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By Barton G.

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A−k b−k−1 converges and thus A0 . . A−k b−k−1 tends almost surely to 0 as k → ∞. 2 But since b−k−1 = ωη−k−1 e1 + ωeq+1 , it follows that A0 . . A−k b−k−1 can be decomposed into two positive terms, and we have 2 lim A0 . . A−k ωη−k−1 e1 = 0, k→∞ lim A0 . . s. 26) holds for i = q + 1. Now we use the equality 2 e1 + βi eq+1 + eq+i+1 , A−k eq+i = βi η−k i = 1, . . 29) with ep+q+1 = 0 by convention. For i = 1, this equality gives 0 = lim A0 . . A−k eq+1 ≥ lim A0 . . 26) is true for i = q + 2, and, by induction, it is also true for i = q + j, j = 1, .

Thus, periods with high volatility level (corresponding 2 ) will be followed by periods where the fluctuations have a smaller amplitude. 7 illustrate the volatility clustering for simulated GARCH models. Large absolute values are not uniformly distributed on the whole period, but tend to cluster. 5, are also second-order stationary. Even if the absolute values can be extremely large, these processes are not explosive, as can be seen from these figures. 56 for the N(0, 1) distribution, as will be established below) lead to explosive paths.

Isk E(ηt−i . . ηt−i ) 1 1 −···−ik ∞ = φ0s 1 + = (As µ2s )k k=1 φ0s . 44) This shows that St is almost surely finite. ik ≥1 2 2 φi1 . . φik ηt−i . . ik ≥1 2 2 φi0 . . φik ηt−i . . ηt−i . 0 0 −···−ik Therefore, the following recursive equation holds: ∞ S t = φ0 + 2 φi St−i ηt−i . 40) is then obtained by setting t = St ηt . 44). 40), such that E| t |2s < ∞. iq+1 ≥1 := St,q + Rt,q . 2 2 φi1 . . φik ηt−i . . ηt−i 1 1 −···−ik 2 2 φi1 . . φiq+1 ηt−i . . s. 43). Moreover, because the solution is nonanticipative, t is independent of ηt for all t > t.

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An introduction to Advanced Field Theory by Barton G.


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