MAXIMAL INEQUALITIES AND STRONG LAW OF LARGE NUMBERS FOR AANA SEQUENCES

Size: px
Start display at page:

Download "MAXIMAL INEQUALITIES AND STRONG LAW OF LARGE NUMBERS FOR AANA SEQUENCES"

Transcription

1 Commu Korea Math Soc 26 20, No, pp 5 6 DOI 0434/CKMS20265 MAXIMAL INEQUALITIES AND STRONG LAW OF LARGE NUMBERS FOR AANA SEQUENCES Wag Xueju, Hu Shuhe, Li Xiaoqi, ad Yag Wezhi Abstract Let {X, } be a sequece of asymptotically almost egatively associated radom variables ad S = X i I the paper, we get the precise results of Hájek-Réyi type iequalities for the partial sums of asymptotically almost egatively associated sequece, which geeralize ad improve the results of Theorem 24 Theorem 26 i Ko et al [4] I additio, the large deviatio of S for sequece of asymptotically almost egatively associated radom variables is studied At last, the Marcikiewicz type strog law of large umbers is give Itroductio Defiitio A fiite collectio of radom variables X, X 2,, X is said to be egatively associated NA, i short if for every pair of disjoit subsets A, A 2 of {, 2,, }, Cov{fX i : i A, g : j A 2 } 0, wheever f ad g are coordiatewise odecreasig such that this covariace exists A ifiite sequece {X, } is NA if every fiite subcollectio is NA Defiitio 2 A sequece {X, } of radom variables is called asymptotically almost egatively associated AANA, i short if there exists a oegative sequece q 0 as such that CovfX, gx +, X +2,, X +k q [V arfx V argx +, X +2,, X +k ] /2 Received September 27, Mathematics Subject Classificatio 60E0, 60F5, 60F0 Key words ad phrases Hájek-Réyi iequality, asymptotically almost egatively associated sequece, strog law of large umbers, large deviatio Supported by the NNSF of Chia 08700, , Provicial Natural Sciece Research Project of Ahui Colleges KJ200A005, Talets Youth Fud of Ahui Provice Uiversities 200SQRL06ZD, Youth Sciece Research Fud of Ahui Uiversity 2009QN0A 5 c 20 The Korea Mathematical Society

2 52 WANG XUEJUN, HU SHUHE, LI XIAOQIN, AND YANG WENZHI for all, k ad for all coordiatewise odecreasig cotiuous fuctios f ad g wheever the variaces exist The family of AANA sequece cotais NA ad idepedet sequeces as special cases A AANA sequece of radom variables meas roughly that asymptotically the future is almost egatively associated with the preset A example of a AANA sequece which is ot NA was costructed by Chadra ad Ghosal [] Sice the cocept of AANA sequece was itroduced by Chadra ad Ghosal [], may applicatios have bee foud See for example, Chadra ad Ghosal [] derived the Kolmogorov type iequality ad the strog law of large umbers, Chadra ad Ghosal [2] obtaied the almost sure covergece of weighted averages, Ko et al [4] studied the Hájek-Réyi type iequality, ad Wag et al [5] established the law of the iterated logarithm for product sums Recetly, Yua ad A [6] established some Rosethal type iequalities for imum partial sums of AANA sequeces The mai purpose of the paper is to further study the Hájek-Réyi type iequalities, which geeralize ad improve the results of Theorem 24 Theorem 26 i Ko et al [4] I additio, the large deviatio ad Marcikiewicz type strog law of large umbers for AANA sequece are studied Throughout the paper, let {X, } be a sequece of AANA radom variables defied o a fixed probability space Ω, F, P Deote S = X i ad IA be the idicator fuctio of the set A For p >, let q = p/p be the dual umber of p C deotes a positive costat which may be differet i various places Lemma cf Yua ad A, [6, Lemma 2] Let {X, } be a sequece of AANA radom variables with mixig coefficiets {q, }, f, f 2, be all odecreasig or oicreasig fuctios, the {f X, } is still a sequece of AANA radom variables with mixig coefficiets {q, } Lemma 2 Let < p 2 ad {X, } be a sequece of AANA radom variables with mixig coefficiets {q, } ad EX = 0 for each If = q2 <, the there exists a positive costat C p depedig oly o p such that 2 E S i p C p E X i p i for all, where C p = 2 p [ 2 2 p p + 6p p = q2 p/q ] We poit out that Lemma 2 has bee studied by Yua ad A [6] But here we give the accurate coefficiet C p Ad Lemma 2 geeralizes ad improves the result of Lemma 22 i Ko et al [4] The followig Khitchie- Kolmogorov type covergece theorem is the immediate byproduct of Lemma ad Lemma 2

3 MAXIMAL INEQUALITIES AND STRONG LAW 53 Corollary Khitchie-Kolmogorov type covergece theorem Let {X, } be a sequece of AANA radom variables with mixig coefficiets {q, } ad = q2 < Assume that 3 V arx <, = the = X EX coverges as Lemma 3 cf Fazekas ad Klesov, [3, Theorem ] Let β, β 2,, β be a odecreasig sequece of positive umbers ad α, α 2,, α be oegative umbers Let r be a fixed positive umber Assume that for each m with m, r l m 4 E l m α l, β l l= the l 5 E X r j 4 l α l β r l= l Lemma 4 cf Yua ad A, [6, Theorem 2] Let {X, } be a sequece of AANA radom variables with EX i = 0 for all i ad p 3 2 k, 4 2 k ], where iteger umber k If = qq/p <, the there exists a positive costat D p depedig oly o p such that for all p/2 6 E S i p D p E X i i p + EXi 2 2 Hájek-Réyi type iequalities for AANA sequece Theorem 2 Let {X, } be a sequece of AANA radom variables with mixig coefficiets {q, } ad {b, } be a odecreasig sequece of positive umbers Assume that EX = 0 for each ad = q2 < The for ay ε > 0 ad ay iteger, 2 P k ε 2p C p E p ε p b p j [ for all < p 2, where C p = 2 p 2 2 p p + 6p p = q2 ] p/q Proof Without loss of geerality, we assume that b 0 = 0 ad i = 0 whe i = It is easy to check that 22 S k = = j b i b i = b i b i j=i

4 54 WANG XUEJUN, HU SHUHE, LI XIAOQIN, AND YANG WENZHI 22 ad k b k b i b i = imply that S k 23 b k ε i k Therefore k S k b k which implies that 24 P k ε S k b k = j=i k i k j=i i k i ε ε i b j b j ε i ε, 2 ε i P i ε 2 By Lemma, we ca see that {X /b, } is still a sequece of AANA radom variables with mixig coefficiets {q, } Thus, by 24, Markov s iequality ad Lemma 2, we ca obtai P k ε 2p ε p E i i The proof of the theorem is complete p 2p C p ε p E p Theorem 22 Let {X, } be a sequece of AANA radom variables with mixig coefficiets {q, } ad {b, } be a odecreasig sequece of positive umbers Assume that EX = 0 for each ad = q2 < The for ay ε > 0 ad ay positive itegers m <, 25 P m k ε 2p C p m E p ε p b p + 2 p E p m b p j=m+ j [ for all < p 2, where C p = 2 p 2 2 p p + 6p p = q2 ] p/q Proof Observe that m k m b m + m+ k j=m+, b p j

