tiết 21 bài 9 nhật bản tự nhiên dân cư và tình hình phát triển kinh tế
A Course in Mathematical Statistics
... mathematical statistics 197 3 Includes index ISBN 0-12- 599 315-3 Mathematical statistics I Roussas, George G First course in mathematical statistics II Title QA276.R687 199 7 96 - 4211 5 5 19. 5—dc20 CIP Printed ... Theorem 187 Exercises 194 Laws of Large Numbers 196 Exercises 198 ix x Contents 8.5 8.6* Chapter Transformations of Random Variables and Random Vectors 212 9. 1 9. 2 9. 3 9. 4 Chapter 10 10.2 Order ... statistics II Title QA276.R687 199 7 96 - 4211 5 5 19. 5—dc20 CIP Printed in the United States of America 96 97 98 99 00 EB Contents To my wife and sons v vi Contents This Page Intentionally Left Blank Contents...
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independent and stationary sequences of random variables
... of limiting tails 190 Formulation On the condition (10 9) 190 192 CONTENTS Derivation of the fundamental integral 192 Application of the method of steepest descents 194 Completion of the ... the proof of Theorem 10 1 197 Chapter 11 Narrow zones of normal attraction 198 Classification of narrow zones by the function h 198 Statement of the theorems 199 On the conditions imposed ... Chapter Refinements of the limit theorems for normal convergence 94 Introduction 94 Some auxiliary theorems 94 The deviation R„ (x) 97 Necessary and sufficient conditions 104 The maximum deviation...
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Báo cáo toán học: " A note on the almost sure limit theorem for self-normalized partial sums of random variables in the domain of attraction of the normal law" pptx
... theorem Mathematische Nachrichten 137, 2 49 256 ( 198 8) [4] Lacey, MT, Philipp, W: A note on the almost sure central limit theorem Statist Probab Lett 9, 201–205 ( 199 0) [5] Ibragimov, IA, Lifshits, M: ... 40, 343–351 ( 199 8) 11 [6] Miao, Y: Central limit theorem and almost sure central limit theorem for the product of some partial sums Proc Indian Acad Sci C Math Sci 118(2), 2 89 294 (2008) [7] ... ) = a.s., ( 19) (20) n ¯2 dk I(Vk > (1 + ε)kl(ηk )) = a.s., k=1 (21) lim n→∞ Dn n ¯2 dk I(Vk < (1 − ε)kl(ηk )) = a.s (22) k=1 by the arbitrariness of ε > Firstly, we prove ( 19) Let < β < 1/2...
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Báo cáo hóa học: "Research Article Recurring Mean Inequality of Random Variables" doc
... pp 293 –310, 199 5 Y L Tong, “Some recent developments on majorization inequalities in probability and statistics,” Linear Algebra and Its Applications, vol 199 , supplement 1, pp 69 90 , 199 4 Y ... no 2, pp 330–340, 198 8 M Shaked and J G Shanthikumar, Stochastic Orders and Their Applications, Probability and Mathematical Statistics, Academic Press, Boston, Mass, USA, 199 4 M Shaked, J G Shanthikumar, ... classical mathematical inequalities,” Journal of Inequalities and Applications, vol 1, no 1, pp 85 98 , 199 7 M Wang, “The mean inequality of random variables,” Mathematical Inequalities & Applications,...
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Independent And Stationary Sequences Of Random Variables ppt
... m(x+2„) Setting 21= =A,,=0, there exists A=A(n) such that m(x+A) = nm(x) If p/q (2.2.8) (2.2 9) is any positive rational in its lowest terms, define (p/q) = (p) - A (q) then (2.2 9) implies that ... c (k) such that F (x) - G (x) < k + cl(k) E (1 2) T whenever Tl >1 c (k) For proof, see [ 19] (page 214 ) Theorem 5.4 Let T, b, e be constants, F and G functions of bounded variation, f and g ... the property that u(t)=11-it for jtj> T Then by virtue of (1 9) , 32 PROBABILITY DISTRIBUTIONS ON THE REAL LINE Ilh(1-k)ll < Ilf -91 1 Ilull Chap Ill-kll {Ilfll+Ilgll} Ilull { < 4(Var F+Var +...
