Probability Theory Independence Interchangeability Martingales 3rd Edition Yuan Shih Chow, Henry Teicher - Solutions

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(Klass) Let (X) be id with EX,-0 and E, Prix,b)
Hint: Let PX >>0 and ELX < for 1>> x>0. The latter implies, while the former entails Px Ce Pimaxxis Csap-C027-1->0,c>a
(Derman-Robbins) (X) tid. with E X, nonexistent does not preclude PIS
SX1 is a random walk with X, having a symmetric stable distribu- tion of characteristic exponenta, show that Se also has this distribution. Hence, if Is <
The limit distribution of Exercise 5 has density f(x)=(2x)-240 This is actually the stable density function corresponding to a 17-a
In coin tossing, prove that the probability that the mih return to equilibrium (equal number of heads and tails) occurs before w's tends to 2[1-0x usual, is the standard normal d.f.
Adf. F(with cf. is said to be in the domain of attraction (resp. domain of normal attraction) of a (stable) distribution G (with cf. o) if for suitable constants A., B. (resp. for some A, and B, B) ). Show that every stable beim distribution belongs to its own domain of (normal) attraction
Prove that all proper stable distributions are continuous and infinitely differentiable
Prove that if X has a stable distribution EX for Hint: IX-X, X. where
Let X,-,,.> 1, where (1) are bounded lid. rvs and Show that ever log logo) (log) exp (24) expl2in log n) and that 1,, X, 0 when L. Show that the mass of a proper stable dif. is confined to (0, x) (resp. (-x, 0)) if --1 (resp-1
Let (1) be independent, symmetric rvs and X-Xala where < < symmetric rvs with the same joint dfs as responding partial sums and 0 Then then Xare independent. X. 21). Let S. S. & be the coe- and
Let X.-M. that (1), 1,b>1, where [Y) are bounded Lid random variables. Prove provided . Compare with Exercise 5.2.8
If (X,) are independent rxs with variances and (EX), where h, to, then (1)
Show that Theorem 2 extends to the case where (X.21] are martingale differences with constant conditional variances
Let [X] be independent rvs with EX, SEX where () EX < holds. Compare (*) with (35) when r-2 EX!--and Prove that the classical SLLN
(i) rather, log log ny, gives necessary and sufficient conditions for (*) in terms of when X-Onlog log n) a..
Let (X1) be independent rxs with EX, EX-satisfying (28) (i) Show that if entlog n. 60, then (*) (1/m),
Define a sequence of rvs for which (1) but (X) diverges .. revealing limitations to the "Kronecker lemma approach."
Necessary and sufficient conditions for the strong law for independent (X) with cannot be framed solely in terms of variances Hit: EX-EX Consider < and P log log) 2 [1 PX-01 22
Let (X1) be independent vs with EX, EX and, Prove the result of Rakov that 1 How: Apply Corollary 2 to Y, X. 1 sjsn
Let the lid. rvs (Z, (01) and PZ, 21] be independent of Y, where Y is uniformly distributed on 11-1 Then X, y-Z, 21, are, interchangeable Es with (1)x,0 but EX,X, does not exist.
Apropos of Theorems 3 and 4, prove that (i) if a logn. PX-01-1 (a) No. 0PX --N-1) ifa,-log-finite and P, designates a Poisson r.v. with mean Pse Theorem 3.1.21
If two of the last three conditions of Exercise 4 obtain but the third is violated, construct a sequence X2 11 of interchangeable processes for which Se
If for each 1 (XI) are interchangeable rvs with E X.-Q. EX2-1 EX and EX EXXI. EX-then
Suppose that for each 12X-1.2) constitute interchangeable random variables with EX-EX-1, EX < and set m) ELX))[X] EX- when, is the -algebra relative to which (X21) are conditionally Lid. If , and S.-X. then No. t
If (X,, 21) are interchangeable rvs with EX, EX = 1, EIX,!" < , Cov(XX)-0-Cov(XX) then Corollary 94 holds.
