J. Stat. Appl. Pro. Lett. 2, No. 1, (2015) 15
|
|
- Antony Jefferson
- 5 years ago
- Views:
Transcription
1 J. Stat. Appl. Pro. Lett. 2, No. 1, Joural of Statistics Applicatios & Probability Letters A Iteratioal Joural Martigale Method for Rui Probabilityi a Geeralized Ris Process uder Rates of Iterest with Homogeous Marov Chai Premiums ad Homogeous Marov Chai Iterests Phug Duy Quag Departmet of Mathematics, Foreig Trade Uiversity, 91- Chua Lag, Ha oi, Viet Nam Received: 16 May 2014, Revised: 1 Jul. 2014, Accepted: 7 Jul Published olie: 1 Ja Abstract: This paper gives upper bouds for rui probabilities of geeralized ris processes uder rates of iterest with homogeous Marov chai premiums ad Hemogeeous Marov chai Iterests. We assume that premium ad rate of iterest tae a coutable umber of o-egative values. Geeralized Ludberg iequalities for rui probabilities of these processes are derived by the Martigale approach. Keywords: Supermartigale, Optioal stoppig theorem, Rui probability, Homogeeous Marov chai. 1 Itroductio I recet years, the classical ris process has bee exteded to more practical ad real situatios. For most of the ivestigatios treated i ris theory, it is very sigificat to deal with the riss that rise from moetary iflatio i the isurace ad fiace maret, ad also to cosider the operatio ucertaities i admiistratio of fiacial capital. Teugels ad Sudt [9,10] cosidered the effects of costat rate o the rui probability uder the compoud Poisso ris model. Yag [12] built both expoetial ad o expoetial upper bouds for rui probabilities i a ris model with costat iterest force ad idepedet premiums ad claims. Xu ad Wag [11] give upper bouds for rui probabilities i a ris model with iterest force ad idepedet premiums ad claims with Marov chai iterest rate. Cai [1] studied the rui probabilities i two ris models, with idepedet premiums ad claims with the iterest rates is formed a sequece of i.i.d radom variables. I Cai [2], the author studied the rui probabilities i two ris models, with idepedet premiums ad claims ad used a fst order autoregressive process to model the rates of i iterest. I Cai ad Dicso [3], the authors give Ludberg iequalities for rui probabilities i two discrete- time ris process with a Marov chai iterest model ad idepedet premiums ad claims. Feglog Guo ad Digcheg Wag [4] used recursive techique to build Ludberg iequalities for rui probabilities i two discrete- time ris process with the premiums, claims ad rates of iterest have autoregressive ovig average ARMA depedet structures simultaeously. P. D. Quag [5] used martigale approach to build upper bouds for rui probabilities i a ris model with iterest force ad idepedet iterest rates ad premiums, Marov chai claims. P. D. Quag [6] used martigale approach to build upper bouds for rui probabilities i a ris model with iterest force ad idepedet iterest rates, Marov chai claims ad Marov chai premiums. P. D. Quag [7] used martigale approach to build upper bouds for rui probabilities i a ris model with iterest force ad idepedet premiums, Marov chai claims ad Marov chai iterests. P. D. Quag [8] also used recursive approach to build upper bouds for rui probabilities i a ris model with iterest force ad Marov chai premiums, Marov chai claims, while the iterest rates follow a fst-order autoregressive processes. I this paper, we study the models cosidered by Cai ad Dicso [3] to the case homogeous marov chai premiums, homogeous marov chai iterests ad idepedet claims. The mai differece betwee the model i our paper ad the Correspodig author quagmathftu@yahoo.com Natural Scieces Publishig Cor.
2 16 P. D. Quag: Martigale Method for Rui Probabilityi a Geeralized Ris Process... oe i Cai ad Dicso [3] is that premiums, iterests i our model are assumed to follow homogeeous Marov chais. Geeralized Ludberg iequalities for rui probabilities of these processes are derived by the Martigale approach. 2 The Model ad the Basic Assumptios I this paper, we study the discrete time ris models with X =X 0 are premiums, Y =Y 0 are claims, I=I 0 are iterests ad X,Y ad I are assumed to be idepedet. To establish probability iequalities for rui probabilities of these models, we cosider two style of premium collectios. O oe had of the premiums are collected at the begiig of each period the the surplus process which ca be rearraged as U 1 U 1 = u. 1 with iitial surplus U1 o = u>0 ca be writte as U 1 = U I +X Y, 1 1+I + X Y j=+1 1+I j. 2 O the other had, if the premiums are collected at the ed of each period, the the surplus process surplus U 2 o which is equivalet to = u>0 ca be writte as U 2 = u. U 2 1 with iitial U 2 =U X 1+I Y, 3 1+I + [X 1+I Y ] where throughout this paper, we deote b t=a x t = 1 ad b t=a x t = 0 if a>b. j=+1 1+I j. 4 I this paper, we cosider models 1 ad 3, i which X =X 0 is a homogeeous Marov chai, X tae values i a set of o - egative umbers G X =x 1,x 2,...,x m,... with X o = x i ad where 0 p i j 1, + p i j = 1. p i j = P [ X m+1 = x j Xm = x i ],m N,xi,x j G X, We also assume that I =I 0 is homogeeous Marov chai, I tae values i a set of o - egative umbers G I = i 1,i 2,...,i,... with I o = i r ad where 0 q rs 1, + q rs = 1. s=1 q rs = P[I m+1 = i s X m = i r ],m N,i r,i s G I, I additio, Y =Y 0 is sequece of idepedet ad idetically distributed o egative cotiuous radom variables with the same distributio fuctio Fy=PY o y. Based o the previous assumptios, we defie the fiite time ad ultimate rui probabilities i model 1 respectively, by ψ 1 u,x i,i r =P U 1 < 0 ψ 1 u,x i,i r = lim ψ 1 U1 u,x i,i r =P, 5 < 0 U 1 U1. 6 Natural Scieces Publishig Cor.