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Independent And Stationary Sequences Of Random Variables - Chapter 1 pptx
... 6-algebra 9JI with respect to which each of the variables X (t) is measurable This is the a-algebra generated by the events of the form DISTRIBUTIONS AND DISTRIBUTION FUNCTIONS 19 {(X (t1), ... c (k) such that F (x) - G (x) < k + cl(k) E (1 2) T whenever Tl >1 c (k) For proof, see [ 19] (page 214 ) Theorem 5.4 Let T, b, e be constants, F and G functions of bounded variation, f and g ... the property that u(t)=11-it for jtj> T Then by virtue of (1 9) , 32 PROBABILITY DISTRIBUTIONS ON THE REAL LINE Ilh(1-k)ll < Ilf -91 1 Ilull Chap Ill-kll {Ilfll+Ilgll} Ilull { < 4(Var F+Var +...
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Independent And Stationary Sequences Of Random Variables - Chapter 2 ppt
... m(x+2„) Setting 21= =A,,=0, there exists A=A(n) such that m(x+A) = nm(x) If p/q (2.2.8) (2.2 9) is any positive rational in its lowest terms, define (p/q) = (p) - A (q) then (2.2 9) implies that ... Setting t=Bkn ;l Bn3 in (2.2.20) and using (2.2 19) , we obtain the impossible equation e - `k =1 Hence (Bn/Bkn) is bounded, and then (2 2. 19) and (2 2.20) yield exp(-c ItI")=exp (_ctl()ak)(1+o(1)), ... Hence am xl(~, s) W asm _ S=1 , -r 00 exp {limn+- tm - r"exp irirK(a) f3} Tr o (2 9) Setting r = m/a in (2.3 .9) and comparing with (2.3 6), we have m { S - x (~ S -")} exp {-limn+ 2ir7rK(a)f3}...
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Independent And Stationary Sequences Of Random Variables - Chapter 4 ppt
... co Hl (x) dx < oo (4.4 .9) K J 00 Go Therefore for all j > the function hi h2 'f is absolutely integrable, and rn(x) = 1In(x) -9( x)= (m) a j b m-j j - ma m l/2logm °° m f 21 - oo BN B x h (t pmj(x) ... ItI > Tn3 f„(t)I dt + I9(t)I dt ItI > Tn3 (4 6) As we have seen, the last two integrals are o(n -2) By Theorem 2.1 (2), 00 Tn3 J (ItI + ItI 6)e - *t2 dt = o(n - 2) If~(t) -9( t)I dt \ o(n z) Tn ... zero, and it is therefore more natural to use the L metric 4 LIMIT THEOREMS IN THE L, METRIC 1 29 Go HPn - g I I l= _ I Pn (x) - g (x) I dx , or more generally the LP metric 11P - gMI P ={ ~...
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Independent And Stationary Sequences Of Random Variables - Chapter 5 pot
... +E2) = -2 = O(8 +E 2)+o(n ) , where - T f- T I fn(t) gn(t) dt , t T E _ d fn(t) - 9n(t) ~- T d t dt, and a3(it)3 9n(t) _ ~ e"xdGn(x)=e-2t2 + e -2t2 6Uni Arguing as in the derivation of (3 3.4), ... f(t) gn(t) t2 dt+2 Jn (t) gn (t) t dt = E +E , say ESTIMATES OF 11F„-`ell, 1 49 The estimate EI -1 ) = o(n (5 3 .9) is proved in just the same way as (5.3 8) To estimate e2 we split the integral ... remains only to prove (5 3) From Theorem we have, for I tI < Tn = 015P3, the inequalities Ifn(t)-e 2 ,21 < 6JtI3p3n- e- r2 (5 4) ESTIMATES OF 11F.-OII, 147 and Ifn(t) - (e Z`2)~I < t P3n - ( 6t4+2t+...
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Independent And Stationary Sequences Of Random Variables - Chapter 6 pptx
... Theorem 6.1 If x>,0 and x = o (n ) as n + oo, then P(Z„>x) - exp x3 - 7x n2 \n ) 1-G(x) 1+O C (6.1 .9) x+l~ n2 and P (Zn < - x) G(-x) x3 = ex p I- (6.1 10) - x n2 ~ ( n2 n2 (1+0(,X,+,)) Here (z) is ... restrict it to the narrower interval [0, n"], where a < Then it is unnecessary to include in (6.1 9) the whole power series ,Z (z)=Ao+Alz+ , (6.1 11) since the truncated form PI (z) _ A o + A ... describe as having a "collective" character In the range x = o (n+) the asymptotic expressions (6 9) and (6.1 10) are less valuable, since they are not collective They only have a computational...