Let (X) be independent rvs with E X, 0, EX2 I which obey the Lindeberg condition. If,-[k] greatest integer sjuk for.....k and S.- Xprove that (kn) (kn)") converges in distribution to the k-dimensional normal with mean vector rero and covariance matris (ou). where -mini Conclude that
prove that sisisk (-))) converges in distribution to the k-dimensional normal distribution with mean vector zero and covariance matrix (g). Hint Recall Exercise 8.5.6.
Let X. (XX) 2 1,be Lid random vectors with EXP Cov(XX) s-1.1 s/s
9. C.LT. for U-statistics: Let X be id and ox.) a real, symmetric function of its marguments. If E(XX) and E(X, U-Nowhere U E[E XX-] Him: A C.LT. applies to X)-8, then Xand Ex since the components of the sum are kid with mean 0 and variance of, while U-10 via EU-13-1)
Prove that the Lispounox condition of Corollary 9.1.1 is more stringent the larger the value of a 80) For what positive values of x, if any, does a CLT. hold for iid.symmetric random variables with Fix)-1-142x21, F(x)=10x1 (ii) Does a C.LT. hold for independent (N.) with P(X-1)-1.PIX, *}
Let (X) be independent with PIX.-0 (1-(-) > Again, Sn has a limiting normal distribution despite the Lindeberg condition being vitiated
5. Let (1) beid. rvs with finite variance (say I) and let be nonzero constants with of. Show that the weighted i.id...s 1.1) obey the central limit theceem, ie. (1), N, if,-) and EY-0
Failure of the Lindeberg condition does not preclude asymptotic normality. Let be Lid with E Y-QEY-1;let (2,1 be independent with PIZ,-)- 1/2 PIZ-0-1-(1) and (2) independent of (Y). Prove that if X.- +Z S-X, then SN, and the Lindeberg condition cannot hold. Explain why this does not contravene
Let the random vectors X, X. where X-(XX) a possibly fictitious random vector with d.f. F. If (20) are k-dimensional random vectors whose ith component is gxsisk 20 where . Isisk) are continuous functions on R, then Y,
Verify that if FF, the marginal dfs FF, Isjsk
Prove the k-dimensional analogue of the continuity theorem (Theorem 8.3.3)
Prove Theorem 8.5.1 and deduce the one-to-one correspondence between difs and cfs on R
Prove the Cramer-Levy theorem (Theorem 843) in R. Hint: Use the result for k-1. 8 Generalize Theorem 8.44 to the multinomial distribution
If X.-(XX 21, is a sequence of random vectors for which every linear combination XNwhere N is a fictitious normal r.x. with mean and variances, prove that the df of X, converges to the normal df. on R" with mean vector and covariance matris
Prove that X-(XX) has a multivariate normal distribution with mean vector (...) and covariance matrix () iffevery linear combination cxcx, has a normal distribution on R' with mean cy and variance c
real, and complex pp. Conversely, a continuous, positive definite function on (-x, x) with 0-1 is a cf. This is Bochner's theorem 4 If F, 1sisk, are dfs on R', show that for any a in [-1.1], F(xx)= [1+T-FA), Fix is a df with the given marginal dis
Show that a cf. is positive definite ic... - 20 for any
Prove that
If 1-) ) as 0 for some x in (0, 2), then PX-0") Hine: Integrate-cosxWF(x)sk over (0) or use Lemma 831
If cfsatisfy -- w +4. wherea, >1,2, then 1)+ with w., positive integers such that x,,+
Prove the Raikov Ottaviani theorem that the family of Poisson difsis factor closed Hot: It may be supposed, without loss of generality, that the support 3, to any factor F) (0.1.2. Rather than cfs utilize the corresponding probability generating functions (w) EwPX-1-1.2
There exist uncountably many absolutely continuous dfs with support (0,x) and identical moments,k1.2. H: For 0 < 0, and a nonnegative integer k, set (k+ya, br+ is in the well-known formula "Tic) fedy Convert this to a similar example with support (-xx)
15. If, in Exercise 14,1 and A-[0.1or [0.1.2...] or [nonnegative rationals). then the mapping of (G GIA) 11 onto (H: Hx)=fFix Geis one-to-one. Then is said to be identifiable Hint: :: G) analytic in 0 <
Let Hx) closed family Fix: G be a G-mixture (Exercise 823) of the additively (Fix: de AcR) (Exercise 8.4.5) Verify that the convolu- tion H, H, is a (G, G)-mixture of Hint: Utilize Exercise
Utilize Exercise 12 to prove the converse portion of Theorem 3.
