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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 t E 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 z x y = + ι , where x and y are real numbers. 2. The modulus (absolute value) | | z of a complex number is 2 2 | | z x y = + . 3. The distance between any two complex numbers 1 z and 2 z is 2 2 1 2 1 2 1 2 ( ) ( ) z z x x y y = + . 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 1 2 1 2 z z z z e e e + = is valid for all complex numbers, 1 z and 2 z . Letting z t = 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 2 3 4 5 1 ... 2! 3! 4! 5! t t t t t ι ι = + ι − + + 2 4 3 5 1 ... ... 2! 4! 3! 5! t t t t t = + + ι + Recall the McLauren series expansion for the sine and cosine functions, 2 4 cos( ) 1 ... 2! 4! t t t = + and 3 5 sin( ) ... 3! 5! t t t t = + From these, we apply deMoivre’s theorem to write cos( ) sin( ) t e t t 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( ) t t = , and sin( ) sin( ) t t = − , so that by simply changing the signs of the above series expansions, we get cos( ) sin( ) t e t t −ι = − ι . As a result, if we add t e ι and t e −ι , we obtain the identity ( ) cos( ) ½ t t t e e ι −ι = + , while if we subtract t e −ι from t e ι , we obtain the identity ( ) sin( ) ½ ½ t t t t e e t e e ι −ι ι −ι = = − ι ι .
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Introduction to Asymptotic Theory ARE 210 page 3 The second equality uses ( ) 2 1 1 1 1 1 ι = = − = −ι . Finally, we obtain the modulus of t e ι as 2 2 cos ( ) sin ( ) 1 t e t 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 ( ) g t are real-valued functions of t , then ( ) ( ) ( ) h t f t g t = + ι defines a complex-valued function of t .
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