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elementary probability for applications

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

1.*6. Prove the assertion made in Exercise 5 of Sec. 5.3 using the methods of thIS sectIOn. [HINT: use Exercise 4 of Sec. 4.3.]
1.*5. In Theorem 7.1.2 let the d.f. of Sn be Fn. Prove that given any E > 0, there exists a 8(E) such that rn :s8(E)=>L(Fn,c:I»:SE, where L is Levy distance. Strengthen the conclusion to read:sup IFn(x) cf>(X) I < €.
1.4. Let {X;} be independent r. v.' s such that maxI < j
1.3. For the double array (2), it is possible that Sn/bn converges in dist.for a sequence of strictly positive constants bn tending to a finite limit. Is it still possible if bn oscillates between finite limits?
1.2. For any sequence of r.v.'s {Yn}, if Yn/bn converges in dist. for an increasing sequence of constants { bn}, then Y n I b~ converges in pr. to 0 if bn= o(b~). In particular, make precise the following statement: "The central limit theorem implies the weak law of large numbers."
1.*1. Prove that for arbitrary r.v.'s {Xnj} in the array (2), the implications(d) => (c) => (b) => (a) are all strict. On the other hand, if the X n / s are independent in each row, then (d) = (c).
1.11. If I is completely monotonic in (0, 00) with 1(0+) = +00, then I is the Laplace transform of an infinite measure fJ., on .9i\:I(A) = r e-AxfJ.,(dx). }[JI{+[HINT: Show that F n (x) :s e2x81(8) for each 8 > 0 and all large n, where Fn is defined in (7). Alternatively, apply Theorem 6.6.4 to I(A
1.*10. If I > 0 on (0,00) and has a derivative I' that is completely mono tonic there, then 1/1 is also completely monotonic.
1.9. Given a function CJ on !Ji>+ that is finite, continuous, and decreasing to zero at infinity, find a CJ-finite measure fJ. on :~+ such that Vt > 0:r CJ(t -s)fJ.(ds) = 1.J[O.t][HINT: Assume CJ(O) = 1 and consider the dJ. 1 -CJ.]
1.*8. In the notation of Exercise 7, prove that for every A, fJ. E .~+:A A!C).) !CM)·
1.7. Let leA) = Jooo e-A.x f(x)dx where fELl (0, (0). Suppose that f has a finite right-hand derivative l' (0) at the origin, then A fCO) lim kfCk), f'(O) = lim A[Af(A) -f(O)].).,--+00
1.6. Let F be an s.dJ. with support in f?I4. Define Go F, for 11 2: 1. Find Gil (A) in terms of F(A).
1.5. Use Exercise 3 to prove that for any d.f. whose support is in a finite interval, the moment problem is determinate.
1.*4. Let {FIl }be s.d.f.'s. If AO > 0 and limn~ooFn (A) exists for all A 2: AO, then {Ell} converges vaguely to an s d f
1.*3. Let F and G be s.d.f.'s. If AO > 0 and F(A) = G(A) for all A ~ AO, then F = G. More generally, if F(nAo) = G(nAo) for integer n ~ 1, then F = G.[HINT: In order to apply the Stone-Weierstrass theorem as cited, adjoin the constant 1 to the family {e-A.x, A ~ AO}; show that a function in Co can
1.* 2. Two uncorrelated r. v.' s with a joint normal d.f. of arbitrary parameters are independent. Extend this to any finite number of r.v.'s.
1.1. If X and Yare independent r. v.' s with normal d.f.' s of the same variance, then X + Y and X -Y are independent.
1.9. Show that in Theorem 6.3.2, the hypothesis (a) cannot be relaxed to require convergence of {In} only in a finite interval It I :::; T.
1.8. Suppose I(t, u) is a function on gr2 such that for each u, 1(·, u) is a ch.f.~ and for each t, I(t,·) is continuous. Then for any d.f. G, exp{I: [/(t, u) -l]dG(U)}is a ch.f.
