Diagonal approximations by martingales
|
|
- Joseph Whitehead
- 5 years ago
- Views:
Transcription
1 Alea 7, Diagoal approximatios by martigales Jaa Klicarová ad Dalibor Volý Faculty of Ecoomics, Uiversity of South Bohemia, Studetsa 3, , Cese Budejovice, Czech Republic address: LMRS, Uiversité de Roue, Aveue de l Uiversité F 7680 Sait Etiee du Rouvray, Frace address: daliborvoly@uiv-rouefr Abstract the paper, we study ad compare several types of approximatios of statioary processes by martigales troductio Let Ω, A, µ be a probability space with a bijective, bimeasurable ad measure preservig trasformatio T For a measurable fuctio f o Ω, f T i i is a strictly statioary process ad reciprocally, ay strictly statioary process ca be represeted i this way We shall deote Uf = f T; U is a uitary operator i L 2 The fuctio f will be supposed to have a zero mea F i i Z, where T F i = F i+, is a icreasig filtratio of sub-σ-fields of A H i deotes the space of F i -measurable ad square itegrable radom variables P i f is the projectio of f oto the space H i H i, ie P i f = Ef T i F 0 Ef T i+ F 0 We suppose the fuctio f to be regular, ie f = + i= P if Let σ = S f, where is the L 2 -orm ad S f = i= f T i For simplicity of otatio we write Q 0 S f for ES f F 0 ad R S f for S f ES f F Defiitio We say that a fuctio f has a martigale approximatio if there exists a martigale differece sequece m T i i Z such that S f m 2 = o t is a classical result that the martigale approximatio implies the cetral limit theorem Gordi 969 his article, Gordi oticed that if f = i=j P if, j, j the for m = P 0 i= Ui f we have f = m + g Ug with g L 2 hece we get a martigale approximatio Received by the editors February 23, 200; accepted May 8, Mathematics Subject Classificatio 60G0, 60F05, 28D05, 60G42 Key words ad phrases Martigale approximatio, wea diagoal approximatio, strog diagoal approximatio, WW approximatio Supported by KONTAKT grat D MEB
2 258 Jaa Klicarová ad Dalibor Volý Defiitio 2 We say that for a fuctio f there exists a wea diagoal approximatio if there is a sequece N, = o of itegers such that i for f = i= P i f we have S f f 0, σ ii for f = m + g Ug with m = g = i= P 0 U i f we have g = oσ i P i U j f i= i j=0 P i U j f f for the array U i m,, of martigale differeces a CLT holds, we get it for S f/σ as well cf Volý, 2006; this geeralises Gordi s result to processes with oliear growth of variaces of partial sums Haa 973 has oticed that if the series of P i f = P 0 U i f is summable the the series of m = i Z P 0U i f coverges ad gives a martigale approximatio We, moreover, get that for ay ǫ > 0 there is a such that for m = i= P 0U i f, limsup S f m 2 / ǫ; this holds if we replace m by f = i= P if as well Defiitio 3 We say that for a fuctio f there exists a strog diagoal approximatio if there is a iteger sequece of d N, d = o, such that for all iteger sequeces N which satisfy d for all ad = o: i for f = i= P i f we have S f f 0, σ ii for f = m + g Ug with m = g = i= P 0 U i f we have g = oσ i P i U j f i= i j=0 P i U j f The Haa s coditio thus implies the strog diagoal approximatio The Haa s coditio implies the wea ivariace priciple; i full geerality the result was proved i Dedecer et al 2007, the martigale approximatio for o adapted processes ca be foud i Volý 993 Heyde has show that if the series m = i Z P 0U i f coverges i L 2 ad m 2 limif S f 2 the a martigale approximatio exists cf Hall ad Heyde, 980 Volý 993 has show that the covergece i L 2 is ot sufficiet for the martigale approximatio ad that a ucoditioal covergece is a sufficiet coditio
