If gx r e erx exists then e ersn e n i1 n erxi

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Unformatted text preview: to minimize q1(A) for a given constraint on q0(A). Typically this is a threshold test, but sometimes, especially if Y is discrete, it is a randomized threshold test. 10 Thresholds for random walks and Chernoff b ounds The Chernoff b ound says that for any real b and any r ￿ ￿ such that gZ (r) = E erZ exists, Pr {Z ≥ b} ≤ gZ (r) exp(−rb); for b > Z , r > 0 Pr {Z ≤ b} ≤ gZ (r) exp(−rb); for b < Z , r < 0 This is most useful when applied to a sum, Sn = X1+· · · Xn ￿ ￿ of I ID rv’s. If gX (r) = E erX exists, then ￿ ￿ E erSn = E n ￿ i=1 n erXi = gX (r) n Pr {Sn ≥ na} ≤ gX (r) exp(−rna); for a > X , r > 0 n Pr {Sn ≤ na} ≤ gX (r) exp(−rna); for a < X , r < 0 11 This is easier to interpret and work with if expressed in terms of the semi-invariant MGF, γX (r) = ln gX (r). Then n gX (r) = enγX (r) and Pr {Sn ≥ na} ≤ exp(n[γX (r) − ra]); for a > X , r > 0 Pr {Sn ≤ na} ≤ exp(n[γX (r) − ra]); 0 r r∗ ro ❅ ❅ ❅ ❅ ❅ ❅ slope X...
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This note was uploaded on 01/13/2012 for the course ELECTRICAL 6.262 taught by Professor Staff during the Fall '11 term at MIT.

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