3 J. Stat. Appl. Pro. Lett. 2, No. 1, / 17 Similarly, we defie the fiite time ad ultimate rui probabilities i model 3 respectively, by ψ 2 u,x i,i r =P U 2 < 0, 7 ψ 2 u,x i,i r = lim ψ 2 U2 u,x i,i r =P < 0 U 2 U2. 8 I this paper, we derive probability iequalities for ψ 1 u,x i,i r ad ψ 2 u,x i,i r by the Martigale approach. 3 Upper Bouds for Rui Probability by the Martigale approach To establish probability iequalities for rui probabilities of model 1, we fst prove the followig Lemma. Lemma 3.1. Let model 1. If ay x i G X,i r G I, Y 1 X 1 1+I 1 1 Xo = x i,i o = i r < 0 ad P Y 1 X 1 1+I 1 1 > 0 X o = x i,i o = i r >0, 9 the there exists a uique positive costat R satisfyig: e R Y 1 X 1 1+I 1 1 Xo = x i,i o = i r = Proof. Defie We have f t= e ty 1 X 1 1+I 1 1 Xo = x i,i o = i r 1, t 0,+ f t= Y 1 X 1 1+I 1 1 e ty 1 X 1 1+I 1 1 Xo = x i,i o = i r, [ f t= Y 1 X 1 1+I 1 1] 2 e ty 1 X 1 1+I 1 1 X o = x i,i o = i r. Hece f t 0. This implies that f t is a covex fuctio with f 0=0 11 ad f 0= Y 1 X 1 1+I 1 1 Xo = x i,i o = i r <0. 12 By P Y 1 X 1 1+I 1 1 > 0 X o = x i,i o = i r >0, we ca fid some costat δ > 0 such that P Y 1 X 1 1+I 1 1 > δ > 0 X o = x i,i o = i r >0. The, we get This implies that f t= e ty 1 X 1 1+I 1 1 Xo = x i,i o = i r 1 e ty 1 X 1 1+I 1 1 Xo = x i,i o = i r e tδ.p.1 Y1 X 1 1+I 1 1 >δ X o =x i,i o =i r Y 1 X 1 1+I 1 1 > δ Xo = x i,i o = i r 1. 1 lim f t=+. 13 t + Natural Scieces Publishig Cor.
4 18 P. D. Quag: Martigale Method for Rui Probabilityi a Geeralized Ris Process... From 11, 12 ad 13 there exists a uique positive costat R satisfyig 10. This completes the proof. Let: R o = if R > 0 : e R Y 1 X 1 1+I 1 1 Xo = x i,i o = i r =1,x i G X,i r G I. Remar 3.1. e R oy 1 X 1 1+I 1 1 Xo = x i,i o = i r 1. To establish probability iequalities for rui probabilities of model 1, we prove the followig Theorem. Theorem 3.1. Let model 1. Uder the coditio of Lemma 3.1 ad R o > 0, the for ay u>0, x i G X,i r G I, Proof. Cosider the process U 1 V 1 ad S 1 = e R ov 1. Thus, we have is give by 2, we let = U 1 ψ 1 u,x i,i r e R ou I j 1 = u+ S 1 +1 = S1 X j Y j e R +1 ox +1 Y +1 j 1+I t 1. With ay 1, we have S 1 +1 X 1,X 2,...,X,Y 1,Y 2,...,Y,I 1,I 2,...,I = S 1 e R ox +1 Y I t 1 X 1,X 2,...,X,Y 1,Y 2,...,Y,I 1,I 2,...,I = S 1 e R ox +1 Y +1 1+I I t 1 X 1,X 2,...,X,I 1,I 2,...,I 1+I t 1, 15. From 0 1+I t 1 1 ad Jese s iequality implies S 1 e R ox +1 Y +1 1+I I t 1 X 1,X 2,...,X,I 1,I 2,...,I S 1 e R ox +1 Y +1 1+I +1 1 t X1,X 2,...,X,I 1,I 2,...,I 1+I 1. I additio, e R ox +1 Y +1 1+I +1 1 X1,X 2,...,X,I 1,I 2,...,I e R ox +1 Y +1 1+I +1 1 X,I e R ox 1 Y 1 1+I 1 1 Xo,I o 1. Thus, we have S 1 +1 X 1,X 2,...,X,Y 1,Y 2,...,Y,I 1,I 2,...,I Hece,,=1,2,... is a supermartigale with respect to the σ - filtratio S 1 I 1 = σx 1,...,X,Y 1,...,Y,I 1,...,I S 1. Natural Scieces Publishig Cor.
