Các định lý giới hạn cho martingale

I HC QUC GIA H NI TRNG I HC KHOA HC T NHIấN T CễNG SN CC NH Lí GII HN CHO MARTINGALE LUN N TIN S TON HC H Ni - 2014 I HC QUC GIA H NI TRNG I HC KHOA HC T NHIấN T CễNG SN CC NH Lí GII HN CHO MARTINGALE Chuyờn ngnh: Lý thuyt xỏc sut v thng kờ toỏn hc Mó s: 62460106 LUN N TIN S TON HC NGI HNG DN KHOA HC GS.TSKH NG HNG THNG H Ni - 2014 LI CAM OAN Tụi xin cam oan õy l cụng trỡnh nghiờn cu ca riờng tụi Cỏc kt qu nờu lun ỏn l trung thc v cha tng c cụng b bt kỡ cụng trỡnh no khỏc Tỏc gi T Cụng Sn LI CM N Tỏc gi xin by t lũng bit n sõu sc nht ti Thy, GS TSKH ng Hựng Thng vỡ s nh hng v s gi m ca Thy nghiờn cu, s nghiờm khc ca Thy hc v s bao dung ca Thy cuc sng dnh cho tỏc gi Tỏc gi xin gi li cm n ti Khoa Toỏn - C - Tin hc, Phũng Sau i hc, Trng i hc Khoa hc T nhiờn - i hc Quc Gia H Ni, ni tỏc gi ó hc v nghiờn cu Tỏc gi xin gi li cm n ti cỏc Thy, Cụ B mụn Xỏc sut v thng kờ, Khoa Toỏn - C - Tin hc, Trng i hc Khoa hc T nhiờn, ni tỏc gi ang cụng tỏc v ging dy, ó giỳp tỏc gi rt nhiu quỏ trỡnh hc hon thnh lun ỏn Trong quỏ trỡnh hc v hon thnh lun ỏn, tỏc gi vụ cựng bit n nhn c s quan tõm giỳp v gúp ý ca GS.TSKH Nguyn Duy Tin, GS.TS Nguyn Hu D, GS.TS Nguyn Vn Hu, GS.TS Nguyn Vn Qung, PGS.TS Phan Vit Th, PGS.TS Trn Hựng Thao, PGS.TS H ng Phỳc, TS Trn Mnh Cng, TS Nguyn Thnh, TS Lờ Vn Dng, TS Lờ Vn Thnh, TS Nguyn Vn Hun Tỏc gi xin chõn thnh cm n ti anh, TS Lờ Vn Dng v nhiu s giỳp , úng gúp quý bỏu Tỏc gi xin c gi li cỏm n ti tt c thy cụ, bn bố ó gúp ý, ng h v ng viờn tỏc gi quỏ trỡnh hc v hon thnh lun ỏn Lun ỏn l mún qu quý giỏ ca tỏc gi dnh tng cha m, hai em gỏi, em r v ngi v sp ci nhng ngi ó luụn bờn cnh ng viờn tỏc gi nhng lỳc khú khn T Cụng Sn MC LC Nhng kớ hiu dựng lun ỏn M u Chng Cỏc kin thc chun b v khỏi nim c bn 1.1 Kin thc chun b 1.2 Mt s dng hi t ca trng cỏc bin ngu nhiờn 1.3 Trng cỏc hiu martingale 1.4 Toỏn t ngu nhiờn 10 10 13 19 22 Chng Lut s ln cho trng cỏc hiu martingale 26 2.1 Lut mnh s ln cho trng hp cỏc -hiu martingale 26 2.2 Lut s ln dng Brunk-Prokhorov 39 2.3 Lut yu s ln cho trng -tng thớch mnh 50 Chng Hi t hon ton v tc hi hiu Martingale 3.1 Hi t hon ton 3.2 Hi t hon ton trung bỡnh 3.3 Tc hi t ca chui ngu nhiờn t ca trng cỏc 57 57 66 76 Chng S hi t ca dóy cỏc martingale toỏn t 88 4.1 Hi t ca dóy martingale toỏn t 88 4.2 S hi t ca tớch cỏc toỏn t khụng b chn c lp 97 Kt lun v kin ngh 111 Danh mc cỏc cụng trỡnh khoa hc ca tỏc gi liờn quan n lun ỏn 112 Ti liu tham kho 113 NHNG K HIU DNG TRONG LUN N Z N N0 R E ã B(E) (, F, P ) Card(A) IA n n+m [m, n) n m nm n m n 2n |n| n |n | 1/ log(x) log+ (x) [x] Tp hp cỏc s nguyờn Tp hp cỏc s nguyờn dng Tp hp cỏc s nguyờn khụng õm Tp hp cỏc s thc Khụng