Prove for any df. with cf. that for all real x and > 0 26 Hi: Apply the Levy inversion formula to FG. where G is the uniform dis- Iribution on [--k,*]
Utilize cfs to give an alternative proof of Corollary 8.1.10.
Prove that Fx+)+ Rx-)-1- where and hence that for F continuous
Prove that line(120 (F(x+)-PLx)] Hint: X. X are independent with common df. F. apply Exercise 8 at x = 0 to X, - X.
If is the cf of the df. F. prove that and if X is an integer valued rx, show that PEX- Give an analogous formula for S-X, where (X) are id integer-valued TV..
Find the cf., if PIX-a) trigonometric identity that PIX - Show by iteration of an elementary and utilize this to state a result on convergence in law of sums of independen
Prove that a real-valued cf. o satisfies gir+sk for > 0 and sufficiently small>0. & Verify that the cf. corresponding to a uniform distribution on (-3) i (sin ars,>0, and that lim ) exists but F. to adf.
If X and Y are independent rvs such that X + Y and X have identical dfs, then Y-0.ac. Hint: Employ the Lvy concentration function of Exercise 8.
(Chon-Robbins) Prove that if F is adf. with F(0) 0, then Gix) []: Fox+ ad it dF(x) x. More generally, G has a finite ith moment iff F has a finite
If, are densities, their convolution is defined by Verily that if F, is an absolutely continuous df. with density f./ 1,2, then F- F, F, is absolutely continuous with density f-
and suppose that G.- FFF.G. Prove that if G is continuous [4-(1)
Let F, be a discrete df with maximum jump g. 42
The Levy concentration function of a df. F is defined by Demonstrate that if F-F, F. Qa2) QQa) min and deduce that F, F, is continuous of F, or F, is continuous
X21) are id with a uniform distribution on [0, 1], that is, F(x)=x for 0ss1, show that nimas, X.-1) X-
The convolution of two dfs is absolutely continuous if at least one of the components is absolutely continuous. The converse is false (Exercise 8.4.6)
The convolution of two discrete distributions F.)-1,2 is discrete, and if the sup- port SIF) contains, points, 1.2. SIF, F) contains at most n, and at least, 1 points
prove that FF, F, despite the nonindependence of X and Y.
Let Q be the planar region bounded by the quadrilateral with vertices (0,0), (1.1) (0, (3, 1) and the triangle with vertices (3,0), (1, 0), (1. If X. Y are jointly un- formly distributed on Q, that is, P is Lebesgue measure on the Borel subsets of
Let A be a Borel subset of R where R (-)21, and for every & Alet Fix: ) be a df. on R such that Fix: A) is a Borel function on RIG is any f whose support CA, show that H(x) Fix: GA) is a df. on R. It is called a G-mixture of the family (FA) or simply a mixture. Convolution is the special case Fix:
Verify that the class of all dfs on R is a complete metric space under the Levy distance
Give an example showing that Theorem 3 s invalid without the nondegeneracy hypothesis
(Kemperman) For any countable partition of (-, ) into Borel sets B 1, 2 define the non-negative, symmetric function by MXX 22 and let (X, X., 21) be iid. random variables with PX8)->0/21, where ,1.Seta, ciju,with > 1.p>1. Prove that if ka->12ks-p and a[1+(klip]>1+pf, then he
Show that if E (XX)-0, for 2sisk EAXXXX) X. X) + (XX)+... XX
Let S and S/21 be as in Exercise 1, where [X, X., 21) are id, and let [21] be constants such that 0h. f(x0 and PLX1>) a
prove that (i) and () if (X. X. 21) are lid with EX-0, then 3,4) is a completely degener ate U-statistic for ki
If (41) is a sequence of events and Act. prove the following version of the Borel-Cantelli theorem { 8-1) is an 2, martingale for some pin [1.2] and 0 then lim, ac. on the set 4 where & E(X) K>0 and apply Exercise 1 to -Xh 9 Let Y be an , tx, and [
IS, 21] is a martingale satisfying ()EX 0 on [71]. is a martingale or positive submartingale, then for any stopping time T. E|Slim... ES.. In particular, if (S1) is bounded. E|S,E sup, then
Prove that if the r.v.'s (21) of Corollary 5 are conditionally id, given then the coordinate r.v.'s (1) are id
Prove that if X) are, interchangeable rvs then
If (X1) are interchangeable rvs and (Y21] is defined by X,- Y+EX,, then (21) constitute uncorrelated interchangeable rvs
If (X21) are, interchangeable rvs with p E(X,-EX,XX, - EX, then EX,X, 0 iff EX, - 0 and p=0.