5 MAXIMAL INEQUALITIES AND STRONG LAW 55 thus 26 P m k ε P m b m ε + P 2 = I + II m+ k j=m+ For I, by Markov s iequality ad Lemma 2, we have p 27 I 2p m ε p b p E 2p C p m E p m ε p b p m ε 2 For II, we will apply Theorem 2 to {X m+i, i m} ad {b m+i, i m} Notig that m+ k = k m X m+j b m+k, j=m+ thus, by Theorem 2, we get 28 II 2p m C p E X m+j p ε/2 p b p = 22p C p m+j ε p j=m+ E p Therefore, the desired result 25 follows from immediately b p j Theorem 23 Let {X, } be a sequece of AANA radom variables with mixig coefficiets {q, } ad = q2 <, {b, } be a odecreasig sequece of positive umbers Deote T = X i EX i for Assume that V ar 29 b 2 < j The for ay r 0, 2, 20 E sup T b ad 2 E sup r T b + 4rC 2 2 r 2 4C 2 V ar b 2 j V ar b 2 j < <

6 56 WANG XUEJUN, HU SHUHE, LI XIAOQIN, AND YANG WENZHI Furthermore, if lim b = +, the lim b E = 0 as, where 2/2 C 2 = q 2 = q 2 < = Proof By the cotiuity of probability ad Theorem 2, we ca see that E sup T r r b = P sup T r 0 b > t dt + P sup T b > t dt + lim P T N > t/r dt + 4C 2 = + 4rC 2 2 r V ar b 2 j N b V ar b 2 j t 2/r dt < So 20 is proved By Lemma 2, we have 22 E T i 2 C 2 E X i EX i 2 = C 2 i = V arx i = where α j = C 2 V ar 0, j =, 2,, By 22 ad Lemma 3, 23 E T i 2 α j V ar i 4 b i b 2 = 4C 2 j b 2 j Thus, by mootoe covergece theorem ad 23, E sup T 2 [ = E T i 2] = lim b 4C 2 lim i V ar b 2 j b i < This completes the proof of 2 Observe that T P > ε = P =m b N=m = lim N P m N m N By Theorem 22 for p = 2 we ca obtai that P T m N b > ε 4C m 2 V ar ε 2 b m E T b T b N j=m+ α j, T i 2 i b i > ε > ε V ar b 2 j

7 MAXIMAL INEQUALITIES AND STRONG LAW 57 Hece, by 29 ad Kroecker s lemma, it follows that lim P T m > ε = 0, ε > 0, which is equivalet to lim are proved =m b b E = 0 as The desired results Remark 2 Hájek-Réyi type iequalities for AANA sequece have bee studied by Ko et al [4] But their results are based o the followig coditios 24 k= σ M/M k /M D k= σ 2 k /2 for some M >, D > 0, ad EXk 2 <, where σ2 k = EX2 k Here Theorem 2 Theorem 23 we remove the coditios above, ad geeralize p = 2 to the case of < p 2 I additio, we give the accurate coefficiet C p So our Theorem 2 Theorem 23 geeralize ad improve the results of Theorem 24 Theorem 26 i Ko et al [4], respectively 3 Large deviatios for AANA sequece I this sectio, we will study the asymptotic behavior of the probabilities 3 P S > x, x > 0, I the followig, we let X p = E X p /p for some p > 0 Theorem 3 Let < p 2 ad {X, } be a sequece of AANA radom variables with = q2 < ad EX i = 0 for all i If there exists a positive costat M < such that X i p M for all i, the for every x > 0, 32 P S i > x C pm p i x p p, where C p is defied i Lemma 2 Proof By Markov s iequality ad Lemma 2, we ca see that P S i > x i p x p E S i p i 33 which implies 32 C p p x p E X i p C pm p x p p,

8 58 WANG XUEJUN, HU SHUHE, LI XIAOQIN, AND YANG WENZHI Theorem 32 Let {X, } be a sequece of AANA radom variables with EX i = 0 for all i If there exists a positive costat M < such that X i p M for all i ad some p 3 2 k, 4 2 k ], where iteger umber k, the for every x > 0, P S i > x 2D pm p i x p p/2, where D p is defied i Lemma 4 Proof By 0 < 2/p < ad C r s iequality, 2/p X i p which implies that Xi 2, p/2 E X i p E Xi 2 By Jese s iequality, we have p/2 EXi 2 E p/2 Xi 2 Therefore, the statemets above ad C r s iequality imply that p/2 35 E X i p + 2 p/2 E X i p 2M p p/2 EX 2 i Combiig Lemma 4 ad 35, 36 E S i p 2D p M p p/2 i It follows from Markov s iequality ad 36 that 37 P S i > x i p x p E this completes the proof of the theorem S i p i 2D pm p x p p/2, 4 Marcikiewicz type strog law of large umbers for AANA sequece Theorem 4 Let {X, } be a sequece of idetically distributed AANA radom variables with = q2 < ad E X p < for 0 < p < 2 Assume that EX = 0 if p < 2 The 4 X /p k 0 as, k=

9 MAXIMAL INEQUALITIES AND STRONG LAW 59 Proof Deote the Y = /p IX /p + X I X < /p + /p IX /p, P X Y = P X /p CE X p <, = = which implies that P X Y, io = 0 by the Borel-Catelli lemma Thus /p k= X k 0 as if ad oly if /p k= Y k 0 as So we oly eed to show that 42 Y /p k EY k 0 as,, ad 43 k= /p EY k 0, k= By Corollary ad Kroecker s lemma, to prove 42, it suffices to show that Y 44 V ar < I fact, V ar = Y /p = = /p C P X /p + C C + C = C + C C + C < = 2/p k= =k = EX 2 I X < /p 2/p EX 2 Ik X p < k k= E X p X 2/p 2 p Ik X p < k k 2/p E X p k 2 p/p Ik X p < k k= Hece 42 holds Next, we will prove 43 It will be divided ito two cases: i If p =, by E X p < ad Lebesgue domiated covergece theorem, we have 45 lim /p P X /p = 0, lim 46 EX I X < /p = lim X ωi X ω < /p P dω Ω = EX = 0

10 60 WANG XUEJUN, HU SHUHE, LI XIAOQIN, AND YANG WENZHI Thus, EY /p P X /p + EX I X < /p 0 as By the Toeplitz lemma, we obtai lim k= EY k = 0 ii If p, by the Kroecker s lemma, to prove 43, it suffices to show that 47 For 0 < p <, = = = EY < /p EY P X /p /p + C + = C + C + C = =j = E X I X < /p /p /p E X Ij X p < j /p E X Ij X p < j j /p E X p j p/p Ij X p < j < For p < 2, by EX = 0, we ca see that EY P X /p /p + = = C + = C + C + C = /p E X I X /p = = EX I X < /p /p j /p E X Ij X p < j + j /p E X p j p/p Ij X p < j + < Thus 47 holds, which implies 43 by Kroecker s lemma We get the desired result Ackowledgemets The authors are most grateful to the editor ad aoymous referees for careful readig of the mauscript ad valuable suggestios which helped i improvig a earlier versio of this paper