gian Banach thc v kh ly Chun trờn khụng gian Banach E -i s Borel cỏc ca E Khụng gian xỏc sut y S phn t ca hp A Hm ch tiờu ca hp A Phn t (1, 1, , 1) Nd Phn t (n1 , n2 , , nd ) Zd Phn t (n1 + m1 , n2 + m2 , , nd + md ) Zd d i=1 [mi , ni ) n1 m1 , n2 m2 , , nd md n m v n = m di=1 (ni mi ) ( tn ti ớt nht mt i d cho ni mi ) Phn t (2n1 , 2n2 , , 2nd ) Phn t (2n1 , 2n2 , , 2nd d ) vi = (1 , , d ) Rd Giỏ tr |n| = n1 n2 nd Giỏ tr n = min{n1 , n2 , , nd } Giỏ tr |n | = n1 n2 nd d vi = (1 , , d ) Rd Phn t (1/1 , , 1/d ) logarit c s e ca x max{log(x), 0} S nguyờn ln nht khụng vt quỏ x M U Lớ chn ti 1.1 Lý thuyt martingale nghiờn cu nhng liờn quan n lý thuyt trũ chi nhng v sau c phỏt trin thnh mt lnh vc toỏn hc cht ch, tr thnh mt mụ hỡnh toỏn hc quan trng cú nhiu ng dng thng kờ, phng trỡnh vi phõn, toỏn kinh t c bit, gn õy ó cú nhiu ng dng thỳ v chng khoỏn, thu hỳt khỏ nhiu nh toỏn hc quan tõm V phng din xỏc sut, martingale l s m rng ca tng cỏc bin ngu nhiờn c lp kỡ vng khụng 1.2 Cỏc nh lý gii hn úng vai trũ quan trng lý thuyt xỏc sut, chỳng c vớ nh nhng viờn ngc ca xỏc sut, Kolmogorov ó tng núi "Giỏ tr chp nhn c ca lý thuyt xỏc sut l cỏc nh lớ gii hn, cỏc kt qu ch yu nht v quan trng nht ca lý thuyt xỏc sut l cỏc lut s ln" Ngy nay, cỏc nh lý gii hn ang l cú tớnh thi s ca lý thuyt xỏc sut 1.3.T nhng nm 1950 tr li õy, cỏc nh lý gii hn ó c nghiờn cu m rng cho dóy bin ngu nhiờn nhn giỏ tr khụng gian Banach Tuy nhiờn i vi trng hp trng cỏc hiu martingale cng nh vi cỏc dóy martingale toỏn t cha c nghiờn cu nhiu Vi cỏc lớ trờn chỳng tụi quyt nh chn ti nghiờn cu cho lun ỏn ca mỡnh l: Cỏc nh lý gii hn cho martingale Mc ớch nghiờn cu Lun ỏn nghiờn cu s hi t cng nh tc hi t ca ca trng cỏc hiu martingale nhn giỏ tr khụng gian Banach, lut mnh s ln Kolmogorov, lut mnh s ln Marcinkiewicz - Zygmund, lut s ln dng Brunk-Prokhorov, lut yu s ln, hi t hon ton v hi t hon ton trung bỡnh ca trng cỏc hiu martingale Lun ỏn nghiờn cu v s hi t ca cỏc dóy toỏn t ngu nhiờn, dóy martingale toỏn t ngu nhiờn cng nh tớch cỏc toỏn t ngu nhiờn c lp khụng gian Banach i tng nghiờn cu i tng nghiờn cu ca lun ỏn l trng cỏc bin ngu nhiờn nhn giỏ tr khụng gian Banach v dóy cỏc toỏn t ngu nhiờn nhn giỏ tr khụng gian Bannach Phm vi nghiờn cu Lun ỏn nghiờn cu cỏc nh lý gii hn nh lut mnh s ln, lut yu s ln, cỏc nh lý v hi t hon ton, hi t hon ton trung bỡnh, tc hi t ca tng cỏc trng hiu martingale, cỏc nh lý v hi t cho dóy cỏc martingale toỏn t ngu nhiờn cng nh tớch vụ hn ca dóy toỏn t ngu nhiờn c lp nhn giỏ tr khụng gian Banach Phng phỏp nghiờn cu Lun ỏn s dng cỏc k thut ca xỏc sut, gii tớch, gii tớch ngu nhiờn, cỏc cụng c ca martingale chng minh cỏc nh lớ hi t Mt s b quan trng nh: B