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Unformatted text preview: t, the eﬀect of the second threshold in (7.32)
was set to 0, and, second, the overshoot in the threshold crossing at α was set to 0 in going 7.5. THRESHOLDS, STOPPING RULES, AND WALD’S IDENTITY 297 from (7.33) to (7.34). It is easy to account for the second threshold by recognizing that
Pr {SN ≤ β } = 1 − Pr {SN ≥ α}. Then (7.32) can be solved, getting
Pr {SN ≥ α} = 1 − E [exp(r∗ SN )  SN ≤ β ]
.
E [exp(r∗ SN )  SN ≥ α] − E [exp(r∗ SN )  SN ≤ β ] (7.36) Accounting for the overshoots is much more diﬃcult. For the case of the simple random
walk, overshoots never occur since the random walk always changes in unit steps. Thus, for
α and β integers, we have E [exp(r∗ SN )  SN ≤ β ] = exp(r∗ β ) and E [exp(r∗ SN )  SN ≥ α] =
exp(r∗ α). Substituting this in (7.36) yields the exact solution
Pr {SN ≥ α} = exp(−r∗ α)[1 − exp(r∗ β )]
.
1 − exp[−r∗ (α − β )] (7.37) Solving the equation ∞ (r∗ ) = 0 for the simple random walk with probabilities p and q
yields r∗ = ln(q /p). This is also valid if X takes on the three values −1, 0, and +1 with
p = Pr {X = 1}, q = Pr {X = −1}, and 1 − p − q = Pr {X = 0}. It can be seen that if α
and −β are large positive integers, then the simple bound of (7.31) is almost exact for this
example. Equation (7.37) is sometimes taken as an approximation for (7.36). Unfortunately,
for many applications, the overshoots are more signiﬁcant than the eﬀect of the opposite
threshold so that (7.37) is only negligibly better than (7.31) as an approximation, and has
the disadvantage of not being a bound.
If Pr {SN ≥ α} must actually be calculated, then the overshoots in (7.36) must be taken
into account. See Chapter 12 of [9] for a treatment of overshoots. 7.5.2 Joint distribution of N and barrier Next we look at Pr {N ≥ n, SN ≥ α}, where again we assume that X < 0 and that ∞ (r∗ ) = 0
for some r∗ > 0. For any r in the region where ∞ (r) ≤ 0 (i.e., for 0 ≤ r ≤ r∗ ), we have
−N ∞ (r) ≥ −n∞ (r) for N ≥ n. Thus, from the Wald identity, we have
1 ≥ E [exp[rSN − N ∞ (r)]  N ≥ n, SN ≥ α] Pr {N ≥ n, S N ≥ α}
≥ exp[rα − n∞ (r)]Pr {N ≥ n, SN ≥ α} Pr {N ≥ n, SN ≥ α} ≤ exp[−rα + n∞ (r)] ; for all r such that 0 ≤ r ≤ r∗ . (7.38) Under our assumption that X < 0, we have ∞ (r) ≤ 0 in the range 0 ≤ r ≤ r∗ , and (7.38) is
valid for all r in this range. To obtain the tightest bound of this form, we should minimize
the right hand side of (7.38). This is the same minimization (except for the constraint
r ≤ r∗ ) as in Figure 7.4, and the result, if α/n < ∞ 0 (r∗ ), is
Pr {N ≥ n, SN ≥ α} ≤ exp[−ro α + n∞ (ro )]. (7.39) where ro satisﬁes ∞ 0 (ro ) = α/n. This is the same as the bound on Pr {Sn ≥ α} in (7.20)
except that r ≤ r∗ in (7.39).
For the special case described in Figure 7.5 where ∞ (r) < 0 for all r < r+ , (7.39) is modiﬁed
in the same way as used in (7.23). 298 CHAPTER 7. RANDOM WALKS, LARGE DEVIATIONS, AND MARTINGALES The bound in (7.39) is strictly tighter than the bound Pr {SN ≥ α} ≤ exp(−r∗ α) if α/n <
∞ 0 (r∗ ). For a given value of α, deﬁne n∗ = α/∞ 0 (r∗ ). We can then rewrite (7.39) as for n > n∗ , α/n = ∞ 0 (ro ) exp [n∞ (ro ) − ro α]
Pr {N ≥ n, SN ≥ α} ≤
.
(7.40) exp [−r∗ α]
n ≤ n∗ The interpretation of (7.40) is that n∗ is an estimate of the typical value of N given that
the threshold at α is crossed. For n greater than this typical value, (7.40) provides a tighter
bound on Pr {N ≥ n, SN ≥ α} than the bound on Pr {SN ≥ α} in (7.31), whereas (7.40)
provides nothing new for n ≤ n∗ . In Section 7.7, we shall derive the slightly stronger result
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that Pr supi≥n Si ≥ α is also upper bounded by the right hand side of (7.40).
We next develop an almost identical upper bound to Pr {N ≤ n, SN ≥ α} by using the
Wald identity for r > r∗ . Here ∞ (r) > 0, so −N ∞ (r) ≥ −n∞ (r) for N ≤ n. It follows that
1 ≥ E [exp[rSN − N ∞ (r)]  N ≤ n, SN ≥ α] Pr {N ≤ n, SN ≥ α}
≥ exp[rα − n∞ (r)] Pr {N ≤ n, SN ≥ α} , so that Pr {N ≤ n, SN ≥ α} ≤ exp[−rα + n∞ (r)].
Optimizing over r as before (except recognizing that r ≥ r∗ ), we get for n < n∗ , α/n = ∞ 0 (ro ) exp [n∞ (ro ) − ro α]
Pr {N ≤ n, SN ≥ α} ≤
. exp [−r∗ α]
n ≥ n∗ (7.41) (7.42) This strengthens the interpretation of n∗ as the typical value of N conditional on crossing
the threshold at α. That is, (7.42) provides information on the lower tail of the distribution
of N (conditional on SN ≥ α), whereas (7.40) provides information on the upper tail. 7.5.3 Proof of Wald’s identity The proof is given under the assumption that the rv’s Xi are discrete with a PMF p(X ).
Extending the proof to arbitrary rv’s requires some careful mathematical analysis but...
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This note was uploaded on 09/27/2010 for the course EE 229 taught by Professor R.srikant during the Spring '09 term at University of Illinois, Urbana Champaign.
 Spring '09
 R.Srikant

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