11 MAXIMAL INEQUALITIES AND STRONG LAW 6 Refereces [] T K Chadra ad S Ghosal, Extesios of the strog law of large umbers of Marcikiewicz ad Zygmud for depedet variables, Acta Math Hugar 7 996, o 4, [2], The strog law of large umbers for weighted averages uder depedece assumptios, J Theoret Probab 9 996, o 3, [3] I Fazekas ad O Klesov, A geeral approach to the strog laws of large umbers, Teor Veroyatost i Primee , o 3, ; traslatio i Theory Probab Appl , o 3, [4] M H Ko, T S Kim, ad Z Y Li, The Hájeck-Rèyi iequality for the AANA radom variables ad its applicatios, Taiwaese J Math , o, 22 [5] Y B Wag, J G Ya, ad F Y Cheg, The strog law of large umbers ad the law of the iterated logarithm for product sums of NA ad AANA radom variables, Southeast Asia Bull Math , o 2, [6] D M Yua ad J A, Rosethal type iequalities for asymptotically almost egatively associated radom variables ad applicatios, Sci Chia Ser A , o 9, Wag Xueju School of Mathematical Sciece Ahui Uiversity Hefei , P R Chia address: wxjahdx2000@26com Hu Shuhe School of Mathematical Sciece Ahui Uiversity Hefei , P R Chia address: hushuhe@263et Li Xiaoqi School of Mathematical Sciece Ahui Uiversity Hefei , P R Chia address: lixiaoqi983@63com Yag Wezhi School of Mathematical Sciece Ahui Uiversity Hefei , P R Chia address: wzyag827@63com

Complete Convergence for Asymptotically Almost Negatively Associated Random Variables

Complete Convergence for Asymptotically Almost Negatively Associated Random Variables Applied Mathematical Scieces, Vol. 12, 2018, o. 30, 1441-1452 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ams.2018.810142 Complete Covergece for Asymptotically Almost Negatively Associated Radom

More information

Research Article Moment Inequality for ϕ-mixing Sequences and Its Applications

Research Article Moment Inequality for ϕ-mixing Sequences and Its Applications Hidawi Publishig Corporatio Joural of Iequalities ad Applicatios Volume 2009, Article ID 379743, 2 pages doi:0.55/2009/379743 Research Article Momet Iequality for ϕ-mixig Sequeces ad Its Applicatios Wag

More information

Mi-Hwa Ko and Tae-Sung Kim

Mi-Hwa Ko and Tae-Sung Kim J. Korea Math. Soc. 42 2005), No. 5, pp. 949 957 ALMOST SURE CONVERGENCE FOR WEIGHTED SUMS OF NEGATIVELY ORTHANT DEPENDENT RANDOM VARIABLES Mi-Hwa Ko ad Tae-Sug Kim Abstract. For weighted sum of a sequece

More information

Research Article Strong and Weak Convergence for Asymptotically Almost Negatively Associated Random Variables

Research Article Strong and Weak Convergence for Asymptotically Almost Negatively Associated Random Variables Discrete Dyamics i Nature ad Society Volume 2013, Article ID 235012, 7 pages http://dx.doi.org/10.1155/2013/235012 Research Article Strog ad Weak Covergece for Asymptotically Almost Negatively Associated

More information

Equivalent Conditions of Complete Convergence and Complete Moment Convergence for END Random Variables

Equivalent Conditions of Complete Convergence and Complete Moment Convergence for END Random Variables Chi. A. Math. Ser. B 391, 2018, 83 96 DOI: 10.1007/s11401-018-1053-9 Chiese Aals of Mathematics, Series B c The Editorial Office of CAM ad Spriger-Verlag Berli Heidelberg 2018 Equivalet Coditios of Complete

More information

A Note on the Kolmogorov-Feller Weak Law of Large Numbers

A Note on the Kolmogorov-Feller Weak Law of Large Numbers Joural of Mathematical Research with Applicatios Mar., 015, Vol. 35, No., pp. 3 8 DOI:10.3770/j.iss:095-651.015.0.013 Http://jmre.dlut.edu.c A Note o the Kolmogorov-Feller Weak Law of Large Numbers Yachu

More information

Review Article Complete Convergence for Negatively Dependent Sequences of Random Variables

Review Article Complete Convergence for Negatively Dependent Sequences of Random Variables Hidawi Publishig Corporatio Joural of Iequalities ad Applicatios Volume 010, Article ID 50793, 10 pages doi:10.1155/010/50793 Review Article Complete Covergece for Negatively Depedet Sequeces of Radom

More information

Convergence of random variables. (telegram style notes) P.J.C. Spreij

Convergence of random variables. (telegram style notes) P.J.C. Spreij Covergece of radom variables (telegram style otes).j.c. Spreij this versio: September 6, 2005 Itroductio As we kow, radom variables are by defiitio measurable fuctios o some uderlyig measurable space

More information

Asymptotic distribution of products of sums of independent random variables

Asymptotic distribution of products of sums of independent random variables Proc. Idia Acad. Sci. Math. Sci. Vol. 3, No., May 03, pp. 83 9. c Idia Academy of Scieces Asymptotic distributio of products of sums of idepedet radom variables YANLING WANG, SUXIA YAO ad HONGXIA DU ollege

More information

COMPLETE CONVERGENCE AND COMPLETE MOMENT CONVERGENCE FOR ARRAYS OF ROWWISE END RANDOM VARIABLES

COMPLETE CONVERGENCE AND COMPLETE MOMENT CONVERGENCE FOR ARRAYS OF ROWWISE END RANDOM VARIABLES GLASNIK MATEMATIČKI Vol. 4969)2014), 449 468 COMPLETE CONVERGENCE AND COMPLETE MOMENT CONVERGENCE FOR ARRAYS OF ROWWISE END RANDOM VARIABLES Yogfeg Wu, Mauel Ordóñez Cabrera ad Adrei Volodi Toglig Uiversity

More information

Research Article On the Strong Laws for Weighted Sums of ρ -Mixing Random Variables

Research Article On the Strong Laws for Weighted Sums of ρ -Mixing Random Variables Hidawi Publishig Corporatio Joural of Iequalities ad Applicatios Volume 2011, Article ID 157816, 8 pages doi:10.1155/2011/157816 Research Article O the Strog Laws for Weighted Sums of ρ -Mixig Radom Variables

More information

Chapter 3. Strong convergence. 3.1 Definition of almost sure convergence

Chapter 3. Strong convergence. 3.1 Definition of almost sure convergence Chapter 3 Strog covergece As poited out i the Chapter 2, there are multiple ways to defie the otio of covergece of a sequece of radom variables. That chapter defied covergece i probability, covergece i

More information

Central limit theorem and almost sure central limit theorem for the product of some partial sums

Central limit theorem and almost sure central limit theorem for the product of some partial sums Proc. Idia Acad. Sci. Math. Sci. Vol. 8, No. 2, May 2008, pp. 289 294. Prited i Idia Cetral it theorem ad almost sure cetral it theorem for the product of some partial sums YU MIAO College of Mathematics