5 J. Stat. Appl. Pro. Lett. 2, No. 1, / 19 Defie T 1 = mi : V 1 < 0 U 1, with V 1 is give by 15. Hece, T 1 is a stoppig time ad T 1 = mi,t 1 is a fiite stoppig time. Therefore, from the optioal stopplig theorem for supermartigales, we have S 1 T 1 S 1 o =e R ou. This implies that e Rou S 1 S 1 T 1 T 1 S 1 T T.1 1 T e R o V 1 T 1.1 T From V 1 < 0 the 16 becomes T 1 I additio, ψ 1 u,x i,i r =P Combiig 17 ad 18 imply that = P e Rou U 1 V 1 This complete the proof. Similarly to Lemma 3.1, we have Lemma 3.2. Lemma 3.2. Let model 3. Ay x i G X,i r G Y, if 1 1 T U1 = PT < 0 U1 < 0 = PT ψ 1 u,x i,i r e R ou. 19 Y 1 1+I 1 1 Xo X 1 = x i,i o = i r < 0 ad P Y 1 1+I 1 1 X 1 > 0 X o = x i,i o = i r >0, 20 the, there exists a uique positive costat R satisfyig e R [Y 1 1+I 1 1 X 1] Xo = x i,i o = i r = Let R o = if R > 0 : Remar 3.2. e R oy 1 1+I 1 1 X 1 X o = x i,i o = i r = 1,x i G X,i r G I. X o = x i,i o = i r 1. e R Y 1 1+I 1 1 X 1 Similarly, we establish probability iequalities for rui probabilities of model 3 by provig the followig Theorem. Theorem 3.2. Let model 3. Uder the coditios of Lemma 3.2 ad R o > 0,the for ay u>0,x i G X,i r G r, ψ 2 u,x i,i r e R ou 22 Natural Scieces Publishig Cor.
6 20 P. D. Quag: Martigale Method for Rui Probabilityi a Geeralized Ris Process... Proof. Cosider the process U 2 give by 4, we let V 2 = U 2 ad S 2 = e R ov 2. Thus, we have 1+I j 1 = u+ S 2 +1 = S2 X j 1+I j Y j e R ox +1 Y +1 1+I +1 1 j 1+I t 1. 1+I t 1, 23 With ay 1, we have S 2 +1 X 1,X 2,...,X,Y 1,Y 2,...,Y,I 1,I 2,...,I = S 2 e R ox +1 Y +1 1+I I t 1 X 1,X 2,...,X,Y 1,Y 2,...,Y,I 1,I 2,...,I = S 2 e R ox +1 Y +1 1+I I t 1 X 1,X 2,...,X,I 1,I 2,...,I. From 0 1+I t 1 1 ad Jese s iequality implies S 2 X 1,X 2,...,X,Y 1,Y 2,...,Y,I 1,I 2,...,I S 2 e R ox +1 Y +1 1+I +1 1 t X1,X 2,...,X,I 1,I 2,...,I 1+I 1. I additio, e R ox +1 Y +1 1+I +1 1 X1,X 2,...,X,I 1,I 2,...,I e R ox +1 Y +1 1+I +1 1 X,I e R ox 1 Y 1 1+I 1 1 Xo,I o 1. Thus, we have S 2 +1 X 1,X 2,...,X,Y 1,Y 2,...,Y,I 1,I 2,...,I Hece,,=1,2,... is a supermartigale with respect to the σ - filtratio S 2 I 2 = σx 1,...,X,Y 1,...,Y,I 1,...,I. S 2 Defie T 2 = mi : V 2 < 0 U 2 with V 2 is give by 23. Hece, T 2 is a stoppig ad T 2 = mi,t 2 is a fiite stoppig time. Therefore, from the optioal stoppig theorem for supermartigales, we have S 2 T 2 S 2 o =e R ou. Natural Scieces Publishig Cor.
