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modern mathematical statistics with applications

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

=+(b) Regard the state as the size of a population and interpret the conditions Poo = 1 and fro > 0 and the conclusion in part (a).
=+(a) Show that P(U"_[X ,, = j i.o.]) =0 for all i.
=+8.8. Suppose that S = (0,1,2 ,... ], Poo =1, and f,o>0 for all i.
=+(c) Show that an irreducible chain is transient if and only if for each i there is a j + i such that f ,; < 1.
=+(b) Take k = i. Show that f .; > 0 if and only if P[X, +i ,..., X ,- +i, X„ =j]>0 for some n, and conclude that i is transient if and only if f .; < 1 for some j + i such that f ,, > 0.
=+8.7. (a) Generalize an argument in the proof of Theorem 8.5 to show that fik =Pik + E; + & Pijfjk- Generalize this further to+ _P[X, + k ,..., X ,-, * k, X„ =j]fjk.j + k
=+8.6. Show by solving (8.27) that the unrestricted random walk on the line (Example 8.3) is persistent if and only if p - ¿-
=+8.5. Call {x}} a subsolution of (8.24) if x; ≤ E,q ,; x, and 0 ≤x, ≤ 1, i € U. Extending Lemma 1, show that a subsolution [x,) satisfies x, So; The solution (0;} of(8.24) dominates all subsolutions as well as all solutions. Show that if x; = E, q ,;* /and - 1 x, ≤ 1, then {\x,l) is a
=+Specialize to the case i = j: in addition to implying that i is transient (Theorem 8.2(i)), a finite value for Ln -, pf,") suffices to determine f ,, exactly.
=+8.3(i)). If j is transient, then fi= C PH) /(1+2 Pc)
=+and prove that if j is transient, then E ., p(") < œ for each i (compare Theorem
=+8.4. Show that B8 [W]k-0 n=1m=1 A-1
=+8.3. Show by example that a function f( X), f(X,) ,... of a Markov chain need not be a Markov chain.
=+8.2. Let Yo, Y ,,... be independent and identically distributed with P[Y ,, = 1] =p, P[Y ,, = 0] = q = 1 - p, p + q. Put X„, = Y ,, + Y„+, (mod2). Show that X0; X1 ,...is not a Markov chain even though P[ X .. + 1 = jlX ,,-, = i] = P[X.+1 =j]. Does this last relation hold for all Markov
=+8.9. 8.5 1 Show for an irreducible chain that (8.27) has a nontrivial solution if and only if there exists a nontrivial, bounded sequence {x;} (not necessarily nonnega-tive) satisfying x, = E, . , P ;; x ;, i +i .. (See the remark following the proof of Theorem 8.5.)
=+8.10. 1 Show that an irreducible chain is transient if and only if (for arbitrary ig)the system y, = E, pijy ,, i + io (sum over all j), has a bounded, nonconstant solution {y ,, i € S).
=+(b) Apply this to the coupled chain in the proof of Theorem 8.6: Ip(?) - PR'l ≤p". Now give a new proof of Theorem 8.9.
=+8.18. (a) Let 7 be the smallest integer for which X, = ig. Suppose that the state space is finite and that the p, are all positive. Find a p such that max;(1 - pii) sp n] ≤p" for all i.
=+probability that @ next occurs n steps later. Such an & is called a recurrent event. If u, is the probability that & occurs at time n, then (8.53) holds. The recurrent event d is called transient or persistent according as f < 1 or f =1, it is called aperiodic if (8.54) holds, and if f = 1, u is
=+Although these definitions and facts are stated in purely analytical terms, they have a probabilistic interpretation: Imagine an event 6 that may occur at times 1,2 ,.... Suppose f ,, is the probability & occurs first at time n. Suppose further that at each occurrence of the system starts anew,
=+(b) Assume that f = 1, set u = En-Inf ,, and assume that(8.54)ged[ n: n ≥1, f ,, > 0] = 1.Prove the renewal theorem; Under these assumptions, the limit u = lim ,, u, exists, and u > 0 if and only if p < œo, in which case u = 1/u.
