Discrete-time stochastic processes

The process sn n 1 is called a markov modulated random

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Unformatted text preview: 1} be a random walk with Sn = X1 + · · · + Xn . We assume throughout that E [X ] exists and is finite. The reader should focus on the case E [X ] = X < 0 on a first reading, and consider X = 0 and X > 0 later. For X < 0 and α > 0, we shall develop upper bounds on Pr {Sn ≥ α} that are exponentially decreasing in n and α. These bounds, and many similar results to follow, are examples of large deviation theory, i.e., probabilities of highly unlikely events. We assumeR throughout this section that X has a moment generating function g (r) = £ § E erX = erx dFX (x), and that g (r) is finite in some open interval around r = 0. As pointed out in Chapter 1, X must then have moments of all orders and the tails of its distribution function FX (x) must decay at least exponentially in x as x → −1 and as R1 x → +1. Note that erx is increasing in r for x > 0, so that if 0 erx dFX (x) blows up for some r+ > 0, it remains infinite for all r > r+ . Similarly, for x < 0, erx is increasing in −r, R so that if x≤0 erx dFX (x) blows up at some r− < 0, it is infinite for all r < r− . Thus if r− and r+ are the smallest and largest values such that g (r) is finite for r− < r < r+ , then g (r) is infinite for r > r+ and for r < r− . The end points r− and r+ can each be finite or infinite, and the values g (r+ ) and g (r− ) can each be finite or infinite. Note that if X is bounded in the sense that Pr {X < −B } = 0 and Pr {X > B } = 0 for some B < 1, then g (r) exists for all r. Such rv’s are said to have finite support and include all discrete rv’s with a finite set of possible values. Another simple example is that if X is a non-negative rv with FX (x) = 1 − exp(−αx) for x ≥ 0, then r+ = α. Similarly, if X is a negative rv with FX = exp(β x) for x < 0, then r− = −β . Exercise 7.7 provides further examples of these possibilities. 288 CHAPTER 7. RANDOM WALKS, LARGE DEVIATIONS, AND MARTINGALES The moment generating function of Sn = X1 + · · · + Xn is given by gSn (r) = E [exp(rSn )] = E [exp(r(X1 + · · · + Xn )] = {E [exp(rX )]}n = {g (r)}n . (7.15) It follows that gSn (r) is finite in the same interval (r− , r+ ) as g (r). First we look at the probability, Pr {Sn ≥ α}, that the nth step of the random walk satisfies Sn ≥ α for some threshold α > 0. We could actually find the distribution of Sn either by convolving the density of X with itself n times or by going through the transform domain. This would not give us much insight, however, and would be computationally tedious for large n. Instead, we explore the exponential bound, (1.38). For any r ≥ 0, in the region where g (r) is finite, i.e., for 0 ≤ r < r+ , we have Pr {Sn ≥ α} ≤ gSn (r)e−rα = [g (r)]n e−rα . (7.16) It is convenient to rewrite (7.16) in terms of the semi-invariant moment generating function ∞ (r) = ln[g (r)]. Pr {Sn ≥ α} ≤ exp[n∞ (r) − rα] ; any r, 0 ≤ r < r+ . (7.17) The first two derivatives of ∞ with respect to r are given by g (r)g 00 (r) − [g 0 (r)]2 . (7.18) [g (r)]2 £§ Recall from (1.32) that g 0 (0) = E [X ] and g 00 (0) = E X 2 . Substituting this into (7.18), we can evaluate ∞ 0 (0) and ∞ 00 (0) as ∞ 0 (r) = g 0 (r) ; g (r) ∞ 00 (r) = ∞ 0 (0) = X = E [X ] ; 2 ∞ 00 (0) = σX . (7.19) The fact that ∞ 00 (0) is the second central moment of X is why ∞ is called a semi-invariant moment generating function. Unfortunately, the higher-order derivatives of ∞ , evaluated at r = 0, are not equal to the higher-order central moments. Over the range of r where g (r) < 1, it is shown in Exercise 7.8 that ∞ 00 (r) ≥ 0, with strict inequality except in the very special (and uninteresting) case where X is deterministic. If X is deterministic, then Sn is also and there is no point to considering a probabilistic model. We thus assume in what follows that X is non-deterministic and thus ∞ 00 (r) > 0 for all r between r− and r+ Figure 7.3 sketches ∞ (r) assuming that X < 0 and r+ = 1 We can now minimize the exponent in (7.17) over r ≥ 0. For simplicity, first assume that r+ = 1. Since ∞ 00 (r) > 0, the exponent is minimized by setting its derivative equal to 0. The minimum (if it exists) occurs at the r, say ro for which ∞ 0 (r) = α/n. As seen from Figure 7.3, this is satisfied with r ≥ 0 only if α/n ≥ X . Thus ©£ §™ Pr {Sn ≥ α} ≤ exp n ∞ (ro ) − ro ∞ 0 (ro ) where ∞ 0 (ro ) = α/n ≥ E [X ] (7.20) Ω∑ ∏æ ∞ (ro ) = exp α 0 − ro . (7.21) ∞ (ro ) 7.4. THRESHOLD CROSSING PROBABILITIES IN RANDOM WALKS ❅0 ❅ ❅ ❅ ❅ ❅ 289 r ∞ (r) slope = E [X ] Figure 7.3: Semi-invariant moment generating function ∞ (r) for a rv X such that E [X ] < 0 and r+ = 1. Note that ∞ (r) is tangent to the line of slope E [X ] < 0 at 0 and has a positive second derivative everywhere. 0 ∞ (r) − rα/n r ∞ (r) ro r∗ ro − ∞ (ro )(n/α) s...
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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.

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