1.7. Construct a ch.f. that vanishes in [-b, -a] and [a, b], where 0 < a
1.6. Show that there is a ch.f. that has period 2m, m an integer ~ 1, and that is equal to 1 -It I in [-1, +1]. [HINT: Compute the Fourier series of such a function to show that the coefficients are positive.]
1.*5. Another proof of Theorem 6.5.3 is as follows. ShO'.v that 70 tdj'(t) = 1 and define the d.f. G on .0'C+ by G(u)r t df' (t)N ext show that Hence if we set rill) f (u, t) = 1 --;; V 0(see Exercise 2 of Sec 6 2), then f(t) =r feu, t)dG(u).J[O,OO)Now apply Exercise 2 of Sec. 6.1.
1.4. If F is a symmetric stable distribution of exponenta, 0 < a < 2, then J~oo Ixlr dF(x) < 00 for r < a and = 00 for r ::::a. [HINT: Use Exercises 7 and 8 of Sec. 6.4.]
1.3. If {X n} are independent r. v.' s with the same stable distribution of exponenta, then I:Z=l Xdn Ija has the same distribution. [This is the origin of the name "stable".]
1.2. Show that the following functions are ch.f.' s:1 1 + It I ' f(t) = { 1 -Itla, 0,{I -Itl, f(t) = _1 41tl'if It I :s 1;if It I :::: 1;if 0< It I :s!;if It I ~ !.0< a :s 1,
1.1. If f is continuous in ~1 and satisfies (3) for each x E .~1 and each T > 0, then f is positive definite.
1.35. Random variables defined on ® are also referred to as defined"modulo I". The theory of addition of independent r.v.'s on ® is somewhat simpler than on ,921, as exemplified by the following theorem. Let {X j, j ~ I}be independent and identically distributed r. v.' s on ® and let S k = L~= 1
1.34. Suppose that the space V is replaced by its closure [0, 1] and the two points° and 1 are identified; in other words, suppose '1/ is regarded as the circumference ® of a circle; then we can improve the result in Exercise 33 as follows. If there exists a function g on the integers such that f
1.*33. f.-Lk~f.-L if and only if I!.ik(·) ~ I p.(.) everywhere.
1.*32. f.-L is equidistributed on the set {jn-I ,° :s j :s n -I} if and only if I p.(j) = ° or 1 according to j f n or j In.
1.31. Prove that If p.(n )1 -I If and only If fJ, has its support in the set{8o + jn-1 ,° :s j :s n -I} for some 80 in (0, n-I ].
1.30. EstablIsh the InVerSIOn formula expressIng f.-L In terms of the f p.(n)'s.Deduce again the uniqueness result in Exercise 29. [HINT: Find the Fourier series of the indicator function of an interval contained in 0t/.]
1.29. Define for each n:f (n \ = [ e27rinx u(dx\ ~t( ) Jw ""')Prove by WeIerstrass ' s approxImatIOn theorem (by tngonometncal polyno mials) that if 11*, (n) = 11*2 (n) for every n > 1 , then J.-L 1 = U2. The conclusion becomes false if 11 is replaced by [0, 1].
1.28. Let Qn be the concentration function of S n = 2:J= 1 X j, where the x /s are independent r.v.'s having a common nondegenerate dJ. F. Then for every h > 0,[HINT: Use Exercise 27 above and Exercise 16 of Sec. 6.1. This result is due to Levy and Doeblin, but the proof is due to Rosen.]In
1.27. If I is any nondegenerate ch. I, then there exist constants A >°and 8 >° such that II (t)1 :s 1 -At2 for It I :s 8.[HINT: Reduce to the case where the d.f. has zero mean and finite variance by translating and truncating.]
1.26. If F is symmetric and J Ixl dF(x) < 00, then n.9>{Sn = j} ~ 00.[HINT: 1 -l(t) = o(ltl) as t ~ 0.]
1.25. If F #-8o, then there exists a constant A such that for every j::?>{Sn = j} :s An -1/2.[HINT: Use a special case of Exercise 27 below.]