More information

for all x ; ;x R. A ifiite sequece fx ; g is said to be ND if every fiite subset X ; ;X is ND. The coditios (.) ad (.3) are equivalet for =, but these

for all x ; ;x R. A ifiite sequece fx ; g is said to be ND if every fiite subset X ; ;X is ND. The coditios (.) ad (.3) are equivalet for =, but these sub-gaussia techiques i provig some strog it theorems Λ M. Amii A. Bozorgia Departmet of Mathematics, Faculty of Scieces Sista ad Baluchesta Uiversity, Zaheda, Ira Amii@hamoo.usb.ac.ir, Fax:054446565 Departmet

More information

Weak Laws of Large Numbers for Sequences or Arrays of Correlated Random Variables

Weak Laws of Large Numbers for Sequences or Arrays of Correlated Random Variables Iteratioal Mathematical Forum, Vol., 5, o. 4, 65-73 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/.988/imf.5.5 Weak Laws of Large Numers for Sequeces or Arrays of Correlated Radom Variales Yutig Lu School

More information

ST5215: Advanced Statistical Theory

ST5215: Advanced Statistical Theory ST525: Advaced Statistical Theory Departmet of Statistics & Applied Probability Tuesday, September 7, 2 ST525: Advaced Statistical Theory Lecture : The law of large umbers The Law of Large Numbers The

More information

Introduction to Probability. Ariel Yadin

Introduction to Probability. Ariel Yadin Itroductio to robability Ariel Yadi Lecture 2 *** Ja. 7 ***. Covergece of Radom Variables As i the case of sequeces of umbers, we would like to talk about covergece of radom variables. There are may ways

More information

An almost sure invariance principle for trimmed sums of random vectors

An almost sure invariance principle for trimmed sums of random vectors Proc. Idia Acad. Sci. Math. Sci. Vol. 20, No. 5, November 200, pp. 6 68. Idia Academy of Scieces A almost sure ivariace priciple for trimmed sums of radom vectors KE-ANG FU School of Statistics ad Mathematics,

More information

Precise Rates in Complete Moment Convergence for Negatively Associated Sequences

Precise Rates in Complete Moment Convergence for Negatively Associated Sequences Commuicatios of the Korea Statistical Society 29, Vol. 16, No. 5, 841 849 Precise Rates i Complete Momet Covergece for Negatively Associated Sequeces Dae-Hee Ryu 1,a a Departmet of Computer Sciece, ChugWoo

More information

Probability and Random Processes

Probability and Random Processes Probability ad Radom Processes Lecture 5 Probability ad radom variables The law of large umbers Mikael Skoglud, Probability ad radom processes 1/21 Why Measure Theoretic Probability? Stroger limit theorems

More information

7.1 Convergence of sequences of random variables

7.1 Convergence of sequences of random variables Chapter 7 Limit Theorems Throughout this sectio we will assume a probability space (, F, P), i which is defied a ifiite sequece of radom variables (X ) ad a radom variable X. The fact that for every ifiite

More information

Research Article On the Strong Convergence and Complete Convergence for Pairwise NQD Random Variables

Research Article On the Strong Convergence and Complete Convergence for Pairwise NQD Random Variables Abstract ad Applied Aalysis Volume 204, Article ID 949608, 7 pages http://dx.doi.org/0.55/204/949608 Research Article O the Strog Covergece ad Complete Covergece for Pairwise NQD Radom Variables Aitig

More information

<, if ε > 0 2nloglogn. =, if ε < 0.

<, if ε > 0 2nloglogn. =, if ε < 0. GLASNIK MATEMATIČKI Vol. 52(72)(207), 35 360 THE DAVIS-GUT LAW FOR INDEPENDENT AND IDENTICALLY DISTRIBUTED BANACH SPACE VALUED RANDOM ELEMENTS Pigya Che, Migyag Zhag ad Adrew Rosalsky Jia Uversity, P.

More information

Hyun-Chull Kim and Tae-Sung Kim

Hyun-Chull Kim and Tae-Sung Kim Commu. Korea Math. Soc. 20 2005), No. 3, pp. 531 538 A CENTRAL LIMIT THEOREM FOR GENERAL WEIGHTED SUM OF LNQD RANDOM VARIABLES AND ITS APPLICATION Hyu-Chull Kim ad Tae-Sug Kim Abstract. I this paper we

More information

A central limit theorem for moving average process with negatively associated innovation

A central limit theorem for moving average process with negatively associated innovation Iteratioal Mathematical Forum, 1, 2006, o. 10, 495-502 A cetral limit theorem for movig average process with egatively associated iovatio MI-HWA KO Statistical Research Ceter for Complex Systems Seoul

More information

7.1 Convergence of sequences of random variables

7.1 Convergence of sequences of random variables Chapter 7 Limit theorems Throughout this sectio we will assume a probability space (Ω, F, P), i which is defied a ifiite sequece of radom variables (X ) ad a radom variable X. The fact that for every ifiite

More information

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 19 11/17/2008 LAWS OF LARGE NUMBERS II THE STRONG LAW OF LARGE NUMBERS

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 19 11/17/2008 LAWS OF LARGE NUMBERS II THE STRONG LAW OF LARGE NUMBERS MASSACHUSTTS INSTITUT OF TCHNOLOGY 6.436J/5.085J Fall 2008 Lecture 9 /7/2008 LAWS OF LARG NUMBRS II Cotets. The strog law of large umbers 2. The Cheroff boud TH STRONG LAW OF LARG NUMBRS While the weak

More information

Distribution of Random Samples & Limit theorems

Distribution of Random Samples & Limit theorems STAT/MATH 395 A - PROBABILITY II UW Witer Quarter 2017 Néhémy Lim Distributio of Radom Samples & Limit theorems 1 Distributio of i.i.d. Samples Motivatig example. Assume that the goal of a study is to

More information

Berry-Esseen bounds for self-normalized martingales

Berry-Esseen bounds for self-normalized martingales Berry-Essee bouds for self-ormalized martigales Xiequa Fa a, Qi-Ma Shao b a Ceter for Applied Mathematics, Tiaji Uiversity, Tiaji 30007, Chia b Departmet of Statistics, The Chiese Uiversity of Hog Kog,

More information

Product measures, Tonelli s and Fubini s theorems For use in MAT3400/4400, autumn 2014 Nadia S. Larsen. Version of 13 October 2014.

