APPENDIX: UPPERBOUNDING e y y > 0 in .NET framework

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9.15 APPENDIX: UPPERBOUNDING e y y > 0
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At this stage, Duf eld and O'Connell [10] invoke the assumed large deviations principle for the process ft 1 St ; t 1; 2; . . .g, say, with good rate function I : R 3 0; I , to conclude the inequality  ! vt 1 St ln P >x lim sup sup t t3I x>y h tx vt !  vt  lim sup sup d inf I z z>x t3I x>y h tx 9:134 for all d > 0. However, from Eq. (9.35) it is easy to conclude that for any d > 0, there exists t* such that   1 S ln P t > x vt t inf I z d;
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9:135
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Unfortunately, t* t* x; d , and this dependence of t* on x precludes taking the supremum supx>y on both sides of Eq. (9.135). Hence, Eq. (9.134) does not follow in a straightforward manner. To remedy this dif culty, we continue with the analysis by means of a slightly different approach; an alternative is offered by Duf eld [11]. Lemma 9.15.1. Under Condition (i), it holds that 1 max ln P St > b h b t 1;...; b=y     vt sup lim inf inf yx L y ; t3I x>y h tx y>0
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9:136
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Proof. Fix y > 0 and x > 0. For each y > 0, the usual Chernoff bound argument gives   1 St >x ln P vt t 1 ln E ey vt =t St e yxvt vt yx Lt y ; t 1; 2; . . . : 9:137
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Under Condition (i), if L y is nite, then for each d > 0, there exists an integer t* t* y; d such that L y d Lt y L y d; t ! t*: 9:138
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Reporting this fact into the Chernoff bound (9.137), we get   1 St >x ln P vt t yx L y d; t ! t*: 9:139
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  1 S ln P t > x h xt t
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vt d yx L y x>y h xt v vt sup t d inf yx L y x>y h xt x>y h xt vt vt d inf yx L y ; t ! t*: x>y h xt h yt sup
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It is now straightforward to conclude that  ! 1 St lim sup sup ln P >x t t3I x>y h xt vt lim inf inf yx L y t3I x>y h xt    9:140
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because d can be made arbitrarily small. The last inequality is automatically satis ed when L y I (in which case the right-hand side is I), and the least upper bound being the sharpest, we readily conclude Eq. (9.136) via Eq. (9.133). j Letting b go to in nity in Eq. (9.131) and making use of Eqs. (9.132) and (9.136) we indeed get e y a y . 9.16 APPENDIX: UPPERBOUNDING f y y > 0
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Fix y > 0. If g y 0, then b y 0 and the bound f y b y immediately holds owing to the fact that f y 0. Hence, from now on we always assume that g y > 0. Fix b > 0 and y > 0. By a simple Chernoff bound argument, we have P St > b e y vt =t b E ey vt =t St evt Lt y ; t 1; 2; . . . : 9:141
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Under Condition (i), if L y is nite, then for each d > 0, there exists a nite integer t* t* y; d such that Eq. (9.138) holds. Hence, by a union bound argument, we get 1 ln f y lim sup b3I h b 1 lim sup ln b3I h b
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