3 Diagoal approximatios by martigales 259 Figure Diagoal ad Wu-Woodroofe approximatio f f T f T 2 f T f 0 T i S f f U m R S f P 2 P P 0 P P 2 0 m 0 g S f f 0 Q 0 S f Defiitio 4 We say that a fuctio f has Wu-Woodroofe approximatio if Q 0 S f = oσ ad R S f = oσ This is equivalet to the existece of a array U i m, of martigale differeces such that S f m = oσ ; for m we ca, i such a case, choose m = i P 0U i f + i= i P 0U i f = =0 i= P 0U i f The Wu-Woodroofe approximatio was itroduced i Wu ad Woodroofe 2004 for adapted processes The oadapted case ca bee foud i Volý 2006 The Wu-Woodroofe approximatio together with liear growth of variaces S f 2 are ecessary but ot sufficiet, cf Klicarová ad Volý, 2009 coditios for the martigale approximatio A coditio which, for processes with liear growth of variace, is ecessary ad sufficiet for the existece of a martigale approximatio, has bee foud by Zhao ad Woodroofe 2008 This coditio, valid for adapted processes, ca be formulated i the followig way: For f = P if 2 Results = lim sup j j S j f f 0 Theorem 2 The wea diagoal approximatio ad the Wu-Woodroofe approximatio are equivalet ad both are implied by the strog diagoal approximatio Now, we shall deal with adapted processes, ie f will be supposed F 0 measurable Theorem 22 There exists a statioary liear process f T i i such that f has a martigale approximatio but o strog diagoal approximatio More precisely, for ay sequece of d = o there exists a sequece such that d, = o ad for f = i= P i f, S f f 0
4 260 Jaa Klicarová ad Dalibor Volý The Propositio shows that i the Zhao-Woodroofe s coditio, the limit of averages caot be replaced by a limit of f + = limsup j j S j f f 3 Proofs 3 Proof of Theorem 2 Let us prove the first part of the theorem At first we prove that the wea diagoal approximatio implies the Wu-Woodroofe approximatio We have Q 0 S f = 0 P i U j f i= i= + 0 P i U j f + i= + j=i+ + = i= + P i U j f P i U j f + i ad + 3 S f f 2 Ug = g Therefore, uder the assumptios of the wea diagoal approximatio, The proof for R S f is similar Q 0 S f = oσ Now, we show that the Wu-Woodroofe approximatio implies the wea diagoal approximatio We ca suppose that f is adapted to the filtratio ad regular, ie f = P if; the o adapted case ca be treated i the same maer cf Volý, 2006 ad the o regular part of f does ot affect the limit behaviour We have σ 2 = S f 2 2 = h where h is a slowly varyig fuctio i the sese of Karamata cf Kalleberg, 997 ad ES f F 0 2 = oσ Recall that the Wu-Woodroofe coditio implies that there is a martigale differece array U i m such that S f m 2 = oσ ; we the have m 2 = σ / + oσ /
