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modern mathematical statistics with applications

Probability And Measure Wiley Series In Probability And Mathematical Statistics 3rd Edition Patrick Billingsley - Solutions

=+. (a) Show that {X„) is a martingale with respect to (5"} if and only if, for all n and all stopping times + such that +
=+(b) Show that, if {X ,, ) is a martingale and + is a bounded stopping time, then E[X,] = E[X,].
=+35.15. 31.9f Suppose that 9.1 % and A E _, and prove that P[A] - IA with probability 1. Compare Lebesgue's density theorem.
=+35.16. Theorems 35.6 and 35.9 have analogues in Hilbert space. For n ≤ co, let P ,, be the perpendicular projection on a subspace M ,,. Then Px - P_x for all x if either (a) M1 CM2 C . . . and M_ is the closure of Un. < M ,, or (b) M. M2 5 ... and M_ = n.
=+35.17. Suppose that 0 has an arbitrary distribution, and suppose that, conditionally on 0, the random variables Y1, Y2 ,... are independent and normally dis-
=+tributed with mean 0 and variance o2. Construct such a sequence (0, Y], Y2 ,.. . ).Prove (35.31).
=+35.18. It is shown on p. 471 that optional stopping has no effect on likelihood ratios.This is not true of tests of significance. Suppose that X1, X2 ,... are indepen-
=+dent and identically distributed and assume the values 1 and 0 with probabili-ties p and 1 - p. Consider the null hypothesis that p = ; and the alternative that p > ;. The usual .05-level test of significance is to reject the null hypothesis if /(X1+.+Xm-in)> 1.645.(35.40)
=+For this test the chance of falsely rejecting the null hypothesis is approximately P[N> 1.645] =05 if n is large and fixed. Suppose that n is not fixed in advance of sampling, and show by the law of the iterated logarithm that, even
=+if p is, in fact, ;, there are with probability 1 infinitely many n for which(35.40) holds.
=+35.19. (a) Suppose that (35.32) and (35.33) hold. Suppose further that, for constants S2, s ,, "E" -of -p 1 and s ,, "ER _ E[Y (, 12 5 .. ]] -> 0, and show that$, ER_YA = N. Hint: Simplify the proof of Theorem 35.11.
=+(b) The Lindeberg-Levy theorem for martingales. Suppose that..., Y_1, YG, Y ,,...is stationary and ergodic (p. 494) and that E[Y2] > and E[Y\Yx-1, Y-2 .... ]=0.
=+Prove that CA_ Y / Vn is asymptotically normal. Hint: Use Theorem 36.4 and the remark following the statement of Lindeberg's Theorem 27.2.
=+35.20. 24.41 Suppose that the o-field $_ in Problem 24.4 is trivial. Deduce from Theorem 35.9 that P[AT-"S ] - PAS_] = P(A) with probability 1, and conclude that T is mixing.
=+36.1. 1 Suppose that [X ,: t € T] is a stochastic process on (0, 5", P) and A E S .Show that there is a countable subset S of T for which P[AlX ,, [ET] =P[AIX ,, t € S] with probability 1. Replace A by a random variable and prove a similar result.
=+36.2. Let T be arbitrary and let K(s, t) be a real function over T x T. Suppose that K is symmetric in the sense that K(s, t) = K(t,s) and nonnegative-definite in the sense that Ci,j-1 K(t1, t,)x;x; ≥ 0 for k ≥ 1, 11, ..., in T, and x1 ,..., Xk
=+. Show that there exists a process [ X ,: t € T] for which (X ,, ..., X, ) has the centered normal distribution with covariances K(( ,, t,), i, j = 1 ...., k.
=+36.3. Let L be a Borel set on the line, let / consist of the Borel subsets of L, and let L' consist of all maps from T into L. Define the appropriate notion of cylinder, and let IT be the o-field generated by the cylinders. State a version of Theorem 36.1 for (L', . T). Assume T countable, and
=+36.4. Suppose that the random variables X1, X2, ... assume the values 0 and 1 and P[X] = 1 i.o.] = 1. Let u be the distribution over (0, 1] of L" _, X ,, /2". Show that on the unit interval with the measure u, the digits of the nonterminating dyadic expansion form a stochastic process with the
=+36.5. 36.3 1 There is an infinite-dimensional version of Fubini's theorem. In the construction in Problem 36.3, let L =/ = (0,1), T= {1,2 ,... ), let / consist of the Borel subsets of I, and suppose that each k-dimensional distribution is the
=+k-fold product of Lebesgue measure over the unit interval. Then IT is a countable product of copies of (0, 1), its elements are sequences x = (x1, x2, ... )of points of (0, 1), and Kolmogorov's theorem ensures the existence on (IT, AT)of a product probability measure T: "[x: x; Sa, isn] =a1 x,
=+(a) Define : I" > IT - IT by 4(*1 ...., x11), ( >1, y2. ... ) = (x1 ,., x1, y1, Y2 .... ).Show that & is measurable " X ST/GT and -1 is measurable IT/X ST. Show that $"1(A ,, X ) = T, where A ,, is n-dimensional Lebesgue measure restricted to I".