=+1. Define u1, u2 ,... recursively by u1 =f, and(8.53)un =fiun -1+ ... +fm-141 +f.(a) Show that f < 1 if and only if E ,, u ,, < m.
=+8.16. Show that the period of j is the greatest common divisor of the set(8.52)[n: n > 1, f/m)> 0].8.17. 1 Recurrent events. Let f1, f2 .... be nonnegative numbers with = D, f.s
=+Show that the chain is irreducible and aperiodic. For which p's is the chain persistent? For which p's are there stationary probabilities?
=+8.15. Suppose that S consists of all the integers and Po .- 1 = Po. o = Po.+1 = ], Pk.k-19, Pk. k+1=P, k ≤-1, Pk.k-1=P, Pk .* +1 =4.k≥1.
=+8.14. Show by example that the coupled chain in the proof of Theorem 8.6 need not be irreducible if the original chain is not aperiodic.
=+8.13. Suppose that {m} solves (8.30), where it is assumed that E; | -, | < 0, so that the left side is well defined. Show in the irreducible case that the ", are either all positive or all negative or all 0. Stationary probabilities thus exist in the irreducible case if and only if (8.30) has a
=+8.12. Show that sup ,, no(i, j) = œ is possible in Lemma 2.
=+The constraint z; ≤ 1 can be dropped: the minimal solution automatically satisfies it, since z, = 1 is a solution.
=+8.11. Show that the P ,- probabilities of ever leaving U for i E U are the minimal solution of the system.z, = E pijzi + E pij, iEU,(8.51)JEU JEU 0 ≤z; ≤ 1, iEU.
=+8.1. Prove Theorem 8.1 for the case of finite S by constructing the appropriate probability measure on sequence space S": Replace the summand on the right in (2.21) by cu Puju, "" Pu ..., and extend the arguments preceding Theorem 2.3. If X ,, (.) = z,(.), then X1, X2 ,... is the appropriate
=+E[ f(X )] > 1 as n -> co, so that u(i) = 1. Thus the support set is M = (0), and for an initial state i > 0 the probability of ever hitting M is fro < 1.For an arbitrary finite stopping time T, choose n so that P.[ 0.Then E[f(X,)] ≤1- fi+ .... PAT
=+13.6. Let § be a o-field in R'. Show that ?' c 9 if and only if every continuous function is measurable 9. Thus 9 is the smallest o-field with respect to which all the continuous functions are measurable.
=+(b) Suppose that A is a Borel set contained in H ,,. Show that A and indeed all the A @ 0(2n +1), have Lebesgue measure 0.
=+(a) Let H, = UT_ _. (S @ 0,), so that H ,, 1 G. Show that the sets H ,, @ 0(2n + 1)e are disjoint for different v.
=+12.5. 1 The construction here gives sets H ,, such that H ,, 1 G and A. (H ,, ) =0. If J. = G - H ,,, then J ,, LØ and A*(J,) = 1.
=+(d) Show that \, (He 0) =0 and A*(H) =1.
=+(c) Suppose that A is a Borel set contained in H. If A( A) > 0, then D(A)contains an interval (0, €); but then some 024 + 1 lies in (0,€) CD(A) CD(H), and so 02k+ 1 =h1 - h2 =h| eh2 = (s, @ 82 ) ℮ (s2 @ "2",) for some h1, h2 in H and some $1, s2 in S. Deduce that $1 =52 and obtain a
=+(b) Take x and y to be equivalent if x 6 y lies in (0 ,,: n = 0, + 1, ... ), which is a subgroup. Let S contain one representative from each equivalence class(each coset). Show that G = U, (S @ 6,), where the union is disjoint. Put H - U.(See ,, ) and show that G - H = Hee.
=+(a) Fix an irrational 6 in G and for n = 0, + 1, +2 ,... let ,, be no reduced modulo 1. Show that 0 ,, @ em= 0n+m, 0 ,, ℮6]=0 ,__, and the 0 ,, are distinct.Show that (02 ,,: n = 0, + 1 ,... ) and (02n + 1: n = 0, + 1 ,... ) are dense in G.