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Independent And Stationary Sequences Of Random Variables - Chapter 7 potx
... < e l we have (zo)(its' _ i nK" (z o ) t2 + nO (t3) (7.2 .21) Consider t in the range n - ' (log n)2 < I t( (7 22) 81 Because of (7.2 .21) we have in this range, 00 Re n ~ K(')(zo)(itY < - 4nK"(z ... exp{-2nr +nT3 A('r)}(1+0(x/n )) (7.4.8) Further, according to (7 19) , M (z o )" exp (- z o a rn) = exp { - Zn r + ni3 )? (i)} (7.4 .9) Because of (7 4.5) and the continuity of M we have IM(zo +it)I ... INEQUALITY 69 Hence IM (zo +lt)I' Iexp(-z6in)I ldzI = SEi SItISrz/h =BIM(zo)I n exp( - zorin)(1 - il(E1)) n (7 11) Since x>1, x/n>(1-rl(E1))' for sufficiently large n, and this, together with (7.4 .9) ...
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Independent And Stationary Sequences Of Random Variables - Chapter 8 pps
... e-h(''-z)dn e -hsdV(~) (( 8) becomes V (h-Z) _ 00 00 e-h',d J- cc - cc V(~_z)dWn(z)= e -h ''dWn+1(n) (8 9) whence the induction succeeds, and (8 2.7) is proved From (8 2.7) and (8.2 6) we have Fn (x) ... the saddle point equation K' (h) - 6r = , (8 5) where i=x/n =o(1) According to (7.2.18) and (7 19) there holds, for sufficiently large n, the equation K (h) - hK' (h) = K (h) - aT = - lz + i (z) ... 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) + Q, ,(y) = Bn - , Qn (y) , (8.3 10) so that ~co 00 exp(-hdn-Iy)dFP(y)=(2n)- exp(-hin#y-iy...
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Independent And Stationary Sequences Of Random Variables - Chapter 9 ppsx
... (9 17) (nal < r and (9 18) show that (9 5.6) is equal to < m) Thus, in f tl < n-°", we have cl nKso + (t) = Bn I n - r"`' (9 9) + Bn -1 = r=so+3 = Bn L, n-r/(so+3) =B, (9 10) r=so+3 using (9 ... - RP (t) , ( 9. 3 .9) where RP (t)1 < sup 14(P) (t) I (9. 3 10) Itk< T Further, 0'(0) =0, 0" (0) _ -1, so that, for suitable so and ltl < so , (9 9) implies that 10(t)1 < 1- 4t2 (9 3.11) If we ... in (9. 3 .9) we need bounds for (q) for large q Now 00 I~,c4)(t)) < J (9 4.1) Ixlg g(x)dx - 00 If k = (1 +2a)/4a , (9 4.2) then (9. 3 2) implies that 00 J - 00 exp (A 1x1 11k )g(x)dx < oo (9 4.3)...