7 J. Stat. Appl. Pro. Lett. 2, No. 1, / 21 This implies that e R ou S 2 T 2 S 2 T 2 S 2 T T.1 2 T e R o V 2 T 2.1 T From V 2 < 0 the 24 becomes T 2 I additio, Combiig 25 ad 26 imply that e Rou ψ 2 u,x i,i r =P Thus, 22 follows by lettig i 27. This completes the proof. = P 1 2 T U 2 V 2 = PT < 0 U2 < 0 U2 = PT ψ 2 u,x i,i r e R ou Coclusio Our mai results i this paper, Theorem 3.1 ad Theorem 3.2 give upper bouds for ψ 1 u,x i,i r ad ψ 2 u,x i,i r by the Martigale approach with homogeous Marov chai premiums ad Hemogeeous Marov chai Iterests. To obtai Therem 3.1 ad Theorem 3.2, fst, we obtai importat prelimiary results, Lemma 3.1 ad Lemma 3.2, which give Ludbergs costats. There remai may ope issues - e.g. a extedig results of this article to cosider X =X 0 ad I=I 0 are homogeous Marov chais, Y =Y 0 is a fst - order autoregressive process; b buildig umerical examples for ψ 1 u,x i,r r ad ψ 2 u,x i,r r by the martigale approach; c Let τ m := if 1 U m < 0 m=1,2 be the time of rui. Ca we calculate or estimate quatities such as τ m. Further research i some of these dectio is i progress. Acowledgemet The authors would lie to tha the ditor ad the reviewers for the helpful commet o a earlier versio of the mauscript which have led to a improvermet of this paper. Refereces [1] J. Cai, Discrete time ris models uder rates of iterest. Probability i the gieerig ad Iformatioal Scieces, 16, 2002, [2] J. Cai, Rui probabilities with depedet rates of iterest, Joural of Applied Probability, 39, 2002, Natural Scieces Publishig Cor.
8 22 P. D. Quag: Martigale Method for Rui Probabilityi a Geeralized Ris Process... [3] J. Cai ad C.M.D. Dicso, Rui Probabilities with a Marov chai iterest model. Isurace: Mathematics ad coomics, 35, 2004, [4] F. Guo ad D Wag, Rui Probabilities with Depedet Rates of Iterest ad Autoregressive Movig Average Structures, Iteratial Joural of Mathematical ad Computatioal Scieces, 62012, [5] P. D. Quag, Upper Bouds for Rui Probability i a Geeralized Ris Process uder iterest force with homogeous Marov chai claims, Asia Joural of Mathematics ad Statistics, Vol 7, No.1, 2014, [6] P. D. Quag, Upper Bouds for Rui Probability i a Geeralized Ris Process uder Rates of Iterest with homogeous Marov chai claims ad homogeous Marov chai premiums, Applied Mathematical Scieces. Vol 8, No. 29, 2014, [7] P. D. Quag, Upper Bouds for Rui Probability i a Geeralized Ris Process uder Rates of Iterest with homogeous Marov chai claims ad homogeous Marov chai Iterests, America Joural of Mathematics ad Statistics, Vol 4, No. 1, 2014, [8] P.D. Quag, Rui Probability i a Geeralized Ris Process uder Rates of Iterest with Depedet Structures, Joural of Statistics Applicatios & Probability Letters, Vol.3, No.3, 2014, 1-9. [9] B. Sudt ad J. L. Teugels, Rui estimates uder iterest force, Isurace: Mathematics ad coomics, 16, 1995, [10] B. Sudt ad J. L. Teugels, The adjustmet fuctio i rui estimates uder iterest force. Isurace: Mathematics ad coomics, 19, 1997, [11] L. Xu ad R. Wag, Upper bouds for rui probabilities i a autoregressive ris model with Marov chai iterest rate, Joural of Idustrial ad Maagemet optimizatio, Vol.2 No.2, 2006, [12] H. Yag, No-expoetial bouds for rui probability with iterest effect icluded, Scadiavia Actuarial Joural, 2, 1999, Natural Scieces Publishig Cor.