=+12.4. 1 The following construction leads to a subset H of the unit interval that is nonmeasurable in the extreme sense that its inner and outer Lebesgue mea-sures are 0 and 1: A, (H)=0 and A*(H) = 1 (see (3.9) and (3.10)). Complete the details. The ideas are those in the construction of a
=+12.3. 1 If A € .91 and A(A) > 0, then the origin is interior to the difference set D(A) =[x-y: x, y = A]. Hint: Choose a bounded open interval / as in Problem 12.2 for 0 = 2. Suppose that |z| < A(/)/2; since ANI and (ANI) +z are contained in an interval of length less than 3A(1)/2 and hence
=+12.2. Suppose that A € 221, A( A) > 0, and 0 < < 1. Show that there is a bounded open interval I such that A( An1) 2 0A(1). Hint: Show that A(A) may be assumed finite, and choose an open G such that A CG and A(A) ≥ 0A(G).Now G = U, I ,, for disjoint open intervals I ,, [A12], and E ,,
=+12.1. Suppose that u is a measure on "' that is finite for bounded sets and is translation-invariant: p( A +x) =u(A). Show that u(A) =«A(A) for some@ ≥0. Extend to R
=+(c) Let v be the extension of vo (see (11.6)) to o(g), and for A € o, define HO(A) = v( AX (0,1]). Show that wo is finitely additive and countably subaddi-tive on the semiring So.
=+(b) Let F be the smallest o-field with respect to which every f in _ is measurable: $ = of H: f Ef, HER']. Let . be the class of A in F for which AX (0,1] € (/). Show that 0 is a semiring and that F=0(0).
=+11.5. ! (a) Assume fef and let fn = (n(f - f A 1) A 1. Show that f(w) ≤ 1 implies f_(w) =0 for all n and f(w) > 1 implies f.(w) =1 for all sufficiently large n. Conclude that for x > 0,(11.7)(0, xf„] 1 [w:f(@)>1] x(0,x].
=+12.6. Suppose that u is nonnegative and finitely additive on 2 and that u( Rf)
=+12.7. Suppose u is a measure on 22 such that bounded sets have finite measure.Given A, show that there exist an F -set U (a countable union of closed sets)and a G3-set V (a countable intersection of open sets) such that U CA c V and"(V-U)=0.
=+13.5. Show of real functions f and g that f(w) + g(w)
=+13.4. 1 Relate the result in Problem 13.3 to Theorem 5.1(ii).
=+13.3. 1 Suppose that f: 22 - R1. Show that f is measurable T-19" if and only if there exists a map ": {' - R' such that " is measurable "" and f = $T. Hint:First consider simple functions and then use Theorem 13.5.
=+(c) Let a'(o") be the a-field in n' generated by ". Show that a(T-'')=T-('('). Prove Theorem 10.1 by taking T to be the identity map from 20 to 2.
=+(b) For given S, Ty', which is the smallest o-field for which T is measurable 9/ S', is by definition the a-field generated by T. For simpie random variables describe o(X ,, ..., X ) in these terms.
=+13.2. (a) For a map T and o-fields 9 and ', define T-19' =[T-'A': A' € 9]and T9 -[A': T-'A' € 9 ]. Show that T -19' and TS are o-fields and that measurability $/ 9 is equivalent to T-19'. and to PCT9.
=+Let A1, A2, ... be a countable covering of 2 by F-sets. Consider the o-field S, = [A: ACA ,,, A € 9 ] in A ,, and the restriction T, of T to A ,,. Show that T is measurable $/9' if and only if T, is measurable 91/ 9 for each n.
=+13.1. Functions are often defined in pieces (for example, let f(x) be x' or x ' as x 2 0 or x < 0), and the following result shows that the function is measurable if the pieces are.Consider measurable spaces (n, 5 ) and (Q', ') and a map T: 0 -0.
=+12.5. Let u, u1, u2 be the measures corresponding to F, F1, F2, and prove that u(A, XA2) = u (A1)2(A2) for intervals A, and A2. This u is the product of", and #2; products are studied in a general setting in Section 18.
=+12.12. Let F, and F2 be nondecreasing, right-continuous functions on the line and put F(x1, x2) = F(x )F (x2). Show that F satisfies the conditions of Theorem
=+12.11. Let G be a nondecreasing, right-continuous function on the line, and put F(x, y) =min{G(x), y). Show that F satisfies the conditions of Theorem 12.5, that the curve C = [(x, G(x)): x ER'] supports the corresponding measure, and that A,(C) = 0.