Product measures, Tonelli s and Fubini s theorems For use in MAT3400/4400, autumn 2014 Nadia S. Larsen. Version of 13 October 2014. Product measures, Toelli s ad Fubii s theorems For use i MAT3400/4400, autum 2014 Nadia S. Larse Versio of 13 October 2014. 1. Costructio of the product measure The purpose of these otes is to preset the

More information

Lecture 19: Convergence

Lecture 19: Convergence Lecture 19: Covergece Asymptotic approach I statistical aalysis or iferece, a key to the success of fidig a good procedure is beig able to fid some momets ad/or distributios of various statistics. I may

More information

Some Tauberian theorems for weighted means of bounded double sequences

Some Tauberian theorems for weighted means of bounded double sequences A. Ştiiţ. Uiv. Al. I. Cuza Iaşi. Mat. N.S. Tomul LXIII, 207, f. Some Tauberia theorems for weighted meas of bouded double sequeces Cemal Bele Received: 22.XII.202 / Revised: 24.VII.203/ Accepted: 3.VII.203

More information

1 Convergence in Probability and the Weak Law of Large Numbers

1 Convergence in Probability and the Weak Law of Large Numbers 36-752 Advaced Probability Overview Sprig 2018 8. Covergece Cocepts: i Probability, i L p ad Almost Surely Istructor: Alessadro Rialdo Associated readig: Sec 2.4, 2.5, ad 4.11 of Ash ad Doléas-Dade; Sec

More information

Self-normalized deviation inequalities with application to t-statistic

Self-normalized deviation inequalities with application to t-statistic Self-ormalized deviatio iequalities with applicatio to t-statistic Xiequa Fa Ceter for Applied Mathematics, Tiaji Uiversity, 30007 Tiaji, Chia Abstract Let ξ i i 1 be a sequece of idepedet ad symmetric

More information

Lecture 8: Convergence of transformations and law of large numbers

Lecture 8: Convergence of transformations and law of large numbers Lecture 8: Covergece of trasformatios ad law of large umbers Trasformatio ad covergece Trasformatio is a importat tool i statistics. If X coverges to X i some sese, we ofte eed to check whether g(x ) coverges

More information

Some limit properties for a hidden inhomogeneous Markov chain

Some limit properties for a hidden inhomogeneous Markov chain Dog et al. Joural of Iequalities ad Applicatios (208) 208:292 https://doi.org/0.86/s3660-08-884-7 R E S E A R C H Ope Access Some it properties for a hidde ihomogeeous Markov chai Yu Dog *, Fag-qig Dig

More information

62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 +

62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + 62. Power series Defiitio 16. (Power series) Give a sequece {c }, the series c x = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + is called a power series i the variable x. The umbers c are called the coefficiets of

More information

Lecture 3 The Lebesgue Integral

Lecture 3 The Lebesgue Integral Lecture 3: The Lebesgue Itegral 1 of 14 Course: Theory of Probability I Term: Fall 2013 Istructor: Gorda Zitkovic Lecture 3 The Lebesgue Itegral The costructio of the itegral Uless expressly specified

More information

Lecture 3 : Random variables and their distributions

Lecture 3 : Random variables and their distributions Lecture 3 : Radom variables ad their distributios 3.1 Radom variables Let (Ω, F) ad (S, S) be two measurable spaces. A map X : Ω S is measurable or a radom variable (deoted r.v.) if X 1 (A) {ω : X(ω) A}

More information

Strong Convergence Theorems According. to a New Iterative Scheme with Errors for. Mapping Nonself I-Asymptotically. Quasi-Nonexpansive Types

Strong Convergence Theorems According. to a New Iterative Scheme with Errors for. Mapping Nonself I-Asymptotically. Quasi-Nonexpansive Types It. Joural of Math. Aalysis, Vol. 4, 00, o. 5, 37-45 Strog Covergece Theorems Accordig to a New Iterative Scheme with Errors for Mappig Noself I-Asymptotically Quasi-Noexpasive Types Narogrit Puturog Mathematics

More information

Generalized Law of the Iterated Logarithm and Its Convergence Rate

Generalized Law of the Iterated Logarithm and Its Convergence Rate Stochastic Aalysis ad Applicatios, 25: 89 03, 2007 Copyright Taylor & Fracis Group, LLC ISSN 0736-2994 prit/532-9356 olie DOI: 0.080/073629906005997 Geeralized Law of the Iterated Logarithm ad Its Covergece

More information

SOME SEQUENCE SPACES DEFINED BY ORLICZ FUNCTIONS

SOME SEQUENCE SPACES DEFINED BY ORLICZ FUNCTIONS ARCHIVU ATHEATICU BRNO Tomus 40 2004, 33 40 SOE SEQUENCE SPACES DEFINED BY ORLICZ FUNCTIONS E. SAVAŞ AND R. SAVAŞ Abstract. I this paper we itroduce a ew cocept of λ-strog covergece with respect to a Orlicz

More information

LAWS OF LARGE NUMBERS WITH INFINITE MEAN

LAWS OF LARGE NUMBERS WITH INFINITE MEAN Joural of Mathematical Iequalities Volume 3, Number 2 (209), 335 349 doi:0.753/mi-209-3-24 LAWS OF LARGE NUMBERS WITH INFINITE MEAN HAIYUN XU, XIAOQIN LI, WENZHI YANG AND FANGNING XU (Commuicated by Z.

More information

LECTURE 8: ASYMPTOTICS I

LECTURE 8: ASYMPTOTICS I LECTURE 8: ASYMPTOTICS I We are iterested i the properties of estimators as. Cosider a sequece of radom variables {, X 1}. N. M. Kiefer, Corell Uiversity, Ecoomics 60 1 Defiitio: (Weak covergece) A sequece

More information

HAJEK-RENYI-TYPE INEQUALITY FOR SOME NONMONOTONIC FUNCTIONS OF ASSOCIATED RANDOM VARIABLES

HAJEK-RENYI-TYPE INEQUALITY FOR SOME NONMONOTONIC FUNCTIONS OF ASSOCIATED RANDOM VARIABLES HAJEK-RENYI-TYPE INEQUALITY FOR SOME NONMONOTONIC FUNCTIONS OF ASSOCIATED RANDOM VARIABLES ISHA DEWAN AND B. L. S. PRAKASA RAO Received 1 April 005; Revised 6 October 005; Accepted 11 December 005 Let

More information

The Choquet Integral with Respect to Fuzzy-Valued Set Functions

The Choquet Integral with Respect to Fuzzy-Valued Set Functions The Choquet Itegral with Respect to Fuzzy-Valued Set Fuctios Weiwei Zhag Abstract The Choquet itegral with respect to real-valued oadditive set fuctios, such as siged efficiecy measures, has bee used i

More information

On Summability Factors for N, p n k

On Summability Factors for N, p n k Advaces i Dyamical Systems ad Applicatios. ISSN 0973-532 Volume Number 2006, pp. 79 89 c Research Idia Publicatios http://www.ripublicatio.com/adsa.htm O Summability Factors for N, p B.E. Rhoades Departmet

More information

MAT1026 Calculus II Basic Convergence Tests for Series

MAT1026 Calculus II Basic Convergence Tests for Series MAT026 Calculus II Basic Covergece Tests for Series Egi MERMUT 202.03.08 Dokuz Eylül Uiversity Faculty of Sciece Departmet of Mathematics İzmir/TURKEY Cotets Mootoe Covergece Theorem 2 2 Series of Real

More information

On Weak and Strong Convergence Theorems for a Finite Family of Nonself I-asymptotically Nonexpansive Mappings

On Weak and Strong Convergence Theorems for a Finite Family of Nonself I-asymptotically Nonexpansive Mappings Mathematica Moravica Vol. 19-2 2015, 49 64 O Weak ad Strog Covergece Theorems for a Fiite Family of Noself I-asymptotically Noexpasive Mappigs Birol Güdüz ad Sezgi Akbulut Abstract. We prove the weak ad

More information

Definition 4.2. (a) A sequence {x n } in a Banach space X is a basis for X if. unique scalars a n (x) such that x = n. a n (x) x n. (4.

