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modern mathematical statistics with applications

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

=+(c) Let N. be the time of the first record after time n. Show that P[ N ,, = n +k]=n(n +k-1)-1(n +k)-1.
=+(b) Show that no record stands forever.
=+. 1 Record values. Let A ,, be the event that a record occurs at time n:max & < . < X ,,(a) Show that A1, A2 ,... are independent and P(A ,, ) = 1/n.
=+(d) Show that Y ,, Y2, ... are independent.
=+(a) Show that T(") is uniformly distributed over the n! permutations.(b) Show that P[Y ,, = r] =1/n, 1 ≤r ≤n.(c) Show that Y, is measurable o(T(")) for k ≤ n.
=+Let T(m)(w) = (T(")(w) ,..., T.") (@) be that permutation (f) ,. . ., (1) of(1 ,. .., n) for which X, (w) < X,(w) < ... < X, (w). Let Y, be the rank of X among X1 ,..., X ,,: Y =r if and only if X; < X ,, for exactly r - 1 values of i preceding n.
=+20.8. Ranks and records. Let X1, X2 ,... be independent random variables with a common continuous distribution function. Let B be the. w-set where X„,(c) =X,(w) for some pair m, n of distinct integers, and show that P(B)=0.Remove B from the space 2 on which the X ,, are defined. This leaves the
=+20.7. Let X0, X1, ... be a persistent, irreducible Markov chain, and for a fixed state j let T1, T2, ... be the times of the successive passages through j. Let Z, = T, and Z ,, = T ., - T ,, - 1, n ≥ 2. Show that Z1, Z2, ... are independent and that P[Z„ = k] = f(*) for n ≥ 2.
=+20.6. Show that X1, X2 ,... are independent if o( X ,,..., X ,, , ) and o(X) are independent for each n.
=+20.5. Suppose that A, B, and C are positive, independent random variables with distribution function F. Show that the quadratic Az2 + Bz + C has real zeros with probability [5/8F(x2/4y) dF(x)dF(y).
=+20.4. The construction in Theorem 20.4 requires only Lebesgue measure on the unit interval. Use the theorem to prove the existence of Lebesgue measure on R.First construct A, restricted to (-n, n] X . . . X (-n, n], and then pass to the limit (n -+ ). The idea is to argue from first principles,
=+20.3. Suppose that a two-dimensional distribution function F has a continuous densityf. Show that f(x, y) =a2F(x, y)/axay.
=+20.2. If X is a positive random variable with densityf, then X -! has density f(1/x)/x2. Prove this by (20.16) and by a direct argument.
=+20.1. 2.11 | A necessary and sufficient condition for a o-field to be countably generated is that $=o(X) for some random variable X. Hint: If =o(A1, A2 ,... ), consider X =Ex-1f(1 )/10 *, where f(x) is 4 for x=0 and 5 for x = 0.
=+define Puf = E, Er(f. , )", Show that Puf EM and f - PMf 1 ?, that g =Pug if gEM, and that f - Puf LM. This defines the general orthogonal projection.
=+(d) Now take $ to be a maximal orthonormal system in a subspace M, and
=+(c) Show that $ is countable if and only if L2 is separable.
=+(b) Let Pf = Eyer(f, , ) ,. Show that f - Pf L $ and hence (maximality)f = Pf. Thus $ is an orthonormal basis.
=+(a) Show that IT= [y: (f.+.) = 0] is countable. Hint: Use ;_ , (f. p.)2 = ||fll2 and the argument for Theorem 10.2(iv).
=+19.9. Drop the assumption that L2 is separable. Order by inclusion the orthonormal systems in L2, and let (Zorn's lemma) $ = [ ,: y € [] be maximal.
=+Prove orthonormality. Hint: Express the sines and cosines in terms of e'" +e -IT, multiply out the products, and use the fact that [2"e"* dx is 2Tr or 0 as m = 0 or m # 0. (For the completeness of the trigonometric system, see Problem 26.26.)
=+19.8. The classical orthonormal basis for L'[0, 2+ ] with Lebesgue measure is the trigonometric system(19.17) (2₸)", TT -1/2 sin nx, TT 1/2 cos nx, n=1,2 ,....
=+(f2-tf1) du ≥ 0, since the integrand is nonnegative.
=+19.7. The Neyman-Pearson lemma. Supposef, and f2 are rival densities and L( jli) is 0 or 1 as j = i or j # i, so that R,(8) is the probability of choosing the opposite density whenf, is the right one. Suppose of 8 that 82(w) = 1 if f2(w) > tff(w)and 82(@) = 0 if fa(w) < tf,(w), where t > 0. Show
=+19.6. Show that a Bayes rule corresponding to p = ( p), .... p.) may not be admissible if p; = 0 for some i. But there will be a better Bayes rule that is admissible.