5 Diagoal approximatios by martigales 26 Note that S f m 2 2 = ES f F U j m P j = ES f F i=j j m P 0 U i f 2 2 U i f 2 2 = by the assumptios, f is adapted hece P j U i f = 0 for j > i We thus have that for ay ǫ > 0 ad big eough, for all but ǫ j {0,,}, m P 0 j Ui f 2 2 < ǫh Let ǫ be oe of such j We the have P 0 U i f P 0 U i f 2 2 < 2ǫh hece for m = P 0 U i f, i=j S f m 2 2 = oσ2 Deote f = P i f To show that S f f 2 2 = oσ 2 it suffices to prove ES f F 0 2 = oσ By the assumptios U j m P j i=j U i f 2 = oσ hece recall that P j U j m = U j m so that j= + j=0 j m P 0 U i f 2 = oσ j m P 0 U i f 2 = oσ Because Uj m 2 2 σ2 = oσ2, we thus get j=0 j P 0 U i f 2 = oσ hece f = m + g Ug where g 2 = oσ ad S f f 2 = oσ So, the f has a wea diagoal approximatio t remais to prove the secod part of the theorem This part follows immediately from the defiitios of the strog diagoal approximatio ad the wea diagoal approximatio
6 262 Jaa Klicarová ad Dalibor Volý 32 Proof of Theorem Costructio of the process Let e H 0 H ad e = U i e i is thus a martigale differece sequece For i, j N N deotes the set of all positive itegers we deote K j = exp j 2 j N j = 2 K i +, b j = j 2 ad put K 0, N 0 = 0, b 0 = We defie fuctios f i, f i for i N: f i := K i = f i := f i T Ni Ki K i e T + =0 + e T, ad put where f 0 = b 0 e f = + i= b i fi + f 0, Mai idea of the proof We will prove that Q0Sf S f 0 Remar that this is a ecessary ad sufficiet coditio for the Wu-Woodroofe approximatio cf Wu ad Woodroofe, 2004 The we will deduce that there is a martigale approximatio the secod part of the proof we will show that for ay sequece of d = o there exists a sequece such that d, = o ad for f = i= P i f, S f f 0 the proof, we shall eed both upper ad lower estimatios of the term S f, a upper estimatio of Q 0 S f ad a lower estimatio of the term S f f We have S f S f Q 0 S f, S f S f Q 0 S f + Q 0 S f, hece for the estimatio of the term S f it suffices to estimate the terms Q 0 S f ad S f Q 0 S f
7 Diagoal approximatios by martigales 263 Figure 32 The costructio of the process f f T 2 f T f f T 2 f T f T f T f 0 P j P j f 0 N N 2 + N 2 f 2 f 0 b K b K b K b K b2 K 2 b2 K 2 b2 K Projectios P j S f Sice Q 0 S f = 0 j= P js f ad S f Q 0 S f = + P js f, i order to estimate the terms Q 0 S f ad S f Q 0 S f we eed to ow estimatios of P j S f From the costructio 3 of the fuctio f we have P j S f = P j + S b i fi + f 0 32 Let us study the fuctios f i From the costructio of f i we ca see that i= ad t follows that f i H Ni H Ni 2K i f i T H Ni+ H Ni 2K i+ S f i H Ni+ H Ni 2K i+ 33 Now, let us fix i ad study P j S b i fi We ca write P j S b i fi = y,j e j The part y,j o the right had side of the equatio depeds i fact o i, ad j Let us split all possibilities ito the followig three cases Recall that N i+ = N i + 2K i
8 264 Jaa Klicarová ad Dalibor Volý The case of > N i+ f we suppose N i+, the the term y,j is as follows: j : j > N i or j N i+ + such case cf 33 y,j = 0 2 j : N i K i + j N i K i y,j = b i =j +N i+k i 3 j : N i 2K i + 2 j N i K i y,j = b i K i =+2 N i K i j = 4 j : N i + j N i 2K i + t is easily see from the costructio of the fuctio f i that Ki P j S b i fi = b i K i + = 0, =N i+k i+j = so y,j = 0 5 j : N i K i + 2 j N i Alie as i 3 K i y,j = b i 6 j : N i 2K i + 2 j N i K i + t is similar case to 2, we have K i y,j = b i = N i K i j+2 The case of N i