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9.10 EXAMPLES
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Evaluation of g*
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We work under the assumption that the scaling sequence fv*; t 1; 2; . . .g satis es t Condition (ii) of Proposition 9.6.2. Fix y > 0. Setting z 0 in Eq. (9.78) we obtain b y c rin K g y : Next, going back to the expression (9.49) we note that    vt yx L y sup lim inf inf x>y h tx y>0 t3I    vt sup lim inf inf x c rin y x>y h tx 0<y<1 t3I    vt y lim inf inf x c rin t3I x>y h tx so that    vt x c rin a y Kg y lim inf inf t3I x>y h tx 9:79
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with closed forms available for speci c choices of the pmf G (Section 9.10). 9.9.2 Evaluation of g*
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If we assume that ft 1 Stb ct ; t 1; 2; . . .g satis es a large deviations principle with good rate function L*, then blind substitution into Eq. (9.78) yields g inf g y y c rin :
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Unfortunately, the Gartner Ellis theorem [9, Theorem 2.3.6, p. 45] here gives a trivial lower bound (9.34) for ft 1 Stb ct ; t 1; 2; . . .g. This fact, pointed out by Duf eld [11], precludes the use of Proposition 9.6.1 to conclude the lower bound (9.4). Nevertheless, as discussed in Section 9.11, such a lower bound does hold in many cases of interest, with case-speci c arguments needed to establish it. 9.10 EXAMPLES
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The examples considered here are constructed by taking the f1; 2; . . .g-valued rv s to be of the form s st X , where X is an integrable R -valued rv with P X 0 0. 9.10.1 Integrated Tails and Forward Recurrence Times
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Recall that with any R -valued rv X with 0 < E X < I, we can associate the R -valued rv X * whose distribution is the integrated tail distribution of X , namely, 1 x P X * x P X > u du; x ! 0: 9:83 E X 0
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^ For any f1; 2; . . .g-valued rv s, the difference between the f1; 2; . . .g-valued rv s and the R -valued rv s* is emphasized through the relation ^ ^ P s* > x P s > r x r P s r 1 ; r x < r 1; 9:84
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with r 0; 1; . . . . Moreover, if s st X for some R -valued rv with P X 0 0 and 0 < E X < I, simple algebra shows that E X P X * > r E s for all r 1; 2; . . . . Consequently, ^ P s > r $ provided lim P X * > r 1 1: P X * > r 9:87 E X P X * > r E s 9:86 ^ P s > r E X P X * > r 1 E s 9:85
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This will happen if the rv X * belongs to the class v of long-tailed distributions on R [12]. This will be so for all examples considered here, in which case Eq. (9.86) implies ^ v* ln P s > t $ ln P X * > t : t 9:88
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Speci c Cases
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9.10.2.1 Weibull The rv X is a Weibull rv with parameters a; n a > 0, 0 < n < 1 ; that is, P X > x e ax ; and it is well known that P X * > x $ C a; n x1 n e ax
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x ! 0;
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for some constant C a; n determined solely by the parameters a; n . Consequently, v* $ at n , so that h b bn , g y ay n , and K 0. Elementary calculus shows that t  1 n a n c rin g* aeH n 1 n ln c rin ; W n 1 n 9:91