=+19.5. Show that the unit ball in L'((0, 1], ", A) is not weakly compact.
=+(b) Do the same for p = 2.
=+(a) For the case p = oo, find functions f ,, and f such that f ,, goes weakly to f but Ilf - f ,, il„ does not go to 0.
=+19.4. Consider weak convergence in LP((0, 1], @, À).
=+19.3. Show that Theorem 19.3 fails for L"((0, 1], ", A). Hint: Take y(f) to be a Banach limit of nfl/"f(x) dx.
=+(c) Show that LP(2, F, P) is not separable if (Theorem 36.2) there is on the space an independent stochastic process [ X]: 0 s / ≤ 1] such that X, takes the values + 1 with probability ; each (9" is not countably generated).
=+(b) Show that LP((0,1], Ø, u) is not separable if u is counting measure (u is not o-finite).
=+19.2. (a) Show that L"((0, 1], Ø, A) is not separable.
=+19.1. Suppose that u(12) < > and fe L". Show that IfIl, Ilfilo.
=+(b) Show that if (15.12) holds, then so does the italicized condition in part (a).
=+Rather than assume that f is measurable $", one can assume that it satisfies the italicized condition in Problem 15.4(a)-which in case (0, 9, u) is complete is the same thing anyway. For the next three problems, assume that u(!) < œ and that f is measurable Fand bounded.
=+15.5. 1 Show that for positive e there exists a finite partition (A,) such that, if (B;)is any finer partition and w, E B ;, then Ifdu - [(w ) H ( B ) < E .15.6. 1 Show that-1-
=+The limit on the right here is Lebesgue's definition of the integral.
=+15.7. 1 Suppose that the integral is defined for simple nonnegative functions by((E,x,IA) du = E;x,u(A,). Suppose that f ,, and g ,, are simple and nondecreas-ing and have a common limit: 0 ≤f ., t f and 0 ≤ g ,. tf. Adapt the arguments
=+used to prove Theorem 15.1(iii) and show that lim ,, ff ,, du = lim ,, fg ,, du. Thus, in the nonnegative case, ffdu can (Theorem 13.5) consistently be defined as lim, ff, du for simple functions for which 0 Sf, 1 f.
=+15.4. 10.5 15.31 (a) Suppose of f that there exist an F-set A and a function g, measurable F, such that u( A) = 0 and [ f = g] CA. This is the same thing as assuming that u*[ f = g] = 0, or assuming that f is measurable with respect to completed with respect to u. Show that (15.12) holds.
=+The definitions (15.3) and (15.6) always make formal sense (for finite u(n) and supif D), but they are reasonable-accord with intuition-only if (15.12) holds. Under what conditions does it hold?
=+15.3. 3.2 15.21 For A cn, define u"( A) and , ( A) by (3.9) and (3.10) with u in place of P. Show that [*I du = u"( A) andf. I du = u , (4) for every A.Therefore, (15.12) can fail if f is not measurable F. (Where was measurability used in the proof of (15.12)?)
=+To define the integral as the common value in (15.12) is the Darboux-Young approach. The advantage of (15.3) as a definition is that (in the nonnegative case) it applies at once to unbounded f and infinite u.
=+1. faus \'fdu .(15.11)(b) Now assume that f is measurable $ and let M be a bound for |fl.Consider the partition A, = [w: ie < f(w) ≤ (i+1)€], where i ranges from - N to N and N is large enough that Ne > M. Show that E sup f(w) (A1)- I| inff(w) +(A) SEH(1).Conclude that (15.12)[ fdu = f fdu.
=+15.2. 1 (a) Show that C| inff(w)\m(A1) = C| inf (w)|m (B;)if {BJ) refines (A,). Prove a dual relation for the sums in (15.10) and conclude that
=+There are many functions familiar from calculus that ought to be integrable but are of the types in the preceding problem and hence have infinite upper integral.Examples are x-21( .= )(x) and x-1/2I(o. 1(x). Therefore, (15.10) is inappropriate as a definition of ffdu for nonnegativef. The only
=+15.1. Suppose that f is measurable and nonnegative. Show that [*fdp = co if p[w):f(w) > 0] =" or if p[w: f(w) > a] > 0 for all a.
=+The infimum in (15.10), like the supremum in (15.9), extends over all finite partitions(A) of 2 into F-sets.
