3 3 sum and linear composites of random variables

... Expansion of R as a Taylor series 236 Further transformations 238 10 Completion of the proof of sufficiency 240 11 Proof of the necessity 241 12 Completion of the proof 2 43 Chapter 13 Monomial ... processes I 31 5 31 5 Statement of the problem The variance of X l + + X„ 32 1 The variance of the integral $ X(t)dt 33 0 The central limit theorem for strongly mixing sequences 33 3 Sufficient ... 34 0 CONTENTS The central limit theorem for functionals of mixing sequences The central limit theorem in continuous time 11 35 2 36 2 Chapter 19 Examples and addenda 36 5 36 5 36 9 37 0 37 4 38 4...

Ngày tải lên: 08/04/2014, 12:28

... § A refinement of the local limit theorems for the case of normal convergence In this section we assume that the common distribution of the random variables X; has zero mean and finite variance ... 13 As in § 4, it is proved that Il = O(n - 2b), and as in § 3, that 12 = O ( e -nc) for some c > Finally 13 j - 2a) eon l/z Assuming that the X3 have finite moments of order k >, 3, ... Now let X , X2 , be a sequence of independent random variables with distribution F, and denote as before by Fn (x) the distribution function of the normalised sum Z,, = (X + X2 + + Xn - An)/...

Ngày tải lên: 02/07/2014, 20:20

... moment, and write E(XX)=0, E(Xj2)=62, E(X ,3) =a3, EIX;I3=J 33 As before, the distribution function and characteristic function of X;, and the distribution function and characteristic function of the ... ), write i = 27r/h and define S (x) and d,,(t) as in § 3. 3 Theorem 5 .3 If X1 , X2 , are independent random variables with the same distribution F belonging to L h and having finite third ... appears in place of Lam Theorems and 3 are analogues of Theorems 3 and 3 Theorem If the XX have finite third moment, then for all p > 1, (5 1) Pp(Fn, (P) ,IIFn-0IIP < c(P-1)IPCi/PP3n-1 , where...

Ngày tải lên: 02/07/2014, 20:20

... power series constructed by means of the cumulants of the XX , and converging in a neighbourhood of z = 0, which conversely determines the distribution of X X , and x e -2` Z dt G (x) = (27r) ... order (s + 3) , and Z„ and Z ;, are the corresponding normalised sums, then for Ix I < n", n ( PZn x ) ( ( 6.1 14 ) as n + oo Thus the asymptotic behaviour of the tails of the distribution ... possible forms of the limiting tails is closely analogous to the classical problem of characterising the possible limit laws for centralised and normalised sums of independent variables, i e...

Ngày tải lên: 02/07/2014, 20:20

... PROBABILITY DENSITIES 1 63 where 2=6 , 73 = P3, Y4=°4 -35 4 , Y5 = °5 - 1O 35 , etc are the cumulants of Xi and ° j are the moments of Xj Turning now to (7.2.6), we assume that x = o (n4), so ... coefficients of this series determine the first (k +3) moments of Xj (assuming that EXj =O and that 62 = VXj is known) In fact if these coefficients are known, we have the first (k + 3) terms of the ... THEOREM FOR LATTICE VARIABLES 67 Substituting into (7 .3 5) and noting that (1+Br)(1+Bn -1 log n)= 1+O(x/n1'), (7 .3. 5) becomes (27r) -1 exp { in r2 +nit + ni3A(z) } (1 + O (x/n*)) ( 7 .3 7) We now remark...

Ngày tải lên: 02/07/2014, 20:20

... the distribution of the normalised sum (X1 + + X©-mn)/Qn- , (8 .3 8) PROOF OF THE THEOREM 175 so that we can use the theorems of „ 3. 5 From (8 .3 3), a + O (h) , = (8 .3 9) and Fn (y) (y) + ... „ The introduction of auxiliary random variables Since E (exp a JXX j) < oo , we may write, for Jhj

Ngày tải lên: 02/07/2014, 20:20

... 2 < a i (s + 1)/(s + 3) , we consider the moments 3, °4, °s+ and the 'cumulants K3 = 11 3, K4=°4 -3, Ks =°s - lO 3, KS+ • If on the other •, hand a=Z(s+1 )/(s +3 ) we consider ... Z, /3 < and (9 2.5) contradicts (9.2.4) The case of (9.2 3) is treated similarly Theorem 2 For random variables of class (d) the condition (9.2 1) is necessary in order that [0, n" p (n)] and ... -2(so +3) _ (so +3) ! Hso +3( x)e = ao~so +3 e - ' X n(x/n )so +3 (1+0(1)) (S O + 3) ! (9.6.11) COMPLETION OF THE PROOF as n-+ cc, where a§ 189 is a positive constant Thus (9 5. 23) becomes Pn(x)...

Ngày tải lên: 02/07/2014, 20:20

... large From (10 .3 6), m tr nK(t) _ -2nt2 +n I Wr - + B exp(-E1 n2 a) , r! r =3 and Re (nK (t)) < for I ti < n - " Write ( 10 .3 6) (10 .3 7) (10 .3. 8) ( 10 .3 9) 194 CRAMER'S SYSTEM OF LIMITING TAILS ... (A IX;I4a/(2a+ 1) ) } 93 (10 .3 1)

Ngày tải lên: 02/07/2014, 20:20

... 11 2) and (11 3) , H(§) (x) =Bxr - e - xI ,r°- Zr = r Bx1 e (x1 n°- )r ( 11 13. 6) 11 14 COMPLETION OF THE PROOF OF THEOREM 11 2 217 But Jx l n'`-ZI < (11 13. 7) 1/P(n) so that (11 13. 6) is ... (11 10.1) (cf „ 10.6) The sum of the terms in (11 9.11) with < r

