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elementary probability for applications

A Course In Probability Theory 3rd Edition Kai Lai Chung - Solutions

1.13. If t(X) -0, $(X2) -a2 , 0 0] .n=l[HINT: Consider 00 lim(l -r)1/2 L rllg'>[vn= 0]rtI 11=0 as in the proof of Theorem 8.4.6, and use the following lemma: if Pn is a decreasing sequence of positive numbers such thatthen Pn '" n-I / 2 .] ~ 21/2, k=1 Pk
1.*12. If g'>{a(o,oo) < oo} < 1, then Vn --+ v, Ln --+ L, both limits being finite a.e. and having the generating functions.[HINT: ConsIder limm-+oo L~=o g'J{vm -n}rn and use (24) of Sec. 8.5.]
1.11. If -00 < g(X) < 0, then g(X) = £'(-V-), where V = sup Sj.[HINT: If V n max 1 '5.} '5.n S}, then 1let n --+ 00, then t t: O. For the case ct(X)to S. Port.]l
1.*10. Define a sequence of r.v.'s {Y n, n E NO} as follows:Yo = 0, Yn+1= (Yn +Xn+d+, n E NO, where {Xn, n E N} is a stationary independent sequence. Prove that for each n, Y nand M n have the same distribution. [This approach is useful in queuing theory.]
1.*9. Prove thatimplies :?P(M 0, exists and is finite, say = X(A). Now use proposition (E) in Sec. 8.4 and apply the convergence theorem for Laplace transforms (Theorem 6.6.3). M8 n=1 n P[S, >0]
1.8. Prove that the left member of the equation in Exercise 7 is equal to[1 -I=9'(a' = n;S. = 01]-1 .n=l where a' = a[O,oo); hence prove its convergence.
1.7. Prove that{00 1 0; Sn O} exp E n f?>[Sn n=l[HINT: One way to deduce this is to switch to Laplace transforms in (6) of Sec. 8.5 and let A ---+ 00.]
1.6. If M < 00 a.e., then it has an infinitely divisible distribution.
1.5. Prove that n=O n=l and deduce that 9'[M -OJ-exp {i'= ~ 9'[5. > OJ}.n=l
1.*4. Prove (13) of Sec. 8.5 by differentiating (7) there; justify the steps.
1.3. Find an expression for the Laplace transform of Sa. Is the corre sponding formula for the Fourier transform valid?
1.*2. Under the conditions of Theorem 8.4.6, show that[{Sa} = -lim r Sn dqp. n-+oo J{a>n}
1.1. Derive (3) of Sec. 8.4 by considering${raeitSa} =f rn [r eitSndqp r eitSn dqpj.n=l J{a>n-l} J{a>n}
1.18. For an arbitrary random walk, if :?>{Vn EN: Sn > O} > 0, then L :?>{Sn :s 0, Sn+l > O} < 00.n Hence if in addition :?>{Vn EN: Sn :s O} > 0, then L 1:?>{Sn > O} -:?>{Sn+l > O}I < 00;n and consequently L q>{Sn > O} < 00. n n(-It[HINT: For the first series, consider the last time that Sn < 0;
1.17. The basic argument in the proof of Theorem 8.3.2 was the "last time in (-E, E)". A harder but instructive argument usmg the "first time" may be given as followsShow that for 1 :s m M M M L gn(E) :s L f~m) Lgn(2E).Il=m n=m ¥This fonn of condition (iii') is due to Hsu Pei; see also Chung and
1.16. Suppose @v(X) -0,° < cS'(X2) < 00, and f.L IS of the mteger lattIce type, then.9>{Sn2 = 0 i.o.} = 1.
1.15. Generalize Theorem 8.3.3 to (!Il2 as follows. If the central limit theorem applies in the fonn that Sn I .Jli converges in dist. to the unit nonnal, then the random walk is recurrent. [HINT: Use ExercIses I3 and 14 and Exercise 4 of § 4 3 This is sharper than Exercise 11 No proof of Exercise
1.14. Extend Lemma 2 in the proof of Theorem 8.3.3 as follows. Keep condition (i) but replace (ii) and (iii) by(iii) l:~-o Un (m) < cm2l:~_o Un (1);There exists d > 0 such that for every b > 1 and m > m(b):n Then (10) is true. *
1.13 .. Generalize Lemma 1 in the proof of Theorem 8.3.3 to gzd. For d = 2 the constant 2m in the right member of (8) is to be replaced by 4m2 , and"Sn < E" means Sn is in the open square with center at the origin and side length 2E.