Remar that P j S f = 0 for j > sice f is adapted So, if have the case of < N i, the we eed to express the projectio P j S b i fi for egative j oly, especially for j [ N, N 2K ] Now, let us have N i Sice S f i H Ni+ H Ni 2K i+, we are iterested oly i projectios P j for egative j The expressio of the term y,j is as follows 2 j : j > N i or j N i+ + such case cf 33 y,j = 0 22 j : N i + j N i K i y,j = b i =j +N i+k i 23 j : N i K i + j N i y,j = b i K i+j+n i =j +N i+k i 24 j : + 2 Ni K i + j N i K i y,j = b i K i+j+n i = j+ N i K i+
9 Diagoal approximatios by martigales 265 Figure 33 The case of 2K i f f T 2 f T f T N i 2 N i K i 0 N i+ + N i N i K i 6 N i j : N i K i + j 2 Ni K i K i j N i+ y,j = b i =j+n i+k i+ 26 j : N i 2K i + j N i K i K i j N i+ y,j = b i = j N i K i+ 27 j : N i 2K i + j N i 2K i y,j = b i K i = j N i K i+ Observe that if is eve, the the term P Ni K i++ 2 S f i is equal to zero The case of N i < < N i+ We ca split this case ito the followig three parts f 2K i, the we ca apply formulas 6 the case of K i, the formulas 2 27 ca be used Whe is such that K i < < 2K i the we ca use the formulas 2 25 ad 27, where the upper boud i 25 is N i K i such a case the set of j i the formula 26 is empty
10 266 Jaa Klicarová ad Dalibor Volý Figure 34 The case of N i f f T 0 2 N i 22 N i N i K i N i K i N i N i+ + 2 Geeral expressio for positive j As we saw above, the projectio for fixed i P j S b i fi depeds o, j Let us deote P j S b i fi = y,j e j For geeral j, N, we ca thus express the term y,j i,j is such i that N i+ + 2 j N i ; recall that i,j is the oly i for which the projectio
11 Diagoal approximatios by martigales 267 P j S f i is ozero as y,j = b i,j K i,j =j +N i,j +K i,j = b i,j for j : N i,j K i,j + j N i,j K i,j =+2 N i,j K i,j j 34 = 0 i other case for j : N i,j+ + 2 j N i,j K i,j From the previous calculatios we ca deduce that still for positive j P j S f 2 = P j S f 0 + b i,j fi,j 2 = + y,j 2 35 where i,j is such that j belogs to the iterval [ N i,j+ +2, N i,j ] ad y,j is as above Remar that y,j [, 0] The upper estimatio of the term S f Q 0 S f Now, let us estimate the term S f Q 0 S f A upper estimatio of this term ca, usig the previous remars, be easily see the orm of each term P j S f is less or equal to oe, cf 35 to be S f Q 0 S f 2 = P j S f 2 36 The lower estimatio of the term S f Q 0 S f Sice cf 35 P j S f 2 = P j S f 0 + b i,j fi,j 2, for the lower estimatio of S f Q 0 S f 2 we eed to estimate P j S f 0 + b i,j fi,j 2 for 0 j Let us fix N The, there exists a iteger such that N < N + Usig 35 we have S f Q 0 S f 2 = P j S f 2 = P j S f 0 + b i,j fi,j 2 We ca write the term o the right had side as P j S f 2 = N i j=max, N i++ P j S f 0 + b i fi 2 Recall that for positive j, the projectios P j S b i fi are equal to zero for every i except oe of them i,j We get S f Q 0 S f 2 = A i