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Independent And Stationary Sequences Of Random Variables - Chapter 10 potx
... Cramer series „ On the condition (10 9) We first show that (10.1 9) follows from (10 1.12) ; the proof follows that of Theorem 9. 2.2 almost verbatim If (10.1 9) does not hold, there exists a sequence ... Write ( 10.3 6) (10.3 7) (10.3.8) ( 10.3 9) 194 CRAMER'S SYSTEM OF LIMITING TAILS Km (t) _ E r=3 Chap 10 Or t ; r then from (10 3 .9) we have (cf (9 5.2)) pn (x) n-u n2 _ - 7r exp(-4nt +nKm ... replacing (10 9) by the weaker condition 10 DERIVATION OF THE FUNDAMENTAL INTEGRAL E {exp (A IX;I4a/(2a+ 1) ) } 93 (10.3 1)
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Independent And Stationary Sequences Of Random Variables - Chapter 11 doc
... g (11 9) and + to g n ~ 11 INVESTIGATION OF THE FUNDAMENTAL INTEGRAL 203 „ Investigation of the fundamental integral In the notation of Chapter 9, we have the equation (cf (9. 3.7), (9 4.5)) ... XP (n) * - co Thus the sum of (11 9. 9) over < r < m is Bexp(-4n 1-2 "`)= B exp{-c exp (2XP (n))} (11 9. 10) 11 10 COMPLETION OF THE PROOF OF THEOREM 11 21 A similar analysis can be made of ... entire function exp (nK (t)) we set ° = 0 .99 ° (11 17.8) and take jtj
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Independent And Stationary Sequences Of Random Variables - Chapter 12 ppsx
... h so that (12 .9 1) x=mn2 , where < x< n"/P7 (n) , P7 (n) % P2 (n) 20 • (12 .9 2) 12 FURTHER TRANSFORMATIONS From (12 17) this implies that h = n-Zx+9Csn-1 p2 (n)-s-3 , 2 39 ( 12 .9. 3) which in ... Moreover, we can prove as in (12 8) and (12 9) that EIXX -mI < C ( 12 .9 7) (i
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Independent And Stationary Sequences Of Random Variables - Chapter 13 pdf
... logR= Fn hp yp-+ OC7n -9= LKI(h)+6C,n -9, P~ p=2 K hp-1 = E yp (P-1) p=2 + OC7 n-8 = mIKI (h)+OC7n-8 , (13 5) (13 6) 252 CRAMER'S SYSTEM OF LIMITING TAILS K 62 hp p (P-2)1 + 9C, n -2 = p=2 -' _ ... particular Y2 = 1+n Thus m[K] = dh -2o L[K] (h) + 9C, n - (13.3 8) Following „ 8.3, we take x = FRO, where < x < n"/P (n) (13.3 9) (cf (12 .9. 2), the notation of Chapter 12 being retained) From ... 3 .9) and (12 10.5), 1- Fn (x) = { 1- (x) } exp x~ ~= n2 K~ x n2 x x (1+Bn-2)(1+Bh)(1+Bx-1) (13 4.4) But x > n2 and h = Bn- x = Bn"- 2/ p (n), so that for the values of x described in (13 3 .9) ,...
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Independent And Stationary Sequences Of Random Variables - Chapter 14 docx
... use (14 2.14), (14 2.15) and (14 2.18) to rewrite (14 9) in the form P(Sn > y) = nP(HI) P(X1 Y) (1 +o(1)) _ (HI ) = nP(X >y)(1 +0(1)) (14.2. 19) Thus, for y >, y,,, P(S,, > y) = A,,ny - ©(1 +o(1)) ... view of the discussion, > y/n} P(Sn > y) = nP(H1 ) P(Sn > yI H I ) +Bqn ny - a (14 2.8) (14 2 .9) We now investigate the expression P(Sn>YIH1), (14 2.10) 14 PROBABILITY OF VERY LARGE DEVIATIONS ... Consider the characteristic function 00 (14.3.4) 0(t) _ e`txg(x)dx , f- RADIAL EXTENSIONS 14 2 59 which by (14 5) and (14 6) is differentiable for all t at least (a-1) times We now introduce the...
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Independent And Stationary Sequences Of Random Variables - Chapter 15 pptx
... 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 any other member Then the number ... have k EEk a -EEka k > 21, which gives a contradiction Thus Lemma 15 2.1 shows that there can be at most (,In) sums of the form y- E k ak lying in any interval (x, x +21) Lemma 15 2.3 Under ... 15.3 shows that sup ID 1(x) (x) I - Fl x n) k (k = ID,- F1 I < pk(1 _ p )n-k - (~)k e - np
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Independent And Stationary Sequences Of Random Variables - Chapter 16 ppsx
... P(PA)=P(A), A E9J2cc (3) Up to events of probability zero, Tt ( Tt U Ak = U Tt (Ak) , = n k Tt (Ak) , (n k Ak Ak E9X , STATIONARY PROCESSES V(q) = V(A) 287 A c- 9W , , Tt(T`A) = T - `(TtA)=A AE9J * ... role will be played by the a-algebras 9Nt 00 (Itl < oo) , =nmt t 9JJ1_ cc clearly, for s>0, Every stationary process defines a family of mappings Tt of9N 0,, into itself given by the following ... Xrs)EA} = {(Xt1+t, , Xrs +t)EA} The set 91 of events of the form (16 2.1) is dense in 9J1 cc in the metric (16.2.2) We can therefore extend Tt from 91 to T1 as its unique continuous ex., tension...
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