1.*12. Prove that no truly 3-dimensional random walk, namely one whose.:-ommon distribution does not have its support in a plane, is recurrent. [HINT:There exists A >° such that A(3 )2 1 11 ~tiXi Il(dx)is a strictly positive quadratic fonn Q in (tI, f2, (3)' If 3 L Itil < A-I, i=I then 8.3
1.*11. Prove that in :!J?2 if the common distribution of the random vector(.\, Y) has mean zero and finite second moment, namely:leX) = 0,
1.10. Generalize Exercises 6 and 7 above to :!J?;J.
1.*9. Prove that the random walk with J(t)=e-1t1a , O
1.*8. Prove that the random walk with J(t) = e-1tl (Cauchy distribution) is recurrent.
1.*7. If there exists a 8 > 0 such that 10 dt sup < 00,then the random walk is not recurrent. [HINT: Use Exercise 3 of Sec. 6.2 to show that there exists a constant C(E) such that 11 -cos E-IX C(E) lIfE jU .'/)(ISIlI < E) :::: C(E) 2 f.Ln (dx) :::: --du Jut dt, :,7i'i x 2 0 -u where f.L1l is the
1.* 6. If there exists a 8 > 0 such that (the integral below being real valued)lim) = 00, rt1 -0 1 -rf(t)then the random walk is recurrent.
1.5. For a recurrent random walk that is neither degenerate nor of the lattice type, the countable set of points {Sn (w), n E N} is everywhere dense in,1£1 for a.e.w. Hence prove the following result in Diophantine approximation:if y is irrational, then given any real x and E > 0 there exist
1.* 4. Assume that 9{X 1= O} < 1. Prove that x is a recurrent value of the random walk if and only if 00 E 9i'){ISn xl < E} 00 for every E > O.1l=1
1.3. Prove the Remark after TheOIem 8.3.2.
1.*2. If a random walk in ql(;J is recurrent, then every possible value is a recurrent value.
1.1. Generalize Theorems 8.3.1 and 8.3.2 to r!Ad . (For d ~ 3 the general ization is illusory; see Exercise 12 belo..,,,.)
1.14. In an independent process where all Xn have a common bound,{{ex} < 00 implie£ l{Su} < 00 for each optional ex [cf. Theorem 5.5.3].
1.13. State and prove the analogue of Theorem 8.2.4 with a replaced by a[O,oo)' [The inclusion of° in the set of entrance causes a small difference.]
1.12. Prove the Corollary to Theorem 8.2.2.
1.*11. Let {Xn, 11 E }V} be a stationary' independent proeess and {ak' kEN}a sequence of strictly increasing finite optional r.v.'s. Then {Xetk+1, kEN} is a stationary independent process with the same common distribution as the original process [Tbis is the gambling-system theorem first given by
1.10. Generalize Theorem 8.2.2 to the case where the domain of definition and finiteness of a is ~ with° < g>(~) < 1. [This leads to a useful extension of the notion of independence. For a given A in '?7 with gz>(A) > 0, two events A and M, where M C b.., are said to be independent relative to b..
1.9. Find an example of two optional r.v.'s a and f3 such that a < f3 but f3 -a is not optional.
1.*8. Find an example of two optional r.v.'s a and f3 such that a .:::: f3 but::!/r; jJ ;/// However, if y is optional relative to the post-a process and f3 =a + y, then indeed ~; :J ~fJ. As a particular case, 97ifk is decreasing (while;/jt~k is increasing) as k increases.
1.7. If a and f3 are any two optional r.v.'s, then Xf3+/r et w) = Xet(W)+~(T"W)+/W);(rf3 0 ret)(w) = r~(T"w)+et(w)(w) #-r~+et(w) in general.