=+12.10. Of minor interest is the k-dimensional analogue of (12.4). Let I, be (0, 1] for 1 20 and (1, 0] for 1 ≤ 0, and let A, = / ,, x . .. x/, . Let (x) be +1 or -1 according as the number of i, 1 sis k, for which x; < 0 is even or odd. Show that, if F(x) = (x)u(A,), then (12.12) holds for
=+12.9. The minimal closed support of a measure u on 2 is a closed set C ,, such that C. CC for closed C if and only if C supports u. Prove its existence and uniqueness. Characterize the points of C ,, as those x such that p(U) > 0 for every neighborhood U of x. If k = 1 and if u and the function
=+12.8. 2.19₸ Suppose that u is a nonatomic probability measure on (RK, @*) and that p( A) > 0. Show that there is an uncountable compact set K such that KCA and u(K)=0.
=+Show that un is finitely additive and countably subadditive on o.
=+(b) Define a set function v, on , by(11.6)"o(f, g] = A(g-f).
=+(a) If f ≤g (f.g €_z), define in 2XR1 an "interval"(11.5)(f.8]=[(w.t): f(@)
=+(b) Find an example in which A ,, ! A, u( A ,, ) = 00, and A = Ø.
=+10.3. (a) In connection with Theorem 10.2(ii), show that if A ,, ! A and u( A,)
=+10.2. On the o-field of all subsets of Q={1,2 ,... } put u( A) = Eke 42 -* if A is finite and u( A) = otherwise. Is u finitely additive? Countably additive?
=+10.1. Show that if conditions (i) and (iii) in the definition of measure hold, and if u(A) < œ for some A € }, then condition (ii) holds.
=+9.7. Show that (9.35) is true if S ,, is replaced by IS,| or maxk ≤ ,, Sk or max k sn Skl-
=+(9.29) to give a simple proof that P[S ,, > (3n log n)1/2i.o.] = 0.
=+9.6. Weakened versions of (9.36) are quite easy to prove. By a fourth-moment argument (see (6.2)), show that P[S ,, > n3/4(log n)(1 +€)/4i.o.] =0. Use
=+9.5. 1 Suppose X ,, takes the values +1 with probability ; each, and show that P[S ,, = 0 i.o.] = 1. (This gives still another proof of the persistence of symmetric random walk on the line (Example 8.6).) Show more generally that, if the X ,, are bounded by M, then P[IS ,, I ≤ M i.o.] = 1.
=+9.4. From (9.35) and the same result for (-X,), together with the uniform bounded-ness of the X ,,, deduce that with probability 1 the set of limit points of the sequence {S, (2n log log n)~1/2} is the closed interval from - 1 to +1.
=+9.3. Relabel the binomial parameter p as 0 = f(p), where f is increasing and continuously differentiable. Show by (9.27) that the distinguishability of 0 from 0 + A0, as measured by K, is (AG)2/8p(1 -p)(f'(p))2 +O(A0)3. The leading coefficient is independent of 0 if f(p) = arcsinyp.
=+9.2. In the Bernoulli case, (9.21) gives P[S, 2 np + x] = exp - nk p + 2, p)(1+0(1)] .where p where x ,, = a ,, Vnpq . Resolve the apparent discrepancy. Use (9.25) to compare the two expressions in case x ,, /n is small. See Problem 27.17.
=+9.1. Prove (6.2) by using (9.9) and the fact that cumulants add in the presence of independence.
=+8.38. 5.121 Consider an irreducible, aperiodic, positive persistent chain. Let T, be the smallest n such that X ,, = j, and let m ;, = E,[7;]. Show that there is an r such that p= PiX, + j. ..., X, -, + j, X, = i] is positive; from ff"+") > pf (") and m, < co, conclude that m .; < > and my = > ;-
=+most one n 2 0 such that (i, i + 1 ,..., i +n) €/ ,. If there is no such n, then E{ f(X,)] =0. If there is one, then E [ f(X)] = P;[(X) ...., X„)= (i ...., i + n)]f(i+ n).and hence the only possible values of E,[f( X,)] are 0, f(i), pif(i+ 1) =f(i)A], P;P1+1f(i + 2) = f(i)A,A+ ......Thus v(i)
=+10.4. The natural generalization of (4.9) is(10.6)u (lim inf An) ≤ lim inf u ( A„)≤ lim sup u( An) ≤ w lim sup An).Show that the left-hand inequality always holds. Show that the right-hand inequality holds if (U. , , A) < @ for some n but can fail otherwise.