Ngày tải lên: 02/07/2014, 20:20

... methods of Chapter „ An upper bound for the probability of a large deviation We now proceed to the proof of the sufficiency part of Theorem 12 2, assuming (12 3) and ~f 3= `r4= where or = O s +3= 0, ... 12 .3 3) and define el (~) = el , = , (ICI,< n 2a /P3 (n)) ( kkI > n 2, /P3 (n)) (12 .3. 4) Let X1 be independent random variables with probability density d17(x) = R -l e (hx)p(x) , dx (12 .3. 5) ... cumulants of Xi Under the conditions (12 1) and (12 .3 1) we have 72 = y =O 1+n -20 , (j =3, 4, , s +3) , (12.8. 13) where s is the greatest integer with S+ l z < s +3 a Thus s+4 2-a s +3 ' (12 14) and...

Ngày tải lên: 02/07/2014, 20:20

... normalised sum ( 13 2.2) Zn = Sn/6n- , and a modified random variable Z n = Zn + Yn /6n ( 13 3) , where Yn is a random variable, independent of the Xj , and having a normal distribution with mean and ... ( 13 4.2) Since r=n -2 x, the substitution of ( 13 4.2) into ( 13 4.1) gives x3 1-Fn (x) _ (2~) -2 x -1 e -2 x exp ~,[2KI x x n2 n2 x (1+Bn -2)(1+Bh)+6p n3 , ( 13. 4 .3) or, because of ( 13 3.9) and ... ( 13. 3 12) From ( 13 4), h>n- -ZE If z' = z - 9C, n - , we easily find (cf (8 .3 6)), LK](h)-h d L K1 (h) = -'-z'2+z '3~ [2K](z')+Bn -3 , ( 13 13) where )± K3 is a truncated Cramer series By virtue of...

Ngày tải lên: 02/07/2014, 20:20

... most likely to be because exactly one of the summands is very large ; the probability of S§ being large as a result of an accumulation of moderately large summands is comparatively small Since the ... are constants, with A a > 0, and s>0 The class of such probability densities we call (A) Such variables have only a finite number of moments, and the role of the linear functionals aj , b i is ... ek k=r +0(~ -K-1 ) (14 .3 7) l+ The question of the differentiable of the radial extensions of (14 5) therefore reduces to that of the radial extensions of 00 e it4 ~k (14 .3 8) -00 ( +Ok 60...

Ngày tải lên: 02/07/2014, 20:20

... shall assume that the distribution function F(x) of the variables X3 is continuous and strictly increasing As in ± 2, X1 = F -1 (~r), where (~;) is a sequence of independent random variables, ... Is'- (15.2.1) The proof of this theorem requires a number of auxiliary results Lemma 15.2.1 Let 21 be a set of n elements, and K a class of subsets of 21 that no member of K is contained in ... respectively over Ixk1 < -1 and lXkl,> n' Then EIP,,(k)-17(k)I =E'IPn (k)-17(k)I+E"IPn (k)-17(k)I and we examine separately the two sums E' and E" (I) E" From (15 3. 6) and (15 3. 7) and Chebyshev's inequality,...

Ngày tải lên: 02/07/2014, 20:20

... STRUCTURE OF L,p AND LINEAR TRANSFORMATIONS 297 Theorem 16 6.1 The space L,,,, consists exactly of the random variables of the form (16.6.1) In fact, we can say more than this Because of (16 ... description of the process Z, and to this end we construct stochastic integrals of which (16 5.1) is a particular case For a detailed account of such integrals see Chapter IX of [31 ] A random process ... values of t are considered, a semigroup of isometric operators) In the discrete time case, this is the cyclic group of powers of the restriction U of Ti In fact, because of (16 2 .3) and (16.2...

Ngày tải lên: 02/07/2014, 20:20

... Y is the sum of the spectral function Fy( N) (.?) of Y (N) and Fz( N) (A) of Z(N) The process Y1N' is linearly regular and Fy(N) (A) is absolutely continuous Thus the total variation of the singular ... oo, where ~ N, 17N are random variables of...

Ngày tải lên: 02/07/2014, 20:20

... 18 STATEMENT OF THE PROBLEM 31 7 will ensure that the ~ j are almost independent, and the study of E ~ j may be related to the well understood case of sums of independent random variables If, ... sine (2TA) dF(A) f -00 Ae The proof of (18 .3 3) is just the same as that of (18.2 3) Theorem 18 .3 If lim R (t) = 0, t- ao then either lim V {S(T)} = T- o0 (Do 33 2 CENTRAL LIMIT THEOREM FOR STATIONARY ... h(2r-1)< (1+E3)s < c4(1+E3)1-1 If E and e are chosen so small that (1+E3)(2+E)2-1-+b < p < , we obtain Yr < c5/(1 - P) = c3'' Thus, for this choice of e, 18 SUFFICIENT CONDITIONS 34 3 a2r < (...

Ngày tải lên: 02/07/2014, 20:20

... The distribution of values of sums of the form Ef (2k x) Let f (t) be a periodic function of the real argument t, with period 1, and consider the distribution of the values of the sum n Sn (t) _ ... A of the Ualgebra Rx of subsets of 3r We choose an initial distribution it (A), a probability measure on (X, Rx) Using the Kolmogorov extension theorem, we can find a sequence of random variables ... (A) = (log 2)-1 dt fA lt (19.4 3) It is evident that the a„ (t) are random variables, but the proof of the properties of the sequence (a„) is quite complicated, and we need some simple facts about...

Ngày tải lên: 02/07/2014, 20:20