=+16.1. 13.91 Suppose that (12) < œ and f ,, are uniformly bounded.(a) Assume f ,, - f uniformly and deduce jf ,, du - ffdu from (16.5).
=+(b) Use part (a) and Egoroff's theorem to give another proof of Theorem 16.5.
=+(c) Show forf. = nl(0,n-10) - n.(1 -1,24-1) that thef, can be integrated to the limit (Lebesgue measure) even though the f ,, are not uniformly integrable.
=+(b) On the unit interval with Lebesgue measure, letf. = (n/log n)/(0 .- 1) for n 2 3. Show that thef, are uniformly integrable (and ff, du -> 0) although they are not dominated by any integrable g.
=+16.7. (a) Show that if |f ., i ≤g and g is integrable, then {f,) is uniformly integrable.Compare the hypotheses of Theorems 16.4 and 16.14.
=+16.6. Suppose that f(w, - ) is, for each w, a function on an open set W in the complex plane and that f( ., z) is for z in W measurable § and integrable.Suppose that A satisfies (16.9), that f(w, . ) is analytic in W for @ in A, and that for each z in W there is an integrable g( ., z0) such that
=+(b) If u is Lebesgue measure on the unit interval I, (a,b) - (0,1), and f(w, t) = I(o.)(w), then part (i) applies but part (ii) does not. Why? What about(16.32)?
=+The natural way to check (16.32), however, is by the mean-value theorem, and this requires (for « € A) a derivative throughout a neighborhood of 10.
=+(16.32)≤81(w)for w € A and 0 < |h| < 8, where 8 is independent of w and g, is integrable.Then ₺'(to) = ff'(w, to)u(da).
=+(a) Part (i) is local: there can be a different set A for each to. Part (ii) can be recast as a local theorem. Suppose that for w € A, where A satisfies (16.9),f(w, t) has derivative f'(w, to) at to; suppose further that
=+16.5. About Theorem 16.8:
=+(b) Deduce Lebesgue's dominated convergence theorem from part (a).
=+jb, du -> jbdu. Suppose, finally, that the first two sequences enclose the third:a, sf. sb, almost everywhere. Show that the third may be integrated to the limit.
=+16.4. (a) Suppose that functions an, b ,, f ,, converge almost everywhere to func-tionsa, b.f, respectively. Suppose that the first two sequences may be integra-ted to the limit-that is, the functions are all integrable and fa ,, du -> fadu,
=+16.3. Suppose that f ,, are integrable and sup ,, ff ,, du < wo. Show that, if f ,, 1f, then f is integrable and ff, du -> ffdu. This is Beppo Levi's theorem.
=+16.2. Prove that if 0 sf. of almost everywhere and ff, du SA < 0, then f is integrable and ff du ≤ A. (This is essentially the same as Fatou's lemma and is sometimes called by that name.)
=+16.8. Show that if f is integrable, then for each e there is a 8 such that u( A)
=+These problems outline alternative definitions of the integral and clarify the role measurability plays. Call (15.3) the lower integral, and write it as[ [du = sup [int () (A1)w-(15.9)[WEAP 3to distinguish it from the upper integral[du = inf [ sup (w) ( A ).(15.10)
=+14.9. Show that (14.24) and (14.25) are everywhere infinitely differentiable, although not analytic.
=+(b) Extend to the general case. Hint: Let F (x) be -n or F(x) or n as F(x) < - nor -n ≤F(x)
=+(a) Extend to the case of bounded F.
=+14.2. For distribution functions F, the second proof of Theorem 14.1 shows how to construct a measure u on (R', ') such that p(a, b] = F(b) - F(a).
=+-cach nonempty one contains a rational.
=+14.1. The general nondecreasing function F has at most countably many discontinu-ities. Prove this by considering the open intervals sup F(w), inf F(u))uCx
=+13.17. Let S = {0,1), and define a map T from sequence space S" to [0,1] by Tw -Ex _, a.(@)/2". Define a map U of [0, 1] to S" by Ux = (df( x), d2(x) ,... ), where the da(x) are the digits of the nonterminating dyadic expansion of x(and d (0) = 0). Show that T is measurable 6/9 and that U is
=+13.16. Let H, be the union of the intervals ((i - 1)/24, i/2*] for i even, 1 ≤i ≤ 2 *.Show that if 0 < f(@) ≤ 1 for all w and Ax =f-1(H,), then f(a) =ER _, /A (@)/24, an infinite linear combination of indicators.