Comparison of Minimum Initial Capital with Investment and Non-investment Discrete Time Surplus Processes
The 22 d Aual Meetig i Mathematics (AMM 207) Departmet of Mathematics, Faculty of Sciece Chiag Mai Uiversity, Chiag Mai, Thailad Compariso of Miimum Iitial Capital with Ivestmet ad -ivestmet Discrete Time
More informationThe Martingale Method for Probability of Ultimate Ruin Under Quota -(α, β) Reinsurance Model
J. Stat. Appl. Pro. 5, No. 3, 411-419 2016) 411 Joural of Statistics Applicatios & Probability A Iteratioal Joural http://dx.doi.org/10.18576/jsap/050305 The Martigale Method for Probability of Ultimate
More informationOn forward improvement iteration for stopping problems
O forward improvemet iteratio for stoppig problems Mathematical Istitute, Uiversity of Kiel, Ludewig-Mey-Str. 4, D-24098 Kiel, Germay irle@math.ui-iel.de Albrecht Irle Abstract. We cosider the optimal
More informationSelf-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 informationA Note on Positive Supermartingales in Ruin Theory. Klaus D. Schmidt
A Note o Positive Supermartigales i Rui Theory Klaus D. Schmidt 94-1989 1 A ote o positive supermartigales i rui theory Klaus O. SeHMIOT Semiar für Statistik, Uiversität Maheim, A 5, 0-6800 Maheim, West
More informationMi-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 informationON 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 informationResearch 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 informationPrecise 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 informationLecture 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 informationAn Introduction to Randomized Algorithms
A Itroductio to Radomized Algorithms The focus of this lecture is to study a radomized algorithm for quick sort, aalyze it usig probabilistic recurrece relatios, ad also provide more geeral tools for aalysis
More informationSolutions to HW Assignment 1
Solutios to HW: 1 Course: Theory of Probability II Page: 1 of 6 Uiversity of Texas at Austi Solutios to HW Assigmet 1 Problem 1.1. Let Ω, F, {F } 0, P) be a filtered probability space ad T a stoppig time.
More informationLecture 7: Properties of Random Samples
Lecture 7: Properties of Radom Samples 1 Cotiued From Last Class Theorem 1.1. Let X 1, X,...X be a radom sample from a populatio with mea µ ad variace σ
More informationGeneralized Semi- Markov Processes (GSMP)
Geeralized Semi- Markov Processes (GSMP) Summary Some Defiitios Markov ad Semi-Markov Processes The Poisso Process Properties of the Poisso Process Iterarrival times Memoryless property ad the residual
More informationIntroduction to Extreme Value Theory Laurens de Haan, ISM Japan, Erasmus University Rotterdam, NL University of Lisbon, PT
Itroductio to Extreme Value Theory Laures de Haa, ISM Japa, 202 Itroductio to Extreme Value Theory Laures de Haa Erasmus Uiversity Rotterdam, NL Uiversity of Lisbo, PT Itroductio to Extreme Value Theory
More informationResearch 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 informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall Midterm Solutions
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.65/5.070J Fall 0 Midterm Solutios Problem Suppose a radom variable X is such that P(X > ) = 0 ad P(X > E) > 0 for every E > 0. Recall that the large deviatios rate
More informationProperties 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 informationAsymptotic 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 informationA New Solution Method for the Finite-Horizon Discrete-Time EOQ Problem
This is the Pre-Published Versio. A New Solutio Method for the Fiite-Horizo Discrete-Time EOQ Problem Chug-Lu Li Departmet of Logistics The Hog Kog Polytechic Uiversity Hug Hom, Kowloo, Hog Kog Phoe: +852-2766-7410
More information7.1 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 informationLecture 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 informationMASSACHUSETTS 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 informationResearch Article Approximate Riesz Algebra-Valued Derivations
Abstract ad Applied Aalysis Volume 2012, Article ID 240258, 5 pages doi:10.1155/2012/240258 Research Article Approximate Riesz Algebra-Valued Derivatios Faruk Polat Departmet of Mathematics, Faculty of
More informationHAJEK-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 informationRademacher Complexity
EECS 598: Statistical Learig Theory, Witer 204 Topic 0 Rademacher Complexity Lecturer: Clayto Scott Scribe: Ya Deg, Kevi Moo Disclaimer: These otes have ot bee subjected to the usual scrutiy reserved for
More informationUniform Strict Practical Stability Criteria for Impulsive Functional Differential Equations
Global Joural of Sciece Frotier Research Mathematics ad Decisio Scieces Volume 3 Issue Versio 0 Year 03 Type : Double Blid Peer Reviewed Iteratioal Research Joural Publisher: Global Jourals Ic (USA Olie
More informationSeed and Sieve of Odd Composite Numbers with Applications in Factorization of Integers
IOSR Joural of Mathematics (IOSR-JM) e-issn: 78-578, p-issn: 319-75X. Volume 1, Issue 5 Ver. VIII (Sep. - Oct.01), PP 01-07 www.iosrjourals.org Seed ad Sieve of Odd Composite Numbers with Applicatios i
More informationDecoupling Zeros of Positive Discrete-Time Linear Systems*
Circuits ad Systems,,, 4-48 doi:.436/cs..7 Published Olie October (http://www.scirp.org/oural/cs) Decouplig Zeros of Positive Discrete-Time Liear Systems* bstract Tadeusz Kaczorek Faculty of Electrical
More informationMAXIMAL INEQUALITIES AND STRONG LAW OF LARGE NUMBERS FOR AANA SEQUENCES
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
More informationBerry-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 informationSummary and Discussion on Simultaneous Analysis of Lasso and Dantzig Selector
Summary ad Discussio o Simultaeous Aalysis of Lasso ad Datzig Selector STAT732, Sprig 28 Duzhe Wag May 4, 28 Abstract This is a discussio o the work i Bickel, Ritov ad Tsybakov (29). We begi with a short
More informationHÖLDER 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 informationApproximation 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 informationConfidence interval for the two-parameter exponentiated Gumbel distribution based on record values
Iteratioal Joural of Applied Operatioal Research Vol. 4 No. 1 pp. 61-68 Witer 2014 Joural homepage: www.ijorlu.ir Cofidece iterval for the two-parameter expoetiated Gumbel distributio based o record values
More informationStatistical Analysis on Uncertainty for Autocorrelated Measurements and its Applications to Key Comparisons
Statistical Aalysis o Ucertaity for Autocorrelated Measuremets ad its Applicatios to Key Comparisos Nie Fa Zhag Natioal Istitute of Stadards ad Techology Gaithersburg, MD 0899, USA Outlies. Itroductio.