1.*24. If J x dF(x) = 0, J x2 dF(x) = (J2, then for each integer j:1 n 1/2~}J{SIl = j} ~ -Jfii'(J 2Jr[HINT: Proceed as in Theorem 6.4.4, but use (15).]
1.*23. If {Xn} is a sequence of independent and identically distributed r.v.'s, then there does not exist a sequence of constants {en} such that Ln (Xn -en)converaes a e lInless the common d f is degenerate b , In Exercises 24 to 26, let Sn = LJ-l Xj, where the Xjs are independent r.v.'s with a
1.22. Let f(s, t) be the ch.f. of a 2-dimensional p.m. v. If If(so, to)1 = 1 for some (s-o, to) #-(0,0), what can one say about the support of v?
1.21. The span of an integer lattice distribution is the greatest common divisor of the set of all differences between points of jump.
1.20. Show by using (14) that I cos tl is not a ch.f. Thus the modulus of a ch.f. need not be a ch.f., although the squared modulus always IS.
1.*19. Reformulate Exercise 18 in terms of dJ.'s and deduce the following consequence. Let F n be a sequence of dJ.' s an, a~ real constants, bn > 0, b;! > O. If where F is a nondegenerate dJ., then and ~o.[Two dJ.'s F and G such that G(x) = F(bx +a) for every x, where b > 0 and a IS real, are saId
1.*18. Let f and g be two nondegenerate chJ.'s. Suppose that there exist real constants all and bn > 0 such that for every t:Then an ~a, bll~b, where a is finite, 0 < b < 00, and g(t) = eita / b f(t/b).[HINT: Use Exercises 16 and 17.]
1.*17. Suppose Cn is real and that eCnit converges to a limit for every t in a set of strictly positive Lebesgue measure. Then Cn converges to a finite limit. [HINT: Proceed as in Exercise 16, and integrate over t. Beware of any argument using "logarithms", as given in some textbooks, but see
1.*16. Suppose bn > 0 and I/(bnt)1 converges everywhere to a ch.f. that is not identically 1, then bn converges to a finite and strictly positive limit.[HINT: Show that it is impossible that a subsequence of bn converges to 0 or to +00, or that two subsequences converge to different finite limits.]
1.*IS.1f 1111(1)1----+ 1 for every t as n----+ 00, and F" is the dJ. corre sponding to In, then there exist constants an such that F n (x + an )480. [HINT:Symmetrize and take all to be a median of F n.J
1.14. If 11 (t) I 1, 11 (t') I -1 and tit' is an irrational number, then f is degenerate. If for a sequence {tkJ of nonvanishing constants tending to 0 we have If (tdl = I, then f IS degenerate.
1.13. The converse part of Theorem 6.4.1 is false for an odd k. Example.F is a discrete symmetric dJ. with mass C /n2log n for integers n > 3, where o is the appropriate constant, and k = 1. [HINT: It is well known that the series I:: sin nt n log n 11 converges uniformly in t.]
1.*12. Let {Xj ' j > I} be independent, identically distributed r.v.'s with mean 0 and variance 1 Prove that both 11;=1 and ;=1 converge in dist. to
1.11. Let X and Y be independent with the common dJ. F of mean 0 and variance 1. Suppose that eX + Y)/~ also has the dJ. F. Then F -. [HINT:Imitate Theorem 6.4.5.]
1.10. Suppose F satisfies the condition that for every 1] > 0 such that as A ---+ 00,] dF(x) _ a(e 71,1).Ixl>A rThen all moments of F are finite, and condition (6) in Theorem 6.4.5 is satisfied.
1.9. Suppose that e-cltla, where c > 0,0 < a < 2, is a chJ. (Theorem 6.5.4 below). Let {X j, j > I} be independent and identically distributed r. v.' s with a common chJ. of the form as t ---+ O. Determine the constants band () so that the ch.f. of S n / bn e converges to g_ltl a .