=+13.15. Consider Lebesgue measure A restricted to the class $ of Borel sets in (0, 1].For a fixed permutation n1, n2 ,... of the positive integers, if x has dyadic expansion .x1x2 ..., take Tx = . xx ..... Show that T is measurable /9 and that AT" = A.
=+13.14. Show by example that u o-finite does not imply u. T" o-finite.
=+13.13. 1 Suppose that the circular Lebesgue measure of A satisfies u( A) > 1 - n-1 and that B contains at most n points. Show that some rotation carries B into A: 0B CA for some 0 in C.
=+C. Show that @ is generated by the arcs of C. Circular Lebesgue measure is defined as u = AT 1. Show that u is invariant under rotations: u[0z: z € A] =p( A) for A E 6 and 0 € C.
=+13.12. Circular Lebesgue measure. Let C be the unit circle in the complex plane, and define T: [0, 1) -> C by To = e2*Iw. Let $ consist of the Borel subsets of [0, i), and let A be Lebesgue measure on Ø. Show that € =[ A: T 'A € ¢] consists of the sets in 92 (identify R2 with the complex
=+nk so that p( B(k)) < €/24, and put B = U- ; B(k)13.10. 1 Show that Egoroff's theorem is false without the hypothesis u( A)
=+13.9. Suppose that f ,, and f are finite-valued, F-measurable functions such that f.(w)-f(w) for w E A, where (A) < c (u a measure on ). Prove Egoroff's theorem: For each € there exists a subset B of A such that u(B) k"' for some i z n. Show that B(*) | /Ø as n to, choose
=+14.3. (a) Suppose that X has a continuous, strictly increasing distribution function
=+F. Show that the random variable F(X) is uniformly distributed over the unit interval in the sense that P[F(X) ≤ u] =u for 0 ≤u ≤ 1. Passing from X to F(X) is called the probability transformation.
=+(c) Show that, if F ., = > F and F is everywhere continuous, then F.(x) - F(x)uniformly in x. What if F is continuous over a closed interval?
=+(b) Let 8, (€) = sup[ F(x) - F(y): [x -y| ≤ €] be the modulus of continuity of F. Show that d(F, G) < < implies that sup, |F(x) - G(x)| 5€ +8;(€).
=+14.8. 14.51 (a) Show that if a distribution function F is everywhere continuous, then it is uniformly continuous.
=+(c) By means of Problem 14.6 give a new construction of a nonmeasurable function and a nonmeasurable set.
=+(b) Define f arbitrarily on S, and define it elsewhere by f(n, x] + .. . +n, x;)= n. f(x,) + ... +n . f(x). Show that f satisfies Cauchy's equation but need not satisfy f(x)=xf(1).
=+(a) By Zorn's lemma find a maximal such S. Show that it is a Hamel basis. That is, show that each real x can be written uniquely as x = n, x 1 + . . . +n, x, for distinct points x; in S and integers n ;.
=+14.7. 1 Consider sets S of reals that are linearly independent over the field of rationals in the sense that n1 x1 + . . . +nk X =0 for distinct points x, in S and integers n, (positive or negative) is impossible unless n; = 0.
=+14.6. 12.31 A Borel function satisfying Cauchy's equation [A20] is automatically bounded in some interval and hence satisfies f(x) =xf(1). Hint: Take K large enough that A[x: x > s, If(x)| ≤ K] > 0. Apply Problem 12.3 and conclude that f is bounded in some interval to the right of 0.
=+14.5. The Levy distance d( F, G) between two distribution functions is the infimum of those e such that G(x - €) - < < F(x) { G(x + €) + e for all x. Verify that this is a metric on the set of distribution functions. Show that a necessary and sufficient condition for F = F is that d( F ., F)
=+(b) Show that if F is continuous at each point of F-'A, then P[F(X)€ A] is at most the Lebesgue measure of A.
=+(a) Show that for every Borel set A, P[F(X) € A, X EC] is at most the Lebesgue measure of A.
=+14.4. 1 Let C be the set of continuity points of F.
=+(c) Show that P[F(X) < u] = F(((u) - ) and hence that the result in part (a)holds as long as F is continuous.
=+(b) Show that the function (u) defined by (14.5) satisfies F(o(u)-)≤u ≤ F(o(u)) and that, if F is continuous (but not necessarily strictly increasing), then F((u)) =u for 0
=+13.8. A real function f on the line is upper semicontinuous at x if for each € there is a 8 such that |x -y| < 8 implies that f(y)
=+16.9. 1 Suppose that u(f) < 0, Show that {f } is uniformly integrable if and only if (i) {If,| du is bounded and (ii) for each e there is a 8 such that p(A)

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