More informationOptimally Sparse SVMs
A. Proof of Lemma 3. We here prove a lower boud o the umber of support vectors to achieve geeralizatio bouds of the form which we cosider. Importatly, this result holds ot oly for liear classifiers, but
More informationNotes 19 : Martingale CLT
Notes 9 : Martigale CLT Math 733-734: Theory of Probability Lecturer: Sebastie Roch Refereces: [Bil95, Chapter 35], [Roc, Chapter 3]. Sice we have ot ecoutered weak covergece i some time, we first recall
More informationIntroducing a Novel Bivariate Generalized Skew-Symmetric Normal Distribution
Joural of mathematics ad computer Sciece 7 (03) 66-7 Article history: Received April 03 Accepted May 03 Available olie Jue 03 Itroducig a Novel Bivariate Geeralized Skew-Symmetric Normal Distributio Behrouz
More informationA survey on penalized empirical risk minimization Sara A. van de Geer
A survey o pealized empirical risk miimizatio Sara A. va de Geer We address the questio how to choose the pealty i empirical risk miimizatio. Roughly speakig, this pealty should be a good boud for the
More informationStat 319 Theory of Statistics (2) Exercises
Kig Saud Uiversity College of Sciece Statistics ad Operatios Research Departmet Stat 39 Theory of Statistics () Exercises Refereces:. Itroductio to Mathematical Statistics, Sixth Editio, by R. Hogg, J.
More informationIt 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 informationConvergence 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 informationReview 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 informationApproximating the ruin probability of finite-time surplus process with Adaptive Moving Total Exponential Least Square
WSEAS TRANSACTONS o BUSNESS ad ECONOMCS S. Khotama, S. Boothiem, W. Klogdee Approimatig the rui probability of fiite-time surplus process with Adaptive Movig Total Epoetial Least Square S. KHOTAMA, S.
More informationLimit 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 informationEstimation 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 informationComparison Study of Series Approximation. and Convergence between Chebyshev. and Legendre Series
Applied Mathematical Scieces, Vol. 7, 03, o. 6, 3-337 HIKARI Ltd, www.m-hikari.com http://d.doi.org/0.988/ams.03.3430 Compariso Study of Series Approimatio ad Covergece betwee Chebyshev ad Legedre Series
More informationACO Comprehensive Exam 9 October 2007 Student code A. 1. Graph Theory
1. Graph Theory Prove that there exist o simple plaar triagulatio T ad two distict adjacet vertices x, y V (T ) such that x ad y are the oly vertices of T of odd degree. Do ot use the Four-Color Theorem.
More informationChapter 6 Principles of Data Reduction
Chapter 6 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 0 Chapter 6 Priciples of Data Reductio Sectio 6. Itroductio Goal: To summarize or reduce the data X, X,, X to get iformatio about a
More informationResearch Article Some E-J Generalized Hausdorff Matrices Not of Type M
Abstract ad Applied Aalysis Volume 2011, Article ID 527360, 5 pages doi:10.1155/2011/527360 Research Article Some E-J Geeralized Hausdorff Matrices Not of Type M T. Selmaogullari, 1 E. Savaş, 2 ad B. E.