=+(b) Let f be a function measurable T and, for simplicity, bounded. Define
=+f. (x+1, x=+2 ... ) = f .... [{(y .... >> x+1 ... ) dy. ... dyn;in other words, integrate out the coordinates one by one. Show by Problem
=+34.18, martingale theory, and the zero-one law that(36.30)except for x in a set of T-measure 0.
=+(c) Adopting the point of view of part (a), let g.(x1, ..., x,) be the result of integrating the variable (ym+), y+2) ... ) out (with respect to w) from f(x) ,.;., x ,., y„.+1, ... ). This may suggestively be written as f(x ..... Y+1,Y+2 .... )dyn+ dyn+2 ....
=+Show that g(x) ,..., x„) ->f(x1, x2 .... ) except for x in a set of 1-measure 0.
=+36.6. (a) Let T be an interval of the line. Show that " fails to contain the sets of:linear functions, polynomials, constants, nondecreasing functions, functions of bounded variation, differentiable functions, analytic functions, functions con-
=+tinuous at a fixed tp, Borel measurable functions. Show that it fails to contain the set of functions that: vanish somewhere in T, satisfy x(s)
=+(b) Let C be the set of continuous functions on T = [0, 0). Show that A E RT and ACC imply that A = Ø. Show, on the other hand, that A € 927 and
=+C CA do not imply that A = RT.
=+36.7. Not all systems of finite-dimensional distributions can be realized by stochastic processes for which 22 is the unit interval. Show that there is on the unit interval with Lebesgue measure no process [ X ,: t ≥ 0] for which the X, are independent and assume the values 0 and 1 with
=+36.8. Here is an application of the existence theorem in which T is not a subset of the line. Let (N, , v) be a measure space, and take T to consist of the .4-sets of finite v-measure. The problem is to construct a generalized Poisson process,
=+a stochastic process [X 4: A € T ] such that (i) X , has the Poisson distribution with mean v( A) and (ii) X4 ....., X, are independent if A1 ,..., A ,, are disjoint. Hint: To define the finite-dimensional distributions, generalize this construction: For A, B in T, consider independent random
=+37.1. 36.21 Show that K(s, t) = min{s, t) is nonnegative-definite; use Problem 36.2 to prove the existence of a process with the finite-dimensional distributions prescribed for Brownian motion.
=+37.2. Let X(t) be independent, standard normal variables, one for each dyadic rational t (Theorem 20.4; the unit interval can be used as the probability space). Let W(0) = 0 and W(n) = ER_ X(k). Suppose that W(t) is already defined for dyadic rationals of rank n, and put+--IN 1-
=+Prove by induction that the W(r) for dyadic t have the finite-dimensional distributions prescribed for Brownian motion. Now construct a Brownian motion with continuous paths by the argument leading to Theorem 37.1. This avoids an appeal to Kolmogorov's existence theorem.
=+37.3. 1 For each n define new variables W (t) by setting W.(k/2") = W(k/2") for dyadics of order n and interpolating linearly in between. Set 8, =sup, s "|Wn+ ((t)- W,(r)|, and show that-+-max-0≤k
=+The construction in the preceding problem makes it clear that the difference here is normal with variance 1/2"+2. Find positive x ,, such that Ex ,, and EP[8 ,, 2x„] both converge, and conclude that outside a set of probability 0, W.(t, w) converges uniformly over bounded intervals. Replace
=+37.4. 36.6 1 Let T = [0,00), and let P be a probability measure on (RT, $7 ) having the finite-dimensional distributions prescribed for Brownian motion. Let C consist of the continuous elements of RT.
=+(a) Show that P.(C)=0, or P*(RT-C) - 1 (see (3.9) and (3.10)). Thus completing (R7, 97, P) will not give C probability 1.
=+(b) Show that P*(C) = 1.
=+37.5. Suppose that [W ,: t ≥ 0] is some stochastic process having independent, sta-tionary increments satisfying E[W,] - 0 and E[W,2] =t. Show that if the finite-dimensional distributions are preserved by the transformation (37.11), then they must be those of Brownian motion.