Borel-Cantelli, Bt ng thc Kolmogorov, Bt ng thc Doob, B Toeplitz, lý thuyt toỏn t tt nh, cỏc tớnh cht v thỏc trin toỏn t, nguyờn lý th úng cng c s dng chng minh cỏc kt qu í ngha khoa hc v thc tin í ngha khoa hc: gúp phn lm phong phỳ thờm cỏc kt qu v s hiu bit v hi t ca chui ngu nhiờn, lut mnh s ln ca trng cỏc bin ngu nhiờn nhn giỏ tr khụng gian Banach, cng nh cỏc kt qu ca toỏn t ngu nhiờn í ngha thc tin: lun ỏn gúp phn phỏt trin lý thuyt v cỏc nh lớ gii hn ca trng bin ngu nhiờn nhn giỏ tr khụng gian Banach lý thuyt xỏc sut Tng quan v cu trỳc lun ỏn 7.1 Tng quan lun ỏn Cỏc nh lớ gii hn xỏc sut úng vai trũ quan trng phỏt trin lý thuyt, thc hnh xỏc sut v thng kờ Chớnh vỡ vy m cỏc nh lý v gii hn ó thu hỳt nhiu nh khoa hc nghiờn cu v m rng u tiờn phi k n lut s ln: Lut s ln u tiờn ca Bernoulli c cụng b nm 1713 V sau, kt qu ny c Poisson, Chebyshev, Markov, Liapunov m rng Tuy nhiờn, phi n nm 1909 lut mnh s ln mi c Borel phỏt hin Kt qu ny ca Borel c Kolmogorov hon thin vo nm 1926, ta thng gi l lut s ln dng Kolmogorov ng thi Kolmogorov cng ch rng trng hp dóy cỏc i lng ngu nhiờn c lp cựng phõn b thỡ iu kin cn v ca lut mnh s ln l cỏc bin ngu nhiờn ú cú moment tuyt i cp mt hu hn Kt qu ny ó c Marcinkiewicz v Zygmund m rng (gi l lut s ln dng Marcinkiewicz-Zygmund) Brunk (1948) v Prokhorov (1950) ó khỏi quỏt iu kin dng Kolmogorov vi moment bc cao hn v thu c lut mnh s ln dng Brunk-Prokhorov Lut s ln tip tc c m rng bi nhiu tỏc gi nh Tien, Quang, Hung, Thanh, Huan, Dung, Stadtmulle, Rosalsky, Volodin (xem [47],[48],[49],[16],[14],[67],[50],[30]) bng cỏch lm nh iu kin c lp ca dóy bin ngu nhiờn (nh nghiờn cu trng hp dóy cỏc hiu martingale, cho cỏc hp c lp, v hp martingale), nghiờn cu cho trng hp ch s nhiu chiu, hoc xem xột trờn cỏc khụng gian khỏc Trong lun ỏn ny chỳng tụi tip tc nghiờn cu cỏc nh lý lut s ln cho trng cỏc hiu martingale, trng hp cỏc -hiu martingale nhn giỏ tr khụng gian Banach p-kh trn, trng cỏc bin ngu nhiờn -tng thớch mnh nhn giỏ tr khụng gian Bannach p-kh trn nh lý gii hn cũn c nghiờn cu di dng chui ngu nhiờn, u tiờn c bit n vi cỏc nh lý hai chui, ba chui sau ú l cỏc nghiờn cu v tc hi t ca chui c lp, chui hiu martingale, (xem [51],[52],[64]) Cỏc khỏi nim khỏc nh hi t hon ton, hi t hon ton trung bỡnh cng c nhiu tỏc gi quan tõm, nghiờn cu (nh [31], [34], [7],[53],[10]) Trong lun ny chỳng tụi nghiờn cu v hi t hon ton, hi t hon ton trung bỡnh, v ỏnh giỏ tc hi t ca chui cỏc trng hiu martingale nhn giỏ tr khụng gian p-kh trn Khỏi nim toỏn t ngu nhiờn nh l mt m rng ca ma trn ngu nhiờn c gii thiu cỏc cụng trỡnh ca Skorokhod [56] v c khỏ nhiu tỏc gi quan tõm nghiờn cu nh Thng, Thnh, [73], [69], [74] Trong lun ny chỳng tụi tip tc