1.8. IfO < a < 1 and J Ixla dF(x) < 00, then f(t) -1 = o(ltla ) as t -+ 0.For 1 < a < 2 the same result is true under the additional assumption that J x dF(x) = 0. [HINT: The case 1 < a < 2 is harder. Consider the real and imaginary parts of f (t) -1 separately and write the latter as 1 sintxdF(x)
1.7. Let f be the ch.f. of the dJ. F. Suppose that as t -+ 0, where a < a < 2, then as A -+ 00,[HINT: Integrate ~xl>A (l cos tx) dFex) < ct* over t in (0, A).]
1.*6. Prove that in Theorem 6.4.4, Sn/(JJ1i does not converge in proba bility. [HINT: Consider Sn/CJ..,fii and S2n/CJ.j2n.]
1.5. Let Xk have the Poisson distribution with parameter 2 Prove that[X).. -J....]/J.... 1/2 converges in dist. to as J.... -+ 00.
1.* 4. Let Xn have the binomial distribution with parameter en, PI!), and suppose that n Pn -+ J.... > 0. Prove that Xn converges in dist. to the Poisson d.f.WIth parameter X. (In the old days thIS was called the law of small numbers.)
1.3. Let q7>{X = k} = Pb 1 < k < .e < 00, 2:i=1 Pk = 1. The sum Sn of n independent r.v.' s having the same distribution as K is said to have a multinomial distribution. Define it explicitly. Prove that [Sn -g(Sn )]/(J(Sn)converges to o.
1.1· {U(ISn I) 2 l' @ (S: ) ~ 1m (c r::: = 1m (0 r::: = -(J 11--+00 V n n--+oo V n 7f[If we assume only q7>{X 1 #-o} > 0, £'( IX 11) < 00 and £'(X 1) = 0, then we have t(IS n I) > C.fii for some constant C and all n; this is known as Hornich's inequality.] [HINT: In case (J2 = 00, if limn 0'(1
1.*2. Let {Xn} be independent, identically distributed with mean° and vari ance (J2,° < (J2 < 00. Prove that
1.*1. If f is the ch.f. of X, and . f (t) -1 -2:2 11m = --> -00,then cf(X) = ° and e(X2) = (J2. In particular, if f(t) = 1 + 0(t2) as t -+ 0, then f = 1.
1.12. In the notation of Exercise 11, even if SUPtE~J I in (t) gn (t)1 ) 0, it does not follow that (Fn, Gn) ---+ 0; indeed it may ---+ 1. [HINT: Let I be any ch.f. vanishing outside (-1,1), Ij(t) = e !/ljt I(mjt), gj(t) = e injt I(mjt), and Fj , Gj be the corresponding dJ.'s. Note that if mjn7l
1.*11. Let Fn, Gn be the d.f.'s of /-Ln, Vn, and In, gn their ch.f.'s. Even if SUPXEg;>J IFn(x) Gn(x)I----+ 0, it does not follow that (fn,gnb ---+ 0; indeed it may happen that (In, gn h = 1 for every n. [HINT: Take two step functions"out of phase".]
1.10. Using the strong law of large numbers, prove that the convolution of two Cantor dJ.' s is still singular. [Hll\1'f: Inspect the frequency of the digits in the sum of the corresponding random series; see Exercise 9 of Sec. 5.3.]
1.9. Rewrite the preceding fonnula as sint C t)C t ~ -t=II cos 22k III cos 22k k=l k=l Prove that either factor on the right is the ch.f. of a singular distribution. Thus the convolution of two such may be absolutely continuous. [HINI. Use the same r.v.'s as for the Cantor distribution in Exercise
1.*8. Interpret the remarkable trigonometric identity. 00 Sillt = Ilcos ~t 2n n=l in tenns of ch.f.'s, and hence by addition of independent r.v.'s. (This is an example of Exercise 4 of Sec. 6.1.)
1. 7. If Fn---+ F and Gn ---+ G, then Fn * Gn ---+ F * G. [A proof of this simple result without the use of ch.f.'s would be tedious.]
1.* 6. If the sequence of ch.f. 's {In} converges unifonnly in a neighborhood of the origin, then {In} is equicontinuous, and there exists a subsequence that converges to a ch.f. [HINT: Use Ascoli-Arzela's theorem.]