=+37.6. Show that n,>00[W]: > > t] contains only sets of probability 0 and 1. Do the same for n.>00[W]: 0 <
=+37.7. Show by a direct argument that W( ., w) is with probability 1 of unbounded variation on [0,1]: Let Y, = E ?_ , [W(i2-") - W((i - 1)2"")|. Show that Y, has mean 2"/"E[W]]] and variance at most Var[ W ]. Conclude that _P[Y ,,< n]
=+37.8. Show that the Poisson process as defined by (23.5) is measurable.
=+37.9. Show that for T = [0,0) the coordinate-variable process [Z ,: t € T] on (R™, @™)is not measurable.
=+37.10. Extend Theorem 37.4 to the set [t: W(t,@) =@].
=+37.11. Let +, be the first time the Brownian motion hits @ > 0: 7 = inf[t: W, ≥ @].
=+Show that the distribution of 7, has over (0, 00) the density(37.46)
=+Show that E[+ ] = ", Show that ", has the same distribution as a2/NV2, where N is a standard normal variable.
=+37.12. 1 (a) Show by the strong Markov property that T, and Ta+g -T. are independent and that the latter has the same distribution as 7g. Conclude that ha * hg =ha+p. Show that Br, has the same distribution as Ta 6
=+(b) Show that each h, is stable-see Problem 28.10.
=+37.13. 1 Suppose that X1, X2 ,... are independent and each has the distribution(37.46).
=+(a) Show that (X, + . . . +X )/n2 also has the distribution (37.46). Contrast this with the law of large numbers.
=+(b) Show that P[n "2 max , " Xk ≤x] - exp(-@v2/mx ) for x> 0. Relate this to Theorem 14.3.
=+37.14. 37.11 + Let p(s, t) be the probability that a Brownian path has at least one zero in (s, t). From (37.46) and the Markov property deduce p(s,1)- arcos VE.(37.47)Hint: Condition with respect to W ,.
=+37.15. 1 (a) Show that the probability of no zero in (r, 1) is (2/m) arcsin vt and hence that the position of the last zero preceding 1 is distributed over (0,1)with density ₸-1(t(1 -t))-1/2.
=+(b) Similarly calculate the distribution of the position of the first zero follow-ing time 1.
=+(c) Calculate the joint distribution of the two zeros in (a) and (b).
=+. 1 (a) Show by Theorem 37.8 that inf, ., Y, (u) and inf, , Z (u) both converge in distribution to inf,
=+(b) Let A.(s, t) be the event that S ,, the position at time k in a symmetric random walk, is 0 for at least one k in the range sn ≤ k ≤ tn, and show that P(A„(s, t) -+ (2/IT) arcos vs/t....
=+(c) Let T ,, be the maximum k such that k ≤n and S, = 0. Show that T /n has asymptotically the distribution with density Tr-1((1 -t))-1/2 over (0,1). As
=+this density is larger at the ends of the interval than in the middle, the last time during a night's play a gambler was even is more likely to be either early or late than to be around midnight.
=+37.17. 1 Show that p(s, t) = p(t-1, s-1) = p(cs, ct). Check this by (37.47) and also by the fact that the transformations (37.11) and (37.12) preserve the properties of Brownian motion.
=+37.18. Deduce by the reflection principle that (M ,, W,) has density 2(2y =x)-(2y-x)2 exp -21 on the set where y 2 0 and y 2x. Now deduce from Theorem 37.8 the corresponding limit theorem for symmetric random walk.
=+37.19. Show by means of the transformation (37.12) that for positive a and b the probability is 1 that the process is within the boundary -at < W, < bt for all sufficiently large t. Show that a/(a +b) is the probability that it last touches above rather than below.
=+37.20. The martingale calculation used for (37.39) also works for slanting boundaries.For positivea, b, r, let . be the smallest t such that either W, = - a + rt or W, = b + rt, and let p(a,b, r) be the probability that the exit is through the upper barrier-that W, = b + IT.
=+(a) For the martingale Ya ,, in the proof of Lemma 2, show that E[Y .. ] =1.Operating formally at first, conclude that(37.48)E[e8W ,- 921/2]= 1.
=+Take 0 =2r, and note that OW, - - 02, is then 2rb if the exit is above(probability p(a,b, r)) and -2ra if the exit is below (probability 1 -p(a,b, r)).Deduce 1- 22ra p(a,b, r) - 2rb _ e -2ra
=+(b) Show that p(a,b, r) -> a/(a +b) as r-0, in agreement with (37.39).
=+(c) It remains to justify (37.48) for 0 =2r. From E[Y .., ] =1 deduce E[2[W2-+2℃)] = 1
=+(37.49)for nonrandom o. By the arguments in the proofs of Lemmas 2 and 3, show that (37.49) holds for simple stopping timesa, for bounded ones, for a = T An, for o = T.