For any 22, find interchangeable rvs X.. X) which cannot be embedded in a collection of a + 1 interchangeable rvs IX. X. X). Hint: Recall Exercise 6.3.10
If a r.v. X is independent of the lid. rvs (X1), show that Y, X+X. 1, are interchangeable r.vs. Express that joint df. of (YY) in terms of the dfs of X and X
X (Xis a random vector whose df. F is absolutely continuous relative to Lebesgue measure with demity/find the conditional density of (X.....X) given Xx. where 1sj cP{Y>c) for any c>0. Ye L. and sub-e-algebra 9.
Fix) lim Fix) 1, 2, prove that the df. F, is a mixture of the family Fix, 1x) in the sense of Exercise 8.2.3.
If in Example
If the events (B) of (... P) constitute a finite or e-partition of Qin and have positive probabilities, prove Bayes' theorem, ie, for any event A PI48P8)
Let 9 be a semi-algebra of events and X, Yr.v.s with |E X
Let 14, 21 be o-partition of in Verify that 1) and EXISx.then X JP PEA where the parenthetical quantity is an arbitrary real number if P:4, -0 Show that Corollary 5 remains true if EIX] < and C.
If X is a rx with EXIs and a algebra of events with eX) and indepen dent classes, then EX EX. ac. In particular, if (N) are independent rvs with EX21. then EX, XX) EIX, XX EX ac
Set functions (21) on (F.) are uniformly absolutely continuous relative to if for all >04)
If (F) is a finite measure space and I is a continuous linear functional on >1, there exists ge. where (1/p)+(1/4)-1, such that (f)- Sadu for all fe, This is known as the (Riesz) representation theorem Hint: Let be finite. Since 1, F.) for all Ae, a set function on Fis defined by v[4] () It is
Let V denote a linear functional on, ie, range V-(-x, x) and Viaf by) a()+ big) for allf. ge2, It is continuous whenever 1-1, 0, and V is hounded if is C,1, for all fe where C, is a finite constant. Prove that a continuous linear functional is bounded Hint: Otherwise, there existe,with 1.1, o(1)
If v is a signed measure on a measurable space (F) with total variation (see Exercise 6) and X is integrable relative to t, one may define j X dv-Xd [x de Prove that if is finite, (4) sup{lf, X dv X is measurable and IXIS
A completely additive set function von a measurable space (2) which assumes at most one of the values + and - and satisfies v!)-0 is sometimes called a signed measure. If v is a signed measure on a measurable space (F) and "[4] = sup(B): 48e, then y" is a measure satisfying v(4) 2 (A). AF Likewise,
If Fix: a)-(-a)(xasxs the corresponding measure Fis absolutely continuous relative to Lebesgue measure with Radon Nikodym derivative fo) (hay and F is called the miform distribution on[a b] When a 0-1,FI-1 coincides with Lebesgue measure on [0, 1] and a r.v. with d.f. Fis said to be uniformly
If (F.) is a measure space, i=1,2, then p, is absolutely continuous relative to
Two measures are called equivalent, denoted v. if each is absolutely con- tinuous with respect to the other. Verify that this is an equivalence relation. If (a) is a probability space, X, is a nonnegative, random variable and is the indefinite integral of X, 1-1.2, then, if [X-01A[X-03-0, the two
If and are measures such that is both p-continuous and p-singular, then

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Probability Theory Independence Interchangeability Martingales 3rd Edition Yuan Shih Chow, Henry Teicher - Solutions

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