Definition 4.2. (a) A sequence {x n } in a Banach space X is a basis for X if. unique scalars a n (x) such that x = n. a n (x) x n. (4. 4. BASES I BAACH SPACES 39 4. BASES I BAACH SPACES Sice a Baach space X is a vector space, it must possess a Hamel, or vector space, basis, i.e., a subset {x γ } γ Γ whose fiite liear spa is all of X ad

More information

Unsaturated Solutions of A Nonlinear Delay Partial Difference. Equation with Variable Coefficients

Unsaturated Solutions of A Nonlinear Delay Partial Difference. Equation with Variable Coefficients Europea Joural of Mathematics ad Computer Sciece Vol. 5 No. 1 18 ISSN 59-9951 Usaturated Solutios of A Noliear Delay Partial Differece Euatio with Variable Coefficiets Xiagyu Zhu Yuahog Tao* Departmet

More information

Sequences and Series of Functions

Sequences and Series of Functions Chapter 6 Sequeces ad Series of Fuctios 6.1. Covergece of a Sequece of Fuctios Poitwise Covergece. Defiitio 6.1. Let, for each N, fuctio f : A R be defied. If, for each x A, the sequece (f (x)) coverges

More information

Statistically Convergent Double Sequence Spaces in 2-Normed Spaces Defined by Orlicz Function

Statistically Convergent Double Sequence Spaces in 2-Normed Spaces Defined by Orlicz Function Applied Mathematics, 0,, 398-40 doi:0.436/am.0.4048 Published Olie April 0 (http://www.scirp.org/oural/am) Statistically Coverget Double Sequece Spaces i -Normed Spaces Defied by Orlic Fuctio Abstract

More information

WEAK AND STRONG LAWS OF LARGE NUMBERS OF NEGATIVELY ASSOCIATED RANDOM VARIABLES

WEAK AND STRONG LAWS OF LARGE NUMBERS OF NEGATIVELY ASSOCIATED RANDOM VARIABLES WEAK AND STRONG LAWS OF LARGE NUMBERS OF NEGATIVELY ASSOCIATED RANDOM VARIABLES A Thesis Submitted to the Faculty of Graduate Studies ad Research I Partial Fulfillmet of the Requiremets for the Degree

More information

Theorem 3. A subset S of a topological space X is compact if and only if every open cover of S by open sets in X has a finite subcover.

Theorem 3. A subset S of a topological space X is compact if and only if every open cover of S by open sets in X has a finite subcover. Compactess Defiitio 1. A cover or a coverig of a topological space X is a family C of subsets of X whose uio is X. A subcover of a cover C is a subfamily of C which is a cover of X. A ope cover of X is

More information

The 4-Nicol Numbers Having Five Different Prime Divisors

The 4-Nicol Numbers Having Five Different Prime Divisors 1 2 3 47 6 23 11 Joural of Iteger Sequeces, Vol. 14 (2011), Article 11.7.2 The 4-Nicol Numbers Havig Five Differet Prime Divisors Qiao-Xiao Ji ad Mi Tag 1 Departmet of Mathematics Ahui Normal Uiversity

More information

Solutions of Homework 2.

Solutions of Homework 2. 1 Solutios of Homework 2. 1. Suppose X Y with E(Y )

More information

ECE 330:541, Stochastic Signals and Systems Lecture Notes on Limit Theorems from Probability Fall 2002

ECE 330:541, Stochastic Signals and Systems Lecture Notes on Limit Theorems from Probability Fall 2002 ECE 330:541, Stochastic Sigals ad Systems Lecture Notes o Limit Theorems from robability Fall 00 I practice, there are two ways we ca costruct a ew sequece of radom variables from a old sequece of radom

More information

ki, X(n) lj (n) = (ϱ (n) ij ) 1 i,j d.

ki, X(n) lj (n) = (ϱ (n) ij ) 1 i,j d. APPLICATIONES MATHEMATICAE 22,2 (1994), pp. 193 200 M. WIŚNIEWSKI (Kielce) EXTREME ORDER STATISTICS IN AN EQUALLY CORRELATED GAUSSIAN ARRAY Abstract. This paper cotais the results cocerig the wea covergece

More information

Limit distributions for products of sums

Limit distributions for products of sums Statistics & Probability Letters 62 (23) 93 Limit distributios for products of sums Yogcheg Qi Departmet of Mathematics ad Statistics, Uiversity of Miesota-Duluth, Campus Ceter 4, 7 Uiversity Drive, Duluth,

More information

Proof of a conjecture of Amdeberhan and Moll on a divisibility property of binomial coefficients

Proof of a conjecture of Amdeberhan and Moll on a divisibility property of binomial coefficients Proof of a cojecture of Amdeberha ad Moll o a divisibility property of biomial coefficiets Qua-Hui Yag School of Mathematics ad Statistics Najig Uiversity of Iformatio Sciece ad Techology Najig, PR Chia

More information

Advanced Stochastic Processes.

Advanced Stochastic Processes. Advaced Stochastic Processes. David Gamarik LECTURE 2 Radom variables ad measurable fuctios. Strog Law of Large Numbers (SLLN). Scary stuff cotiued... Outlie of Lecture Radom variables ad measurable fuctios.

More information

Approximation theorems for localized szász Mirakjan operators

Approximation theorems for localized szász Mirakjan operators Joural of Approximatio Theory 152 (2008) 125 134 www.elsevier.com/locate/jat Approximatio theorems for localized szász Miraja operators Lise Xie a,,1, Tigfa Xie b a Departmet of Mathematics, Lishui Uiversity,

More information

Properties of Fuzzy Length on Fuzzy Set

Properties of Fuzzy Length on Fuzzy Set Ope Access Library Joural 206, Volume 3, e3068 ISSN Olie: 2333-972 ISSN Prit: 2333-9705 Properties of Fuzzy Legth o Fuzzy Set Jehad R Kider, Jaafar Imra Mousa Departmet of Mathematics ad Computer Applicatios,

More information

BIRKHOFF ERGODIC THEOREM

BIRKHOFF ERGODIC THEOREM BIRKHOFF ERGODIC THEOREM Abstract. We will give a proof of the poitwise ergodic theorem, which was first proved by Birkhoff. May improvemets have bee made sice Birkhoff s orgial proof. The versio we give

More information

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall 2013 Lecture 6 9/23/2013. Brownian motion. Introduction

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall 2013 Lecture 6 9/23/2013. Brownian motion. Introduction MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/5.070J Fall 203 Lecture 6 9/23/203 Browia motio. Itroductio Cotet.. A heuristic costructio of a Browia motio from a radom walk. 2. Defiitio ad basic properties

More information

Probability for mathematicians INDEPENDENCE TAU

Probability for mathematicians INDEPENDENCE TAU Probability for mathematicias INDEPENDENCE TAU 2013 28 Cotets 3 Ifiite idepedet sequeces 28 3a Idepedet evets........................ 28 3b Idepedet radom variables.................. 33 3 Ifiite idepedet