12 268 Jaa Klicarová ad Dalibor Volý where for i A i = ad N i j= N i++ P j S f 2 = A = N N i j= N i 2K i+ P j S f 0 + b f 2 Remar that all fuctios f i,j cotaied i A i are the same P j S f 0 + b i,j fi,j 2 Let us study the case of i, usig 3,4 recall that N i+ = N i +2K i : A i = Sice = = 2 N i j= N i 2K i+ N i j= N i 2K i+2 N i j= N i K i+2 P j S f 0 + b i,j fi,j 2 P j S f 0 + b i,j fi,j 2 + P Ni 2K i+s f y,j y, Ni K i+ 2 + we have A i 2 N i j= N i K i+2 + y,j 2 + 2y,j + 2y,j + + 2y, Ni K i+ + Usig the expressio of y,j as a sum of the fractios, see 34, puttig l := j + N i +K i, we obtai recall that m =l log m l ad m =l log m log xdx: l K i A i 2 2 K i i 2 l=2 =l 2 K i 2 K i i 2 + 2i K i 2 = l=2 log K i l i 2 + logk i = 2K i 2 K i 2K i 2 i logk i 2 logl + 2K i 2 i 2 2K i logk i 2[xlog x x] Ki + l=2 = 2K i 2 i 2 2K i 2logK i logk i + logk i + 2K i 2 = 2K i 2 K i 2K i 2 i 2 log K i + + logk i Usig log x x = log + x x for x >, we have
13 Diagoal approximatios by martigales 269 A i 2K i 2 2K i 2 i 2 K i + + logk i = 2K i 2 i 2 2K i + logk i = 2K i 4K i + 2 log K i 2 i 2 We have derived recall that K i = exp i 2 Deote C = i= A i We have C = A i = i= A i 2K i + O i= Ki log K i 2K i 4K i + 2 log K i 2 i 2 N N + O i= K 4K i + 2 log K i 2 log K Now, for fiishig of the estimatio of the term S f Q 0 S f, we eed to estimate the term A The estimatio ca be divided ito followig three parts Let = N + K The, cf 34 P j S f 2 = P j S f 0 + b f 2, for j [, N ] Hece, i this case, we estimate the terms P j S f 0 + b f 2 usig the same methods as above ad we obtai usig N K + = ad N = K A = K P j S f 0 + b f 2 K 2K + log K 2 For S f Q 0 S f 2 we have deduced that recall that i such a case K ad log K 2 S f Q 0 S f 2 C + K 2K + log K 2 K + O 2K + log K log K 2 = + o 2 Let N < < N + K We eed to estimate the term N A 2 = P j S f 0 + b f 2 i 2
14 270 Jaa Klicarová ad Dalibor Volý Usig similar argumets as before N < K we get A 2 N 2 2 N 2 2 K K j=n +K + =j K j=n +K + log K j N 2 2 N logk [xlog x x] K K +N = N 2 K 2 N log K +N K K log K +N + N N 2 2 N K +log N 3 2 N We thus have got K K +N + N log K K + N A 2 N 3 2 N So, for S f Q 0 S f 2, we have S f Q 0 S f 2 C + N 3 2 N K + O 3 log K 2 N = + o 3 Let N +K < < N +2K such a case we eed to estimate the term N A 3 = P j S f 0 + b f 2 Sice N > K we divide the term A 3 ito two parts as follows N P j S f 0 + b f 2 = N j= N K + P j S f 0 + b f 2 + N K + P j S f 0 + b f 2 where we ow the estimatio of the first term o the right-had side t is the same estimatio as for the term A So, let us deote by A 32 the secod term o the right-had side This term ca be expressed as
15 Diagoal approximatios by martigales 27 A 32 N K N K A 32 = N K 2 2 K =j+ N K 2 2 = N K 2 2 N K K =j+ log K j 2 N K logk [xlog x x] N K N K log K N K + Now, let us tae a costat c such that 0 < c < ad N K = ck ; we thus have A 32 ck 2 ck 2 log K + ck ck 2 2 ck log c So, for the term A 3 A 3 c + K 2 log K we have derived : 2 = c + K ck ck log c + K + 2K c logc 2 Usig the iequality log c c, we get A 3 c + K 4K 2 We have obtaied for the term S f Q 0 S f 2 recall that C = N + O K log K ad = N + K c + : S f Q 0 S f 2 C + K c K K + O 4 log K 2 K = + o We thus have deduced recall the upper estimate that for each N, S f Q 0 S f 2 = + o 37