nghiờn cu v s hi t ca dóy cỏc toỏn t ngu nhiờn nhn giỏ tr khụng gian Banach Cỏc kt qu ca lun ỏn ó c bỏo cỏo ti Seminar b mụn v ti cỏc hi ngh: Hi ngh khoa hc chỳc mng sinh nht G.S Nguyn Duy Tin (Khoa Toỏn - C - Tin hc, Trng H Khoa hc T nhiờn-HQG H Ni, 2012), hi ngh toỏn hc ton quc ln th 10 (Nha trang, 2013), i hi toỏn hc th gii (ICM) ti Seoul, Hn Quc (2014), hi ngh toỏn ng dng cụng nhip (Math-for-industry) ti Kyushu University, Nht Bn (2014), ó c ng v nhn ng cỏc chớ: Statistics and Probability Letters, Applications of Mathematics, Journal of Inequalities and Applications, Journal of the Korean Mathematical Society, Journal of Probability and Statistical Science, ang c gi ng ti cỏc chớ: An International Journal of Probability and Stochastic Processes, Journal of bulletin of the Korean Mathematical Society 7.2 Cu trỳc lun ỏn Ngoi phn m u, kt lun, danh mc cỏc bi bỏo ca nghiờn cu sinh liờn quan n lun ỏn v ti liu tham kho, lun ỏn c trỡnh by bn chng Chng trỡnh by cỏc khỏi nim v k vng, k vng cú iu kin ca bin ngu nhiờn nhn giỏ tr khụng gian Banach, khỏi nim v trng cỏc hiu martingale, toỏn t ngu nhiờn, dóy toỏn t ngu nhiờn c lp, dóy martingale toỏn t ngu nhiờn, mt s dng hi t ca trng cỏc bin ngu nhiờn v dóy cỏc toỏn t ngu nhiờn nhn giỏ tr khụng gian Banach Chng gm ba mc, mc 2.1 a khỏi nim trng hp cỏc -hiu martingale v trng hp cỏc M-hiu martingale; thit lp lut mnh s ln dng Kolmogorov v Marcinkiewicz - Zygmund cho trng hp cỏc -hiu martingale nhn giỏ tr trờn khụng gian Banach Mc 2.2 thit lp lut s ln dng Brunk-Prokhorov cho trng cỏc hiu martingale Mc 2.3 a khỏi nim trng -tng thớch mnh v thit lp lut yu s ln cho trng cỏc i lng ngu nhiờn -tng thớch mnh Chng gm ba mc, mc 3.1 a cỏc iu kin cho hi t hon ton ca tng trung bỡnh trt ca trng cỏc hiu martingale, t ú i n cỏc lut mnh s ln cho tng trung bỡnh trt cng nh ỏnh giỏ tc hi t ca lut mnh s ln Mc 3.2 trỡnh by cỏc kt qu v hi t hon ton trung bỡnh, cỏc iu kin ca hi t hon ton trung bỡnh cng nh mi quan h gia hi t hon ton trung bỡnh vi hi t h.c.c v hi t trung bỡnh; sau ú ỏp dng cho cỏc nghiờn cu v lut s ln, hi t trung bỡnh v tc hi t ca trng cỏc hiu martingale E-giỏ tr Mc 3.3 trỡnh by v tc hi t ca chui cỏc trng hiu martingale Chng thit lp cỏc iu kin hi t ca dóy cỏc toỏn t ngu nhiờn, toỏn t ngu nhiờn m rng, dóy hiu martingale toỏn t ngu nhiờn b chn khụng gian Banach v nghiờn cu cỏc iu kin hi t ca tớch vụ hn cỏc toỏn t ngu nhiờn c lp TI LIU THAM KHO [1] Adler A., Rosalsky A (1987), "On general strong laws for weighted sums of stochastically dominated random variables", Stoch Anal Appl , pp.1-16 [2] Assouad P (1975), Espaces p-lisses et q-convexes, Inộgalitộs de Burkholder, Sộminaire Maruey-Schwartz, Exp