=+10.5. 3.10 A measure space (2, 7, p) is complete if ACB, BEF, and u( B)=0 together imply that A € 7-the definition is just as in the probability case. Use the ideas of Problem 3.10 to construct a complete measure space (0, 9+,+)such that Fc F and u and u + agree on S.
=+Let A be a real linear functional on a vector lattice / of (finite) real functions on a space 22. This means that if f and g lie in , then so do f V g and f Ag (with values max( f(w), g(w)} and min( f(w), g(w))), as well as af + Bg, and A(af+Bg) =@A(f) +BA(g). Assume further of / that f ./
=+11.4. This and Problems 11.5, 16.12, 17.12, 17.13, and 17.14 lead to proofs of the Daniell-Stone and Riesz representation theorems.
=+11.3. Show that Theorem 11.4(ii) can fail if u(B) =00.
=+(b) Find an example where u is not countably subadditive.
=+(a) Use Lemmas 1 and 2, without reference to Theorem 11.3, to show that u is countably subadditive if and only if it is countably additive.
=+11.2. Suppose that u is a nonnegative and finitely additive set function on a semi-ring .
=+(c) Show that f is a measure on Fand agrees with a on Fo.
=+(b) Suppose that E € 5. If there exist disjoint So-sets A ,, such that ECU ,, A ,, and u( Am) < œ, put A(E) = E ,, A,(EnA,) and prove consistency. Otherwise put A(E)=00.
=+(a) Let u be a measure (not necessarily even o-finite) on a field $0, and let F=o(5%). If A is a nonempty set in 76 and u( A) < co, restrict u to a finite measure u , on the field 900 A, and extend u , to a finite measuref, on the o-field FnA generated in A by So O.A.
=+11.1. The proof of Theorem 3.1 obviously applies if the probability measure is replaced by a finite measure, since this is only a matter of rescaling. Take as a starting point then the fact that a finite measure on a field extends uniquely to the generated o-field. By the following steps, prove
=+10.7. Example 10.5 shows that Theorem 10.3 fails without the o-finiteness condition.Construct other examples of this kind.
=+(b) Show by example that this is false without the condition that there are no"infinite atoms."
=+(a) Suppose that (2, 7, u) has no "infinite atoms," in the sense that for every A in $, if ( A) = , then there is in F a B such that BCA and 0 < p(B)
=+10.6. The condition in Theorem 10.2(iv) essentially characterizes o-finiteness.
=+8.37. 1 Let the chain be as in the preceding problem, but assume that B = 0, so that f/0 = 1 for all i. Suppose that A1, A2 .... exceed 1 and that A, .. . A >>
=+7.13. The functional equation (7.30) and the assumption that Q is bounded suffice to determine Q completely. First, Q(0) and Q(1) must be 0 and 1, respectively, and so (7.31) holds. Let Tox = {x and T,x = ; x + ;; let fox = px and fix =p + qx.Then Q(TL, . T. x) =f ... ... f ., Q(x). If the binary
=+Show that Q_(x) ≤ Q. (x) for all x and all policies w. Such a Tro is optimal.Theorem 7.3 is the special case of this result for p = {, bold play in the role of Tro, and u(x) = 1 or u(x) = 0 according as x =1 or x < 1.The condition (7.34) says that gambling with policy To is at least as good as
=+7.12. Let u be a real function on [0, 1], u(x) representing the utility of the fortune x.Consider policies bounded by 1; see (7.24). Let Q_(F) = E[u( F,)]; this repre-sents the expected utility under the policy T of an initial fortune Fo. Suppose of a policy wo that(7.34)u(x) ≤2m(x), 0 gx≤1,

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