More informationA 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 informationThe random version of Dvoretzky s theorem in l n
The radom versio of Dvoretzky s theorem i l Gideo Schechtma Abstract We show that with high probability a sectio of the l ball of dimesio k cε log c > 0 a uiversal costat) is ε close to a multiple of the
More informationResearch 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 informationThe average-shadowing property and topological ergodicity
Joural of Computatioal ad Applied Mathematics 206 (2007) 796 800 www.elsevier.com/locate/cam The average-shadowig property ad topological ergodicity Rogbao Gu School of Fiace, Najig Uiversity of Fiace
More informationSTRONG 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 informationγn 1 (1 e γ } min min
Hug ad Giag SprigerPlus 20165:79 DOI 101186/s40064-016-1710-y RESEARCH O bouds i Poisso approximatio for distributios of idepedet egative biomial distributed radom variables Tra Loc Hug * ad Le Truog Giag
More informationResearch 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 informationNotes on Snell Envelops and Examples
Notes o Sell Evelops ad Examples Example (Secretary Problem): Coside a pool of N cadidates whose qualificatios are represeted by ukow umbers {a > a 2 > > a N } from best to last. They are iterviewed sequetially
More informationCANTOR SETS WHICH ARE MINIMAL FOR QUASISYMMETRIC MAPS
CANTOR SETS WHICH ARE MINIMAL FOR QUASISYMMETRIC MAPS HRANT A. HAKOBYAN Abstract. We show that middle iterval Cator sets of Hausdorff dimesio are miimal for quasisymmetric maps of a lie. Combiig this with
More informationNEW 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 informationarxiv: v1 [math.pr] 13 Oct 2011
A tail iequality for quadratic forms of subgaussia radom vectors Daiel Hsu, Sham M. Kakade,, ad Tog Zhag 3 arxiv:0.84v math.pr] 3 Oct 0 Microsoft Research New Eglad Departmet of Statistics, Wharto School,
More informationThe Hypergeometric Coupon Collection Problem and its Dual
Joural of Idustrial ad Systes Egieerig Vol., o., pp -7 Sprig 7 The Hypergeoetric Coupo Collectio Proble ad its Dual Sheldo M. Ross Epstei Departet of Idustrial ad Systes Egieerig, Uiversity of Souther
More informationAn analog of the arithmetic triangle obtained by replacing the products by the least common multiples
arxiv:10021383v2 [mathnt] 9 Feb 2010 A aalog of the arithmetic triagle obtaied by replacig the products by the least commo multiples Bair FARHI bairfarhi@gmailcom MSC: 11A05 Keywords: Al-Karaji s triagle;
More informationResearch Article Nonexistence of Homoclinic Solutions for a Class of Discrete Hamiltonian Systems
Abstract ad Applied Aalysis Volume 203, Article ID 39868, 6 pages http://dx.doi.org/0.55/203/39868 Research Article Noexistece of Homocliic Solutios for a Class of Discrete Hamiltoia Systems Xiaopig Wag
More informationB Supplemental Notes 2 Hypergeometric, Binomial, Poisson and Multinomial Random Variables and Borel Sets
B671-672 Supplemetal otes 2 Hypergeometric, Biomial, Poisso ad Multiomial Radom Variables ad Borel Sets 1 Biomial Approximatio to the Hypergeometric Recall that the Hypergeometric istributio is fx = x
More informationResearch 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 informationGeneralization of Contraction Principle on G-Metric Spaces
Global Joural of Pure ad Applied Mathematics. ISSN 0973-1768 Volume 14, Number 9 2018), pp. 1159-1165 Research Idia Publicatios http://www.ripublicatio.com Geeralizatio of Cotractio Priciple o G-Metric
More informationA note on log-concave random graphs
A ote o log-cocave radom graphs Ala Frieze ad Tomasz Tocz Departmet of Mathematical Scieces, Caregie Mello Uiversity, Pittsburgh PA53, USA Jue, 08 Abstract We establish a threshold for the coectivity of
More informationTaylor polynomial solution of difference equation with constant coefficients via time scales calculus
TMSCI 3, o 3, 129-135 (2015) 129 ew Treds i Mathematical Scieces http://wwwtmscicom Taylor polyomial solutio of differece equatio with costat coefficiets via time scales calculus Veysel Fuat Hatipoglu
More informationReconstruction of the Volterra-type integro-differential operator from nodal points
Keski Boudary Value Problems 18 18:47 https://doi.org/1.1186/s13661-18-968- R E S E A R C H Ope Access Recostructio of the Volterra-type itegro-differetial operator from odal poits Baki Keski * * Correspodece:
More information7.1 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 informationEquivalent 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 informationChapter 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 informationLecture 7: October 18, 2017
Iformatio ad Codig Theory Autum 207 Lecturer: Madhur Tulsiai Lecture 7: October 8, 207 Biary hypothesis testig I this lecture, we apply the tools developed i the past few lectures to uderstad the problem
More informationMonte Carlo Optimization to Solve a Two-Dimensional Inverse Heat Conduction Problem
Australia Joural of Basic Applied Scieces, 5(): 097-05, 0 ISSN 99-878 Mote Carlo Optimizatio to Solve a Two-Dimesioal Iverse Heat Coductio Problem M Ebrahimi Departmet of Mathematics, Karaj Brach, Islamic