ZV [3] Borovskykh Y.V., Korolyuk V.S (1997), Martingale Approximation, VSP [4] Brunk H.D (1948), "The strong law of large numbers", Duke Math J 15, pp.181-195 [5] Cabrera M.O (1994), "Convergence of weighted sums of random variables and uniform integrability concerning the weights", Collectanea Mathematica 45(2), pp.121-132 [6] Chattecji S.D (1986), "Martingale convergence and the RadonNikodym theorem in Banach spaces", Math Scand 22, pp.21-41 [7] Chen P., Hu T.C., Volodin V (2006), "A note on the rate of complete convergence for maximus of partial sums for moving average processes in rademacher type Banach spaces", Lobachevskii J Math 21, pp.4555 [8] Christofidesm T.C., Serfling R.J (1990)," Maximal inequalities for multidimensionally indexed submartingale arrays" Ann.Probab 45(3), pp.436-641 113 [9] Choi B.D., Sung S.H (1985), "On convergence of (Sn ESn )/n1/r , < r < for pairwise independent random variables", Bull Korean Math Soc 22(2), pp.79-82 [10] Chow Y.S (1988), "On the rate of moment convergence of sample sums and extremes" Bull Inst Math.Acad.Sin (N.S.) 16, pp.177201 [11] Chow Y S., Teicher H (1997), Probability Theory Independence, Interchangeability, Martingale, Springer, New York [12] Czerebak-Mrozowicz E.B., Klesov O.I., Rychlik Z (2002), "Marcinkiewicz-type strong laws of large numbers for pairwise independent random fields", Probab Math Statist 22(1), pp.127139 [13] Day M.M (1944), "Uniform convexity in factor and conjugate spaces", Ann.of Math 45, pp.375 -385 [14] Dung L.V (2010), "Weak laws of large numbers for double arrays of random elements in Banach spaces" Acta Math Vietnamica 35, pp.387-398 [15] Dung L.V., Son T.C., Tien N.D (2014), "L1 bounds for some martingale central limit theorems" Lithuanian Math J 54(1), pp.46-60 [16] Dung L.V., Tien N.D (2010), "Strong laws of large numbers for random fields in martingale type p Banach spaces" Stat Probab lett 80 (9-10), pp.756-763 [17] Edgar G.A., Louis S (1992), Stopping times and directed processes, 47, Cambridge University, England [18] Fazekas I.; Klesov O (2000), "A general approach to the strong law of large numbers" Theory Probab Appl 45(3), 436-449 [19] Fazekas I., Túmỏcs T (1998), "Strong laws of large numbers for pairwise independent random variables with multidimensional indices", Publ Math Debrecen 53(1-2), pp.149-161 114 [20] Feller W (1971), An introduction to probability theory and its applications, 2, 2nd ed Wiley, New York [21] Gaposhkin V.F (1995), "On the strong law of large numbers for blockwise independent and block-wise orthogonal random variables", Theory Probab Appl 39, pp.667 - 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