1.*6. Prove the following relations:cl = a 0 r~"-I ; r~"-I 0 ret = r~",' X 0 ret X ~k-I+j= ~k+j·
1.5. Vk EN: al + ... + ak is optional. [For the a in (11), this has been called the kth ladder variable.]
1.* 4. If a is optional and 13 is optional relative to the post-a process, then a + 13 is optional (relative to the original process).
1.3. If CYI and CYz are both optional, then so is CYI A CY2, CYI V CY2, CYI -t CY2. If a is optional and !:1 E ~a, then a,6. defined below is also optional:a,6. = {a +00 on !:1 on Q\!:1.
1.*2. For each optional a we have a E ~a and Xa E ~. If a and 13 are both optional and a ::; 13, then ~a C 'Jl{J.
1.* 1. a is optional if and only if 'in EN' {a < n} E4n
1.12. Let {Xn, n ::::. I} be independent r.v.'s with 9{Xn = 4-n} = g'){Xn =_4-1l} = ~. Then the remote field of {Sn, n ::::. I}, where Sn = 2:j=l Xj , is not triviaL
1.11. Show that the conclusion of TheOlem 8.1.4 holds tIUe for a sequence of independent r.v.' s, not necessarily stationary, but satisfying the following condition: for every j there exists a k > j such that Xk has the same distribu tion as Xj . [This remark is due to Susan Hom.]
1.*10. Consider the hi-infinite product space of all bi-infinite sequences of real numbers {WIl' n E N}, where N is the set of all integers in its natural(algebraic) ordering. Define the shift as in the text with N replacing N, and show that it is I-to-l on this space. Prove the analogue of (7).
1.9. Prove that an invariant event is remote and a remote event is permutable.
1.8. Find trivial examples of independent processes where the three numbers ?(r-1 A), ?P(A), ?P(rA) take the values 1,0, 1; or 0, ~, 1.
1.*7. The set {Y 21l E A i.o.}, where A E qj1, is remote but not necessarily invariant; the set {2:j=l Y j A i.o.} is permutable but not necessarily remote.Find some other essentially different examples of these two kinds.
1.* 6. If all > 0, limn--+oo all exists >° finite or infinite, and limn--+oo (an+ 1/ an)= 1, then the set of convergence of {an -1 2:j=l Yj } is invariant. If an ~ +00, the upper and lower limits of this sequence are invariant r.v.'s.
1.5. The set of convergence of an arbitrary sequence of r.v.'s {Yn, n EN}or of the sequence of their partial sums ~'j= 1 Y j are both permutable. Their limits are permutable r.v.'s with domain the set of convergence.
1.4. An r. v. is in variant [permutable] if and only if it belongs to the invariant [permutable] field.
1.3. If A is invariant then A = rA; the converse is false.
1.2. An r v belongs to the all-or-nothing field if and only if it is constant a.e.
1.1. Find an example of a remote field that is not the trivial one; to make It mteresting, insist that the r. v.' s are not identical.
1.*12. ReconSIder ExerCIse 17 of Sec. 6.4 and try to apply Theorem 7.6.3.[HINT: The latter is not immediately applicable, owing to the lack of unifonn convergence. However, show first that if eenit converges for tEA, where meA) > 0, then it converges for all t. This follows from a result due to
1.11. Strengthening Theorem 65 5, show that two infinitely divisible ch.f.' s may coincide in a neighborhood of° without being identical.
1.*10. Some writers have given the proof of Theorem 7.6.6 by apparently considering each fixed t and using an analogue of (21) WIthout the "sUPltl:51l "there. Criticize this "quick proof'. [HINT: Show that the two relations'v't and 'v'm: lim Umn (t) = U /1 (t), m--+oo Yt· lim l(1l (t) -u(t),
1.9. Let f(t) = 1 -t, fk(t) = 1 -t+ (-llitk-1 ,° 1. Then f k never vanishes and converges unifonnly to f in [0, 2]. Let ,JTk denote the distinguished square root of f k in [0, 2]. Show that ,JTk does not converge in any neighborhood of t = 1. Why is Theorem 7.6.3 not applicable? [This example is
1.*8. Give an example to show that in Theorem 7.6.3, if the unifonnity of convergence of kf to f is omitted, then kA need not converge to A. [HINT:kf (t) = exp{27Ti( -1 )kkt(1 + kt)-l }.]
