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