=+27.1. Prove Theorem 23.2 by means of characteristic functions. Hint: Use (27.5) to compare the characteristic function of Che, Zak with explEk Pnk(e" - 1)].
=+27.2. If {X„) is independent and the X ,, all have the same distribution with finite first moment, then n-'S ,, - E[ X, ] with probability 1 (Theorem 22.1), so that n~'S ,, - E[X]]. Prove the latter fact by characteristic functions. Hint: Use(27.5).
=+27.3. For a Poisson variable Y, with mean A, show that (Y - A)/ VA => N as A -00 Show that (22.3) fails for 1 - 1.
=+27.4. Suppose that |X„4| ≤ M ,, with probability 1 and M ,, /s ,, - + 0. Verify Lyapounov's condition and then Lindeberg's condition.
=+27.5. Suppose that the random variables in any single row of the triangular array are identically distributed. To what do Lindeberg's and Lyapounov's conditions reduce?
=+27.6. Suppose that Z1, Z2 ,... are independent and identically distributed with mean 0 and variance 1, and suppose that Xnk = " ,. Z1. Write down the Lindeberg condition and show that it holds if max k ≤r ,, Unk "
=+27.7. Construct an example where Lindeberg's condition holds but Lyapounov's does not.
=+27.8. 22.91 Prove a central limit theorem for the number R ,, of records up to time n.
=+27.9. 6.31 Let S ,, be the number of inversions in a random permutation on n letters. Prove a central limit theorem for S ..
=+27.10. The 8-method. Suppose that Theorem 27.1 applies to {X,), so that Vna '(X ,, -c) = N, where X ,, = n. ""E" _, X ,. Use Theorem 25.6 as in Exam-ple 27.2 to show that, if f(x) has a nonzero derivative atc, then In ( f(X) -
=+f(c))/olf'(c)| = N: X ,, is approximately normal with mean c and standard deviation o/ vn, and f(X,) is approximately normal with mean f(c) and standard deviation |f'(c)lo/ Vn. Example 27.2 is the case f(x)=1/x.
=+27.11. Suppose independent X ,, have density |x|" outside (-1, +1). Show that(n log n) -1/2S ,, = N.
=+27.12. There can be asymptotic normality even if there are no moments at all.Construct a simple example.
=+27.13. Let d.(w) be the dyadic digits of a point @ drawn at random from the unit interval. For a k-tuple (u1 ,..., u.) of O's and 1's, let N.(u1 ,..., u ,; w) be the number of m ≤ n for which (d ,, (w) ,..., dm+k-{(w)) =(u] ,..., u;). Prove a central limit theorem for N.(u ,,..., ug; w). (See
=+27.14. The central limit theorem for a random number of summands. Let X1, X2, ... be independent, identically distributed random variables with mean 0 and vari-ance o2, and let S ,, - X, + . . . + X ,,. For each positive t, let v, be a random variable assuming positive integers as values; it need
=+X„ Suppose that there exist positive constantsa, and 0 such that>> 8 a, as t -+ co. Show by the following steps that Sur -N.(27.27)
=+(a) Show that it may be assumed that 0 = 1 and thea, are integers.
=+(b) Show that it suffices to prove the second relation in (27.27).
=+(c) Show that it suffices to prove (S ,, - S2)/ Va, = 0.
=+(d) Show that P[S ,, - SalzeVa,] SP[Iv ,- a,lze3a,]+P max IS4 -S2,12eva ,.Ik-a,lse30, and conclude from Kolmogorov's inequality that the last probability is at most 2€a2.
=+27.15. 21.21 23.10 23.141 A central limit theorem in renewal theory. Let X1, X2 ,... be independent, identically distributed positive random variables with mean m and variance o2, and as in Problem 23.10 let N, be the maximum n for which S ,, ≤t. Prove by the following steps that
=+N, - tm-1 011/2m-3/2 - N.(a) Show by the results in Problems 21.21 and 23.10 that (Sw, - t)/ VT = 0.
=+(b) Show that it suffices to prove that N ,- Som-1 -(SN, - MN)at1/2m-3/2 011/2m-1/2 > N.
=+(c) Show (Problem 23.10) that N,/t = m -1, and apply the theorem in Prob-lem 27.14.
=+27.16. Show by partial integration that 111 -x2 /2(27.28)as x -o.
=+27.17. t Suppose that X1, X2 ,... are independent and identically distributed with mean 0 and variance 1, and suppose thata, - .. Formally combine the central limit theorem and (27.28) to obtain
=+1 10-a2/2 me - 0}(1+ 34)/2(27.29)P[S,≥a,Vn]~12 TT , where ( ,, - + 0 if a ,, - . For a case in which this does hold, see Theorem 9.4.

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