1.5. Let F be a discrete d.f. with points of jump {a j, j > I} and sizes of jump {b j, j 2: I}. Consider the approximating s.d.f.' s F n with the same jumps v but restricted to j < n. Show that F n ---+ F.
1.4. Let F~, G'l be d.f.'s with ch.f.'s tn and gil' If tn -gil ---+ 0 a.e., then for each f E CK we have J f dFn J f dGn ---+ 0 (see Exercise 10 of Sec. 4.4). This does not imply the Levy distance (F n, Gn)! ---+ 0; find a counterexample. [HINT: Use Exercise 3 of Sec. 6.2 and proceed as in Theorem
1.3. Let F be a gIven absolutely contmuous dJ. and let FIl be a sequence of step functions with equally spaced steps that converge to F uniformly in 2/?!. Show that for the corresponding ch.f.' s we have\In: sup If(t) -fll(t)1 = 1.tEJ/?1
1.*2. Instead of using the Lemma in the second part of Theorem 6.3.2, prove that {L is a p.m. by integrating the inversion formula, as in Exercise 3 of Sec. 6.2. (IntegratIon IS a smoothmg operatIOn and a standard techmque m taming improper integrals: cf. the proof of the second part of Theorem
1.1. Prove the uniform convergence of f n in Theorem 6.3.1 by an inte gratIOn by parts of J eitx dF n (x).
1.1. dF(y)11 00 sin(x -y)t dtl < cll, dF(y)Y,,"X N t Ix-YI::::1/N Nix -)'1+ C2 r dF(y),}O
1.14. There is a deeper supplement to the inversion formula (4) or Exercise] 0 above, due to B Rosen TInder the condition J (1 + log Ixl)dF(x) < 00, -00 the improper Riemann integral in Exercise 1 D may be replaced by a Lebesgue integral. [HINT. It is a matter of proving the existence of the
1.13. The uniqueness theorem holds as well for signed measures [or fUDc tions of bounded variations]. Precisely, if each J-Li, i = 1, 2, is the difference of two finite measures such that then f.-Ll = f.-L2·
1.* 12. Prove Theorem 6.2.2 by the Stone-Weierstrass theorem. [HINT: Cf.Theorem 6.6.2 below, but beware of the differences Approximate uniformly gl and g2 in the proof of Theorem 4.4.3 by a periodic function with "arbitrarily large" period.]
1.11. Theorem 6.2.3 has an analogue in L2. If the ch.f. f of F belongs to L2, then F is absolutely continuous. [HINT: By Plancherel's theorem, there exists cp E L 2 such that r cp(u) du = ~ ['JO e-itX._ 1 f (t) dt.Jo v 2rr Jo -It Now use the inversion formula to show that 1(X F(x) -F(D) = ~ sp(u)
1.10. Prove the following form of the inversion formula (due to Gil Palaez):1 1 iT eitx f (-t) -e-itx f (t) -{F(x+) + F(x-)} = -+ lim . dt. 2 2 6,,0 0 2mt Ttoo[HINT: Use the method of proof of Theorem 6.2.1 rather than the result.]
1.*9. Give a trivial example where the right member of (4) cannot be replaced by the Lebesgue integral 11 00 e-itX1 _ e-itX2 - R . f(t)dt. 2n -00 If But it can always be replaced by the improper Riemann integral: lim 11--2 T-+x S e-itx - e-itx2 R -f (t) dt. it
1.8. Prove that for 0 < r < 2 we have where~r J-cosu f C(r) = I 1,+1 du J-oo u thus C (1) = lin. [HINT:r(r + 1) . rn --n--sm -2 'l: I -cosxt Ixlr = C(r) +1 dt.]OJ It I r
1.*7. If F is absolutely continuous, then limltl_oo f (t) = O. Hence, if the absolutely continuous part of F does not vanish, then liml~CXJ If (t) I < 1. If F is purely discontinuous, then limt_oo f (t) = 1. [The first assertion is the Riemann-Lebesgue lemma; prove It first when F has a denSIty
1.6. For each 11 > 0, we have 11 00(Sint)n+2 1 2 1 u --dt = ((In (t) dt du, n -00 too where ((Jl = ~ 1 [-1.1] and ((In = ((In -1* ((Jl for n > 2.