More informationfor 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 informationA remark on p-summing norms of operators
A remark o p-summig orms of operators Artem Zvavitch Abstract. I this paper we improve a result of W. B. Johso ad G. Schechtma by provig that the p-summig orm of ay operator with -dimesioal domai ca be
More informationComplete 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 informationA note on self-normalized Dickey-Fuller test for unit root in autoregressive time series with GARCH errors
Appl. Math. J. Chiese Uiv. 008, 3(): 97-0 A ote o self-ormalized Dickey-Fuller test for uit root i autoregressive time series with GARCH errors YANG Xiao-rog ZHANG Li-xi Abstract. I this article, the uit
More informationA Further Refinement of Van Der Corput s Inequality
IOSR Joural of Mathematics (IOSR-JM) e-issn: 78-578, p-issn:9-75x Volume 0, Issue Ver V (Mar-Apr 04), PP 7- wwwiosrjouralsorg A Further Refiemet of Va Der Corput s Iequality Amusa I S Mogbademu A A Baiyeri
More informationAxioms of Measure Theory
MATH 532 Axioms of Measure Theory Dr. Neal, WKU I. The Space Throughout the course, we shall let X deote a geeric o-empty set. I geeral, we shall ot assume that ay algebraic structure exists o X so that
More informationThe 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 informationSome 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 informationGamma Distribution and Gamma Approximation
Gamma Distributio ad Gamma Approimatio Xiaomig Zeg a Fuhua (Frak Cheg b a Xiame Uiversity, Xiame 365, Chia mzeg@jigia.mu.edu.c b Uiversity of Ketucky, Leigto, Ketucky 456-46, USA cheg@cs.uky.edu Abstract
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall 2013 Lecture 3 9/11/2013. Large deviations Theory. Cramér s Theorem
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/5.070J Fall 203 Lecture 3 9//203 Large deviatios Theory. Cramér s Theorem Cotet.. Cramér s Theorem. 2. Rate fuctio ad properties. 3. Chage of measure techique.
More informationCentral 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 informationSupplementary Material for Fast Stochastic AUC Maximization with O(1/n)-Convergence Rate
Supplemetary Material for Fast Stochastic AUC Maximizatio with O/-Covergece Rate Migrui Liu Xiaoxua Zhag Zaiyi Che Xiaoyu Wag 3 iabao Yag echical Lemmas ized versio of Hoeffdig s iequality, ote that We
More information(b) What is the probability that a particle reaches the upper boundary n before the lower boundary m?
MATH 529 The Boudary Problem The drukard s walk (or boudary problem) is oe of the most famous problems i the theory of radom walks. Oe versio of the problem is described as follows: Suppose a particle
More informationSTAT 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 informationOn the Variations of Some Well Known Fixed Point Theorem in Metric Spaces
Turkish Joural of Aalysis ad Number Theory, 205, Vol 3, No 2, 70-74 Available olie at http://pubssciepubcom/tjat/3/2/7 Sciece ad Educatio Publishig DOI:0269/tjat-3-2-7 O the Variatios of Some Well Kow
More informationII. EXPANSION MAPPINGS WITH FIXED POINTS
Geeralizatio Of Selfmaps Ad Cotractio Mappig Priciple I D-Metric Space. U.P. DOLHARE Asso. Prof. ad Head,Departmet of Mathematics,D.S.M. College Jitur -431509,Dist. Parbhai (M.S.) Idia ABSTRACT Large umber
More informationCommon 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 informationResearch 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 informationDefinition 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 informationBounds for the Sum of Dependent Risks and Worst. Value-at-Risk with Monotone Marginal Densities
Bouds for the Sum of Depedet Risks ad Worst Value-at-Risk with Mootoe Margial Desities RUODU WANG, LIANG PENG AND JINGPING YANG Jue 23, 202 Abstract I quatitative risk maagemet, it is importat ad challegig
More informationEcon 325/327 Notes on Sample Mean, Sample Proportion, Central Limit Theorem, Chi-square Distribution, Student s t distribution 1.
Eco 325/327 Notes o Sample Mea, Sample Proportio, Cetral Limit Theorem, Chi-square Distributio, Studet s t distributio 1 Sample Mea By Hiro Kasahara We cosider a radom sample from a populatio. Defiitio
More informationLecture 12: Subadditive Ergodic Theorem
Statistics 205b: Probability Theory Sprig 2003 Lecture 2: Subadditive Ergodic Theorem Lecturer: Jim Pitma Scribe: Soghwai Oh 2. The Subadditive Ergodic Theorem This theorem is due
More informationOptimal Two-Choice Stopping on an Exponential Sequence
Sequetial Aalysis, 5: 35 363, 006 Copyright Taylor & Fracis Group, LLC ISSN: 0747-4946 prit/53-476 olie DOI: 0.080/07474940600934805 Optimal Two-Choice Stoppig o a Expoetial Sequece Larry Goldstei Departmet
More informationDisjoint Systems. Abstract
Disjoit Systems Noga Alo ad Bey Sudaov Departmet of Mathematics Raymod ad Beverly Sacler Faculty of Exact Scieces Tel Aviv Uiversity, Tel Aviv, Israel Abstract A disjoit system of type (,,, ) is a collectio
More information