More information

Integrable Functions. { f n } is called a determining sequence for f. If f is integrable with respect to, then f d does exist as a finite real number

Integrable Functions. { f n } is called a determining sequence for f. If f is integrable with respect to, then f d does exist as a finite real number MATH 532 Itegrable Fuctios Dr. Neal, WKU We ow shall defie what it meas for a measurable fuctio to be itegrable, show that all itegral properties of simple fuctios still hold, ad the give some coditios

More information

1 = δ2 (0, ), Y Y n nδ. , T n = Y Y n n. ( U n,k + X ) ( f U n,k + Y ) n 2n f U n,k + θ Y ) 2 E X1 2 X1

1 = δ2 (0, ), Y Y n nδ. , T n = Y Y n n. ( U n,k + X ) ( f U n,k + Y ) n 2n f U n,k + θ Y ) 2 E X1 2 X1 8. The cetral limit theorems 8.1. The cetral limit theorem for i.i.d. sequeces. ecall that C ( is N -separatig. Theorem 8.1. Let X 1, X,... be i.i.d. radom variables with EX 1 = ad EX 1 = σ (,. Suppose

More information

(A sequence also can be thought of as the list of function values attained for a function f :ℵ X, where f (n) = x n for n 1.) x 1 x N +k x N +4 x 3

(A sequence also can be thought of as the list of function values attained for a function f :ℵ X, where f (n) = x n for n 1.) x 1 x N +k x N +4 x 3 MATH 337 Sequeces Dr. Neal, WKU Let X be a metric space with distace fuctio d. We shall defie the geeral cocept of sequece ad limit i a metric space, the apply the results i particular to some special

More information

A NOTE ON INVARIANT SETS OF ITERATED FUNCTION SYSTEMS

A NOTE ON INVARIANT SETS OF ITERATED FUNCTION SYSTEMS Acta Math. Hugar., 2007 DOI: 10.1007/s10474-007-7013-6 A NOTE ON INVARIANT SETS OF ITERATED FUNCTION SYSTEMS L. L. STACHÓ ad L. I. SZABÓ Bolyai Istitute, Uiversity of Szeged, Aradi vértaúk tere 1, H-6720

More information

Math 61CM - Solutions to homework 3

Math 61CM - Solutions to homework 3 Math 6CM - Solutios to homework 3 Cédric De Groote October 2 th, 208 Problem : Let F be a field, m 0 a fixed oegative iteger ad let V = {a 0 + a x + + a m x m a 0,, a m F} be the vector space cosistig

More information

The Borel-Cantelli Lemma and its Applications

The Borel-Cantelli Lemma and its Applications The Borel-Catelli Lemma ad its Applicatios Ala M. Falleur Departmet of Mathematics ad Statistics The Uiversity of New Mexico Albuquerque, New Mexico, USA Dig Li Departmet of Electrical ad Computer Egieerig

More information

STAT Homework 1 - Solutions

STAT Homework 1 - Solutions STAT-36700 Homework 1 - Solutios Fall 018 September 11, 018 This cotais solutios for Homework 1. Please ote that we have icluded several additioal commets ad approaches to the problems to give you better

More information

Common Coupled Fixed Point of Mappings Satisfying Rational Inequalities in Ordered Complex Valued Generalized Metric Spaces

Common Coupled Fixed Point of Mappings Satisfying Rational Inequalities in Ordered Complex Valued Generalized Metric Spaces IOSR Joural of Mathematics (IOSR-JM) e-issn: 78-578, p-issn:319-765x Volume 10, Issue 3 Ver II (May-Ju 014), PP 69-77 Commo Coupled Fixed Poit of Mappigs Satisfyig Ratioal Iequalities i Ordered Complex

More information

ON POINTWISE BINOMIAL APPROXIMATION

ON POINTWISE BINOMIAL APPROXIMATION Iteratioal Joural of Pure ad Applied Mathematics Volume 71 No. 1 2011, 57-66 ON POINTWISE BINOMIAL APPROXIMATION BY w-functions K. Teerapabolar 1, P. Wogkasem 2 Departmet of Mathematics Faculty of Sciece

More information

2.2. Central limit theorem.

2.2. Central limit theorem. 36.. Cetral limit theorem. The most ideal case of the CLT is that the radom variables are iid with fiite variace. Although it is a special case of the more geeral Lideberg-Feller CLT, it is most stadard

More information

The log-behavior of n p(n) and n p(n)/n

The log-behavior of n p(n) and n p(n)/n Ramauja J. 44 017, 81-99 The log-behavior of p ad p/ William Y.C. Che 1 ad Ke Y. Zheg 1 Ceter for Applied Mathematics Tiaji Uiversity Tiaji 0007, P. R. Chia Ceter for Combiatorics, LPMC Nakai Uivercity

More information

Erratum to: An empirical central limit theorem for intermittent maps

Erratum to: An empirical central limit theorem for intermittent maps Probab. Theory Relat. Fields (2013) 155:487 491 DOI 10.1007/s00440-011-0393-0 ERRATUM Erratum to: A empirical cetral limit theorem for itermittet maps J. Dedecker Published olie: 25 October 2011 Spriger-Verlag

More information

HÖLDER SUMMABILITY METHOD OF FUZZY NUMBERS AND A TAUBERIAN THEOREM

HÖLDER SUMMABILITY METHOD OF FUZZY NUMBERS AND A TAUBERIAN THEOREM Iraia Joural of Fuzzy Systems Vol., No. 4, (204 pp. 87-93 87 HÖLDER SUMMABILITY METHOD OF FUZZY NUMBERS AND A TAUBERIAN THEOREM İ. C. ANAK Abstract. I this paper we establish a Tauberia coditio uder which

More information

IJITE Vol.2 Issue-11, (November 2014) ISSN: Impact Factor

IJITE Vol.2 Issue-11, (November 2014) ISSN: Impact Factor IJITE Vol Issue-, (November 4) ISSN: 3-776 ATTRACTIVITY OF A HIGHER ORDER NONLINEAR DIFFERENCE EQUATION Guagfeg Liu School of Zhagjiagag Jiagsu Uiversit of Sciece ad Techolog, Zhagjiagag, Jiagsu 56,PR

More information

Research Article A Note on Ergodicity of Systems with the Asymptotic Average Shadowing Property

Research Article A Note on Ergodicity of Systems with the Asymptotic Average Shadowing Property Discrete Dyamics i Nature ad Society Volume 2011, Article ID 360583, 6 pages doi:10.1155/2011/360583 Research Article A Note o Ergodicity of Systems with the Asymptotic Average Shadowig Property Risog

More information

Notes 5 : More on the a.s. convergence of sums

Notes 5 : More on the a.s. convergence of sums Notes 5 : More o the a.s. covergece of sums Math 733-734: Theory of Probability Lecturer: Sebastie Roch Refereces: Dur0, Sectios.5; Wil9, Sectio 4.7, Shi96, Sectio IV.4, Dur0, Sectio.. Radom series. Three-series

More information

2.1. Convergence in distribution and characteristic functions.

2.1. Convergence in distribution and characteristic functions. 3 Chapter 2. Cetral Limit Theorem. Cetral limit theorem, or DeMoivre-Laplace Theorem, which also implies the wea law of large umbers, is the most importat theorem i probability theory ad statistics. For

