Discrete-time stochastic processes

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Unformatted text preview: (Ø ) ØS Ø 8bE [|X |] Ø ˘n Ø Pr Ø − E [X ]Ø ≥ ≤ ≤ for large enough b. Øn Ø n≤2 56 CHAPTER 1. INTRODUCTION AND REVIEW OF PROBABILITY Exercise § 1.27. Let {Xi ; i ≥ 1} be IID rv’s with mean 0 and infinite variance. Assume that £ E |Xi |1+h = β for some given h, 0 < h < 1 and some given β . Let Sn = X1 + · · · + Xn . a) Show that Pr {|Xi | ≥ y } ≤ β y −1−h b : Xi ≥ b X : −b ≤ Xi ≤ b i −b : Xi ≤ −b hi £ § R1 1−h ˘ Show that E X 2 ≤ 2β b h Hint: For a non-negative rv Z , E X 2 = 0 2 z Pr {Z ≥ z } dz 1− (you can establish this, if you wish, by integration by parts). n o ˘ ˘ ˘ ˘ c) Let Sn = X1 + . . . + Xn . Show that Pr Sn 6= Sn ≤ nβ b−1−h h 1−h i ©Ø Ø ™ n d Show that Pr Ø Sn Ø ≥ ≤ ≤ β (12bh)n≤2 + b1+h . n − ˘ ˘ b) Let {Xi ; i ≥ 1} be truncated variables Xi = e) Optimize your bound with respect to b. How fast does this optimized bound approach 0 with increasing n? Exercise 1.28. (MS convergence → convergence in probability) Assume that {§ n ; n ≥ Y £ 2 = 0. 1} is a sequence of rv’s and z is a number with the property that limn→1 E (Yn − z ) a) Let ε > 0 be arbitrary and show that for each n ≥ 0, £ § E (Yn − z )2 Pr {|Yn − z | ≥ ε} ≤ ε2 b) For the § above, let δ > 0 be arbitrary. Show that there is an integer m such that ε £ E (Yn − z )2 ≤ ε2 δ for all n ≥ m. c) Show that this implies convergence in probability. Exercise 1.29. Let X1 , X2 . . . , be a sequence of IID rv’s each with mean 0 and variance √ √ σ 2 . Let Sn = X1 + · · · + Xn for all n and consider the random variable Sn /σ n − S2n /σ 2n. Find the limiting distribution function for this rv as n → 1. The point of this exercise is √ to see clearly that the distribution function of Sn /σ n is converging but that the sequence of rv’s is not converging. Exercise 1.30. A town starts a mosquito control program and the rv Zn is the number of mosquitos at the end of the nth year (n = 0, 1, 2, . . . ). Let Xn be the growth rate of mosquitos in year n; i.e., Zn = Xn Zn−1 ; n ≥ 1. Assume that {Xn ; n ≥ 1} is a sequence of IID rv’s with the PMF Pr {X =2} = 1/2; Pr {X =1/2} = 1/4; Pr {X =1/4} = 1/4. Suppose that Z0 , the initial number of mosquitos, is some known constant and assume for simplicity and consistency that Zn can take on non-integer values. a) Find E [Zn ] as a function of n and find limn→1 E [Zn ]. b) Let Wn = log2 Xn . Find E [Wn ] and E [log2 (Zn /Z0 )] as a function of n. 1.8. EXERCISES 57 c) There is a constant α such that limn→1 (1/n)[log2 (Zn /Z0 )] = α with probability 1. Find α and explain how this follows from the strong law of large numbers. d) Using (c), show that limn→1 Zn = β with probability 1 for some β and evaluate β . e) Explain carefully how the result in (a) and the result in (d) are possible. What you should learn from this problem is that the expected value of the log of a product of IID rv’s is more significant that the expected value of the product itself. Exercise 1.31. Use Figure 1.7 to verify (1.43). Hint: Show that y Pr {Y ≥y } ≥ R and show that limy→1 z≥y z dFY (z ) = 0 if E [Y ] is finite. R z ≥y z dFY (z ) Q Exercise 1.32. Show that m≥n (1 − 1/m) = 0. Hint: Show that µ ∂ µµ ∂∂ µ ∂ 1 1 1 1− = exp ln 1 − ≤ exp − . m m m Exercise 1.33. Let A1 , A2 , . . . , denote an arbitrary set o events. This exercise shows that of nS o nT S limn Pr m≥n Am = 0 if and only if Pr n m≥n Am = 0. nS o nT S o a) Show that limn Pr = 0 implies that Pr = 0. Hint: First m≥n Am n m≥n Am show that for every positive integer n, n\ [ o n[ o Pr Am ≤ Pr Am . n m≥n m≥n nT S o nS o b) Show that Pr Am = 0 implies that limn Pr Am = 0. Hint: First n m≥n m≥n show that each of the following hold for every positive integer n, n[ o n\ o [ Pr Am = Pr Am . m≥n lim Pr n→1 n[ m≥n k≤n o n\ Am ≤ Pr k≤n m≥k [ m≥k o Am . Exercise 1.34. Assume that X § a zero-mean rv with finite second and fourth moments, £§ £ is i.e., E X 4 = ∞ < 1 and E X 2 = σ 2 < 1. Let X1 , X2 , . . . , be a sequence of IID rv’s, each with the distribution of X . Let Sn = X1 + · · · + Xn . £ 4§ a) Show that E Sn = n∞ + 3n(n − 1)σ 4 . 4 n n− b) Show that Pr {|Sn /n| ≥ ≤} ≤ n∞ +3≤4(n4 1)σ . P c) Show that n Pr {|Sn /n| ≥ ≤} < 1. Note that you have shown that Lemma 1.1 in the proof of the strong law of large numbers holds if X has a fourth moment. d) Show that a finite fourth moment implies a finite second moment. Hint: Let Y = X 2 and Y 2 = X 4 and show that if a nonnegative rv has a second moment, it also has a mean. e) Modify the above parts for the case in which X has a non-zero mean. Chapter 2 POISSON PROCESSES 2.1 Introduction A Poisson process is a simple and widely used stochastic process for modeling the times at which arrivals enter a system. It is in many ways the continuous-time version of the Bernoulli process that was described briefly in Subsection 1.3.5. Recall that...
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