1.5. What is the special case of the inversion fonnula when f = I? Deduce also the following special cases, where a > 0:1 1 00 sin at sin t -2 dt = a /\ 1, n -x t 11 00 sin arCsin t)2 a2 -3 dt = a --for a < 2; 1 for a > 2. n -00 t 4
1.4. If f (t)/t ELI (-00,00), then for each a > 0 such that ±a are points of continuity of F, we have 11 00 sinat F(a) -F(-a) = ---f(t)dt.JT -00 t
1.*3. Prove that fOI each ex > o.I [F(x + u) -F(x -u)] du = -j --t-2 -e ltx f(t)dt. Jo JT -00 ra 1 rOO 1 cos at As a sort of reciprocal, we have l1 a l u 1 00 1 -cos ax -du f (t) dt = 2 dF(x). 2 0 -u -00 X
1.*2. Show that for each T > 0:1 1 00 (1 -cos Tx) cos tx 0 -2 dx = (T -I t I) v .JT -00 X Deduce from this that fOI each T > 0, the function of t given by is a ch.f. Next, show that as a particular case of Theorem 6.2.3, 1 -cos Tx J (1' Itl)eitJ: dt.2 -T Finally, derive the following particularly
1.1. Show that 100(SinX) 2 _ JT --dx--. o x 2
1.*18. Let the support of the p.m. f.L on gel be denoted by supp f.L. Prove that supp (IL * LJ) c10sme of sopp J.L + sopp v, supp EM! * M2 * ) closure of (supp f.i;l + supp f.i;2 + ... )where "+" denotes vector sum.
1.17. Let F be a symmetric d.f., with ch.f. f >° then,00,2 ~oc((JF(h) =J 00 h2~ x2 dF(x) = h .[ e-ht f (t) dt is a sort of average concentration function. Prove that if G is also a d f with ch.f. g > 0, then we have Vh > 0:1 -({JPG(h) < [1 -({JF(h)] + [1 -({JG(h)].
1.16. If° < h)" < 2n, then there is an absolute constant A such that A (' QF(h) < ).. io If (t)1 dt, where f is the ch.f. of F. [HINT: Use Exercise 2 of Sec. 6.2 below.]
1.*15. For a d.f. F and h > 0, define QF(h) = sup[F(x + h) -F(x-)];x QF is called the Levy concentration function of F. Prove that the sup above is attained, and if G is also a d.f., we have
1.14. Find an example of two r.v.'s X and Y with the same p.m. f.L that are not independent but such that X + Y has the p.m. f.L * f.L. [HINT: Take X = Y and use chJ.]
1.13. For any chJ. f we have for every t:R[l -f(t)] > ~R[l -f(2t)].
1.12. Let {Xj , 1 < j < n} be independent r.v.'s each having the dJ. .Find the chJ. of n 20: X]j 1 and show that the conesponding p.d. is 2-n/2P(n/2)-lx(n/2l-1e-x/2 in (0, (0).This is called in statistics the "X2 distribution with n degrees of freedom".
1.11. Let X have the normal distribution . Find the d.f., p.d., and ch.f.OfX2.
1.*10. Let {Xj , j > I} be a sequence of independent r.v.'s having the common exponential distribution with mean 1 lA, A > 0 For given x > 0 let v be the maximum of n such that Sn < x, where So = 0, Sn = 2:J=l Xj as usual. Prove that the r.V. v has the Poisson distribution with mean AX. See Sec.
1.9. Find the nth iterated convolution of an exponential distribution.
1.8. Show that the family of normal (Cauchy, Poisson) distributions is closed with respect to convolution in the sense that the convolution of any two in the family with arbitrary parameters is another in the farruly with some parameter(s ).

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