1.7. Carry out the proof of Theorem 7.6.2 specifically for the "trivial" but instructive case f (t) = eait , where a is a fixed real number.
1.6. Show that the d.f. with density fiCtf(ex)-l xCt 1e-f3x, ex > 0, fi > 0, in(0, (0), and° otherwise, is infinitely divisible.
1.5. Show that f (t) = (1 -b) / (1 -beit ),° < b < 1, is an infinitely divis ible ch.f. [HI1\'T: Use canonical fonn.]
1.4. Give another proof that the right member of (17) is an infinitely divis ible ch.f. by using Theorem 6.5.6.
1.* 3. Let f be a ch.f. such that there exists a sequence of positive integers nk going to infinity and a sequence of ch.f.'s CfJk satisfying f = (CfJk)lIk ; then f is infinitely divisible.
1.2. If j IS an mfimtely dIvISIble ch.f. and X ItS dlstmgUIshed loganthm, r > 0, then the rth power of f is defined to be erA(t). Prove that for each r >°it is an infinitely divisible ch.f.
1.1. Is the convex combination of infinitely divisible ch.f.' s also infinitely divisible?
1.6. Prove that .q'){ISn I > cp(l -8, sn)i.o.} = 1, without use of (8), as follows. Let ek = {w: ISnk (w)1 h = {W:Sn .. , (w) -Sn,(w) > I'(1 -~,Sn'+') }.Show that for sufficiently large k the event ek n f k implies the complement of ek+ 1; hence deduceand show that the product -+ 0 as k -+
1.* 5. The law of the iterated logarithm may be used to supply certain coun terexamples. For instance, if the Xn 's are independent and Xn = ±n liZ/log log 1l with probability ~ each, then Sn/n ) 0 a.e., but Kolmogorov's sufficient conditi on (see case (i) after Theorem 5.4.1) 2: n t (X ~ ) / n 2
1.4. Prove that in Exercise 9 of Sec. 7.3 we have .9>{Sn 0 i.o.} 1.
1.*3. Let {X j, j > I} be a sequence of independent, identically distributed r.v.'s with mean 0 and variance 1, and Sn = 2:j-l Xj. Then 1.[HTh'T: Consider Snk+1 -Snk with nk ""' kk. A quick proof follows from Theorem 8.3.3 below.] Plim 00+11 |Sn(w)| n ! = 0} = 1.
1.*2. Prove that whenever (2) holds, then the analogous relations with Sn replaced by maxl::;m::;n Sm or maxl::;m::;n ISml also hold.
1.1. Show that condition (3) is fulfilled if the X/s have a common d.f.with a finite third moment.
1.4. Prove that for every x > 0: x 1+x x/2 e-y 12 dy J= x
1.*3 'fl . . .lete eXIsts a uIllvetsal constant Al > 0 such that for any sequence of independent, identically distributed integer-valued r. v.' s {X;} with mean 0 and variance 1, we have Al sup IF n (x) - (x)1 :::: 1'2 'x n I where Fn is the d.f. of (2:.1=1 X j )/:j1i. [HINT: Use Exercise 24 of
1.2. If J and g are ch.f.'s such that J(t) = get) for It I < T, then J \F(x) -00 G(x)1 ax ~ T' , Jr This is due to Esseen (Acta Math, 77(1944».
