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Unformatted text preview: MIT OpenCourseWare http://ocw.mit.edu 14.384 Time Series Analysis Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms . More Empirical Process Theory 1 14.384 Time Series Analysis, Fall 2008 Recitation by Paul Schrimpf Supplementary to lectures given by Anna Mikusheva October 24, 2008 Recitation 8 More Empirical Process Theory This section of notes, especially the subsection on stochastic integrals are based on http://cscs.umich. edu/ ~ crshalizi/weblog/472.html . Convergence of Random Walks 1 [ τT ] We never did formally show the weak convergence of to a Brownian motion. Let’s fill that hole √ T t =1 now. Remember the functional central limit theorem. Theorem 1. Functional Central Limit Theorem : If 1. there exists a finitedimensional distribution convergence of ξ T to ξ (as in ( ?? )) 2. ∀ ε,η > there exists a partition of Θ into finitely many sets, Θ 1 ,... Θ k such that lim sup P (max sup ξ T ( τ 1 ) − ξ T ( τ 2 ) > η ) < ε T →∞ i τ 1 ,τ 2 ∈ Θ i   then ξ T ξ ⇒ Finite dimensional convergence follows from a standard central limit theorem. We just need to verify the second condition, stochastic equicontinuity. It is not very hard to do this directly. Θ = [0 , 1] is compact, so we can choose our partition to be a collection of intervals of length δ . This gives P (max sup ξ T ( τ 1 ) − ξ T ( τ 2 ) > η ) ≤ P ( sup ξ T ( τ 1 ) − ξ T ( τ 2 ) > η ) i τ 1 ,τ 2 ∈ Θ i    τ 1 − τ 2  <δ   1 [( τ + δ ) T ] ≤ P ( sup t ) τ ∈ [0 , 1] √ T  t =[ τT ] Chebyshev’s inequality says that if E [ x ] = µ and V ar ( x ) = σ 2 , then P ( x − µ > η ) ≤ σ η 2 2 . Here, [( τ + δ ) T ] [( τ + δ ) T ]   1 1 E [ √ T t =[ τT ] t ] = 0 and V ar ( √ T t =[ τT ] t ) = [ δT T ] σ 2 and these things do not depend on τ , so [( τ + δ ) T ] P ( sup 1 t > η ) ≤ [ δT ] σ 2 √ T  Tη 2 τ ∈ [0 , 1] t =[ τT ] Hence, setting δ ≤ η 2 ε we have the desired result. σ 2 Stochastic Integrals 2 Stochastic Integrals Our interest in stochastic integrals comes from the fact that we would like to say something about the convergence of 1 1 y t t = ξ T ( t/T ) √ T ( ξ T ( t/T ) − ξ T (( t − 1) /T )) T T We know that ξ T ⇒ W , a Brownian motion. If W were differentiable, we’d also know that lim T →∞ √ T ( W ( τT/T ) − W (([ τT ] − 1) /T )) = W ( τ ). Finally, a sum like the above would usually converge to an integral T 1 1 f ( t/T ) g ( t/T ) f ( x ) dg ( x ) T → t =1 We would like to generalize the idea of an integral to apply to random functions such as ξ . There are a few problems. First, W is not differentiable. Nonetheless, we can still look at T 1 lim ξ ( t/T )( W ( t/T ) − W (( t − 1) /T )) T →∞ T t =1 We must verify that this sum converges to something. We will call its limit a stochastic integral, and write 1 it as ξ ( t ) dW ( t ). We then want to define stochastic integrals...
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 '09
 Geurrieri
 Central Limit Theorem, Normal Distribution, Probability theory, Stochastic process, Likelihood function, Likelihoodratio test

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