More information

Probability and Statistics

Probability and Statistics ICME Refresher Course: robability ad Statistics Staford Uiversity robability ad Statistics Luyag Che September 20, 2016 1 Basic robability Theory 11 robability Spaces A probability space is a triple (Ω,

More information

MONOTONICITY OF SEQUENCES INVOLVING GEOMETRIC MEANS OF POSITIVE SEQUENCES WITH LOGARITHMICAL CONVEXITY

MONOTONICITY OF SEQUENCES INVOLVING GEOMETRIC MEANS OF POSITIVE SEQUENCES WITH LOGARITHMICAL CONVEXITY MONOTONICITY OF SEQUENCES INVOLVING GEOMETRIC MEANS OF POSITIVE SEQUENCES WITH LOGARITHMICAL CONVEXITY FENG QI AND BAI-NI GUO Abstract. Let f be a positive fuctio such that x [ f(x + )/f(x) ] is icreasig

More information

Bull. Korean Math. Soc. 36 (1999), No. 3, pp. 451{457 THE STRONG CONSISTENCY OF NONLINEAR REGRESSION QUANTILES ESTIMATORS Seung Hoe Choi and Hae Kyung

Bull. Korean Math. Soc. 36 (1999), No. 3, pp. 451{457 THE STRONG CONSISTENCY OF NONLINEAR REGRESSION QUANTILES ESTIMATORS Seung Hoe Choi and Hae Kyung Bull. Korea Math. Soc. 36 (999), No. 3, pp. 45{457 THE STRONG CONSISTENCY OF NONLINEAR REGRESSION QUANTILES ESTIMATORS Abstract. This paper provides suciet coditios which esure the strog cosistecy of regressio

More information

A FIXED POINT THEOREM IN THE MENGER PROBABILISTIC METRIC SPACE. Abdolrahman Razani (Received September 2004)

A FIXED POINT THEOREM IN THE MENGER PROBABILISTIC METRIC SPACE. Abdolrahman Razani (Received September 2004) NEW ZEALAND JOURNAL OF MATHEMATICS Volume 35 (2006), 109 114 A FIXED POINT THEOREM IN THE MENGER PROBABILISTIC METRIC SPACE Abdolrahma Razai (Received September 2004) Abstract. I this article, a fixed

More information

Estimation of the essential supremum of a regression function

Estimation of the essential supremum of a regression function Estimatio of the essetial supremum of a regressio fuctio Michael ohler, Adam rzyżak 2, ad Harro Walk 3 Fachbereich Mathematik, Techische Uiversität Darmstadt, Schlossgartestr. 7, 64289 Darmstadt, Germay,

More information

It is always the case that unions, intersections, complements, and set differences are preserved by the inverse image of a function.

It is always the case that unions, intersections, complements, and set differences are preserved by the inverse image of a function. MATH 532 Measurable Fuctios Dr. Neal, WKU Throughout, let ( X, F, µ) be a measure space ad let (!, F, P ) deote the special case of a probability space. We shall ow begi to study real-valued fuctios defied

More information

STRONG DEVIATION THEOREMS FOR THE SEQUENCE OF CONTINUOUS RANDOM VARIABLES AND THE APPROACH OF LAPLACE TRANSFORM

STRONG DEVIATION THEOREMS FOR THE SEQUENCE OF CONTINUOUS RANDOM VARIABLES AND THE APPROACH OF LAPLACE TRANSFORM Joural of Statitic: Advace i Theory ad Applicatio Volume, Number, 9, Page 35-47 STRONG DEVIATION THEORES FOR THE SEQUENCE OF CONTINUOUS RANDO VARIABLES AND THE APPROACH OF LAPLACE TRANSFOR School of athematic

More information

Lecture 2. The Lovász Local Lemma

Lecture 2. The Lovász Local Lemma Staford Uiversity Sprig 208 Math 233A: No-costructive methods i combiatorics Istructor: Ja Vodrák Lecture date: Jauary 0, 208 Origial scribe: Apoorva Khare Lecture 2. The Lovász Local Lemma 2. Itroductio

More information

NEW FAST CONVERGENT SEQUENCES OF EULER-MASCHERONI TYPE

NEW FAST CONVERGENT SEQUENCES OF EULER-MASCHERONI TYPE UPB Sci Bull, Series A, Vol 79, Iss, 207 ISSN 22-7027 NEW FAST CONVERGENT SEQUENCES OF EULER-MASCHERONI TYPE Gabriel Bercu We itroduce two ew sequeces of Euler-Mascheroi type which have fast covergece

More information

This section is optional.

This section is optional. 4 Momet Geeratig Fuctios* This sectio is optioal. The momet geeratig fuctio g : R R of a radom variable X is defied as g(t) = E[e tx ]. Propositio 1. We have g () (0) = E[X ] for = 1, 2,... Proof. Therefore

More information

SHARP INEQUALITIES INVOLVING THE CONSTANT e AND THE SEQUENCE (1 + 1/n) n

SHARP INEQUALITIES INVOLVING THE CONSTANT e AND THE SEQUENCE (1 + 1/n) n SHARP INEQUALITIES INVOLVING THE CONSTANT e AND THE SEQUENCE + / NECDET BATIR Abstract. Several ew ad sharp iequalities ivolvig the costat e ad the sequece + / are proved.. INTRODUCTION The costat e or

More information

Entropy Rates and Asymptotic Equipartition

Entropy Rates and Asymptotic Equipartition Chapter 29 Etropy Rates ad Asymptotic Equipartitio Sectio 29. itroduces the etropy rate the asymptotic etropy per time-step of a stochastic process ad shows that it is well-defied; ad similarly for iformatio,

More information

Sequences of Definite Integrals, Factorials and Double Factorials

Sequences of Definite Integrals, Factorials and Double Factorials 47 6 Joural of Iteger Sequeces, Vol. 8 (5), Article 5.4.6 Sequeces of Defiite Itegrals, Factorials ad Double Factorials Thierry Daa-Picard Departmet of Applied Mathematics Jerusalem College of Techology

More information

Research Article Complete Convergence for Maximal Sums of Negatively Associated Random Variables

Research Article Complete Convergence for Maximal Sums of Negatively Associated Random Variables Hidawi Publishig Corporatio Joural of Probability ad Statistics Volume 010, Article ID 764043, 17 pages doi:10.1155/010/764043 Research Article Complete Covergece for Maximal Sums of Negatively Associated

More information

Several properties of new ellipsoids

Several properties of new ellipsoids Appl. Math. Mech. -Egl. Ed. 008 9(7):967 973 DOI 10.1007/s10483-008-0716-y c Shaghai Uiversity ad Spriger-Verlag 008 Applied Mathematics ad Mechaics (Eglish Editio) Several properties of ew ellipsoids

More information

MAS111 Convergence and Continuity

MAS111 Convergence and Continuity MAS Covergece ad Cotiuity Key Objectives At the ed of the course, studets should kow the followig topics ad be able to apply the basic priciples ad theorems therei to solvig various problems cocerig covergece

More information