1.1. If F and G are d.f.' s with finite first moments, then I: IF(x) -G(x)1 dx < 00.[HINT: Use Exercise 18 of Sec. 3.2.]
1.9. Let {Xl > I} be independent r.v.' s with the symmetric Bemoullian distribution. Let Nn (w) be the number of zeros in the first n terms of the sample sequence {S/w),} ::: 1}. Prove that NI1/:Jn converges III dISt. to the same G as in Theorem 7.3.3. (HINT: Use the method of moments. Show that
1.8. Under the same hypothesis as III Theorem 7.3.3, prove that maxI
1.7. Deduce from Exercise 6 that for a symmetric stable LV. X with exponenta, 0 < a < 2 (see Sec. 6.5), there exists a constant c > 0 such that,:7>{IXI > nl/a} ~ cln. [This is due to Feller; use Exercise 3 of Sec. 6.5.]
1.6. If {Xn} are independent, identically distributed symmetric r.v.' s, then for every x ~ 0,
1.* 5. There are a ballots marked A and b ballots marked B. Suppose that these a + b ballots are counted in random order. What is the probability that the number of ballots for A always leads in the countmg?
1.* 4. Give an example of a sequence of independent and identically distributed r. v.' s [X n) with mean 0 and variance 1 and a sequence of positive integer-valued r.v.'s Vn tending to 00 a.e. such that Svjsvn does not converge in distribution. [HINT: The easiest way is to use Theorem 8.3.3 below.]
1.3. Let {X), v), j ~ I} be independent r. v.' s such that the v / s are integer valued, v) -+ 00 a.e., and the central limit theorem applies to (Sn -an)lbn, where Sn = l:J=1 X), an, bl1 are real constants, bn -+ 00. Then it also applies to (Svn aVn)Ibv/l'
1.* 2. Let {X), j ~ I} be a sequence of independent r. v.' s having the Bemoullian d.f. POI + (1 -P )00,0 < P < 1. An r-run of successes in the sample sequence {X)(w), j ~ I} is defined to be a sequence of r consecutive"ones" preceded and followed by "zeros". Let N n be the number of r-runs in the
1.1. Let {X), j ~ I} be a sequence of independent r. v.' s, and j a Borel measurable function of m variables. Then if ~k = j(Xk+I, ... ,Xk+m), the sequence {~b k ~ 1} is (m -I)-dependent.
1.*12. The following combinatorial problem is similar to that of the number of inversions. Let Q and 9 be as in the example in the text. It is standard knowledge that each permutation 2can be uniquely decomposed into the product of cycles, as follows. Consider the permutation as a mapping n from
1.11. Prove that J~oo x2 dF(x) < 00 implies the condition (11), but not vice versa.
1.*10. It is important to realize that the failure of Lindeberg's condition means only the failure of either (i) or (ii) in Theorem 7.2.1 with the specified constants Sn. A central limit theorem may well hold with a different sequence of constants I,et.') IJ , XJ' = ±J' , 0, 1 wim prooaoiliry 12
1.9. Let X j be defined as follows for some Ot > 1:(± 'a J , Xj =0, 1 with probability 6 '2(a 1) each; J 1with probability 1 - 3 '2(a 1) J Prove that Lindeberg's condition is satisfied if and only if Ot < 3/2.
1.8. For each j let X j have the uniform distribution in [-j, j]. Show that Lindeberg's condition is satisfied and state the resulting central limit theorem.
1.*7. Find an example where Lindeberg's condition is satisfied but Liapounov's is not for any 8 > 0.In Exercises 8 to 10 below {Xj , j 2: I} is a sequence of independent r.v.'s.
1.6. Prove that if 8
1.5. Derive Theorem 6.4.4 from Theorem 7.2.1.
1.*4. Prove the sufficiency part of Theorem 7.2.1 Theorem 7.1.2, but by elaborating the proof of the latter.expanSIOn and irx e . (tx)2 1 j ltx+e 2 for Ixl > 17 e = I + ltx ---+ e -for Ix I :s Y}. 2 6 ax . (txf , Itxl3[HINT: Consider without using[HINT: Use the As a matter of fact, Lindeberg's
1.*3. Prove that in Theorem 7.2.1, (i) does not imply (1).r. v. 's with nonnal distributions.]
1.2. Prove that Lindeberg's condition (1) implies that max ani -+ O.
1.1. Restate Theorem 7.1.2 in tenns of nonned sums of a single sequence.

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