TimeSeriesBook.pdf

X remark c2 in the case r 2 the corresponding

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---→ X . Remark C.2. In the case r = 2 , the corresponding definition for random vectors is lim t →∞ E ( k X t - X k 2 ) = lim t →∞ E ( X t - X ) 0 ( X t - X ) = 0 . Theorem C.6 (Riesz-Fisher) . Let { X t } be a sequence of random variables such sup t E | X t | 2 < . Then there exists a random variable X with E | X | 2 < such that X t m.s. ---→ X if and only if E | X t - X s | 2 0 for t, s → ∞ . This version of the Riesz-Fisher theorem provides a condition, known as the Cauchy criterion, which is often easier to verify when the limit is unknown. Definition C.4 (Convergence in Distribution) . A sequence { X t } of random vectors with corresponding distribution functions { F X t } converges in distri- bution , if there exists an random vector X with distribution function F X such that lim t →∞ F X t ( x ) = F X ( x ) for all x ∈ C where C denotes the set of points for which F X ( x ) is continuous. We denote this fact by X t d -→ X . Note that, in contrast to the previously mentioned modes of convergence, convergence in distribution does not require that all random vectors are de- fined on the same probability space. The convergence in distribution states that, for large enough t , the distribution of X t can be approximated by the distribution of X . The following Theorem relates the four convergence concepts. Theorem C.7. (i) If X t a.s. --→ X then X t p -→ X . (ii) If X t p -→ X then there exists a subsequence { X t n } such that X t n a.s. --→ X . (iii) If X t r -→ X then X t p -→ X by Chebyschev’s inequality (Theorem C.3). (iv) If X t p -→ X then X t d -→ X . (v) If X is a fixed constant, then X t d -→ X implies X t p -→ X . Thus, the two concepts are equivalent under this assumption.
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378 APPENDIX C. STOCHASTIC CONVERGENCE These facts can be summarized graphically: X t n a.s. --→ X X t a.s. --→ X = X t p --→ X = X t d --→ X X t r --→ X A further useful theorem is: Theorem C.8. If E X t -→ μ and V X t -→ 0 then X t m.s. ---→ μ and conse- quently X t p -→ μ . Theorem C.9 (Continuous Mapping Theorem) . For any continuous func- tion f : R n -→ R m and random vectors { X t } and X defined on some prob- ability space, the following implications hold: (i) X t a.s. --→ X implies f ( X t ) a.s. --→ f ( X ) . (ii) X t p -→ X implies f ( X t ) p -→ f ( X ) . (iii) X t d -→ X implies f ( X t ) d -→ f ( X ) . An important application of the Continuous Mapping Theorem is the so- called Delta method which can be used to approximate the distribution of f ( X t ) (see Appendix E). A further useful result is given by: Theorem C.10 (Slutzky’s Lemma) . Let { X t } and { Y t } be two sequences of random vectors such that X t d -→ X and Y t d -→ c , c constant, then (i) X t + Y t d ----→ X + c , (ii) Y 0 t X t d ----→ c 0 X . (iii) X t /Y t d ----→ X/c if c is a nonzero scalar. Like the (cumulative) distribution function, the characteristic function provides an alternative way to describe a random variable.
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