16 272 Jaa Klicarová ad Dalibor Volý Estimatio of the term Q 0 S f Now, we will study the term Q 0 S f Recall that 0 0 Q 0 S f 2 = P j S f = P j S f 0 + b i fi j= j= Usig 35 ad a iequality x y 2 x 2 + y 2 for x, y 0 We thus get Q 0 S f 2 where i = 0 j= P js b i fi 2 The estimatio of the terms i ca be divided ito three parts t depeds o the relatio betwee i ad Let i be such that i we still suppose : N < N + ; this meas that N i + 2K i such a case we ca use 5,6 some egative j are also i the part 4 but the projectios i 4 are equal to zero ad puttig l := j + N i + K i i 5 ad l := j + N i + K i 2 i 6 we obtai l= =l + i, i= K i i K i 2 2 i 2 38 i 2 K i i 4 + log K i 2 + log 2 K i j j=2 2 K i i logk i + log 2 K i + K i log 2 K i log 2 j 2 i logk i + K i log 2 K i [xlog 2 x 2xlog x + 2x] Ki 2 i K i + log K i 2K i 5K i i 2 The sum of such i ca be estimated by recall that K j = exp j 2 i= 5K i i 2 5K i= 5K i i 2 6K Let i +, ie N i The, usig expressios 2 27 we get i = N i K i j= N i 2K i P j S b i fi j= N i+ P j S b i fi 2 N i j= N i K i+ P j S b i fi 2 +
17 Diagoal approximatios by martigales 273 hece i 3 K i i 4 j+ =j 2 Usig the iequality i= a i 2 i= a2 i we get K i j+ =j 2 K i j+ =j 2 K i j j + j=2 Ki+ = j + j 3 + log j=k i hece i log i4 Now, we ca sum the terms i ad usig the iequality log i 2 we get + i=+ i + i=+ 3 + i log 3 i=+ 3 + i 2 i 4 4 i = b 2 i +b 2 i +b 2 i +b 2 i K i =K i +j K i 2b 2 i j= + 2 = j+2 K i 2 + b 2 i j+ =K i j+ 2 j=0 j+ =j+ j+ =j b 2 i b 2 i 2 j=0 K i K i j+ =j+ +j =j+ 2 K i j = +K i+ j 2, 2
18 274 Jaa Klicarová ad Dalibor Volý i 2 K i i 4 4 K i i 4 j+ =j+ j+ =j i4 2 2 j=0 j+ =j+ 2 Usig the iequality i= a i 2 i= a2 i we get i 4 i 4 4 i 4 4 i 4 K i j+ =j+ K i j + + j j 4 + log i4 3 t remais the case of i = Accordig to paragraph The case of : N i < < N i+ i the subsectio Projectios P j S f, this estimatio ca be split ito three parts the case of 2K we have For K we have 5K log 4 For K < N + 2K we have, similarly as i the previous case, K Q 0 S b f 2 3 K 2 2 =j 38 we get Q 0 S b f 2 0K 2
19 Diagoal approximatios by martigales 275 t remais to estimate Q 0 S f 2 for N < < K We have Q 0 S b f N 2K + P j S b f 2 + j= N 2K 0 P j S b f 2 + j= N + N j= N K P j S b f 2 K =K j N j= N + N K j= N 2K + K P j S b f 2 + P j S b f 2 + K j+ =K j +2 2 which, lie i 38, ca be estimated by For all it is thus 25K 2 Q 0 S f 2 25 Wu Woodroofe approximatio of the fuctio f Usig the estimatios for terms S f Q 0 S f ad Q 0 S f we obtai Q 0 S f 2 S f o O Sice + recall that log + as +, we have proved Wu-Woodrofe approximatio the Wu-Woodroofe approximatio, we ca tae D,0 = i P 0U i f + i= i P 0U i f the case of a statioary liear process P 0 U i f is a multiple of e hece we get D,0 = c e From S f 2 / cost cf 37 it follows that the c coverge to a costat c ad we get a martigale approximatio with m = ce Strog diagoal approximatio of the fuctio f Now, we show that the fuctio f does ot satisfy the strog diagoal approximatio Tae a sequece d such that d = o The, there exists a sequece of j = l j N j + K j, where N j + K j d j, N j + K j = o ad l j teds to ifiity Defie h j = 0 i= N j K j P i f The S j h j 2 = o j As we proved above S j f 2 = j + o j, hece S j f h j 2 = j o j This shows that for arbitrary d = o the coditio i of Defiitio 3 is ot satisfied
20 276 Jaa Klicarová ad Dalibor Volý Refereces J Dedecer, F Merlevède ad D Volý O the wea ivariace priciple for oadapted sequeces uder projective criteria J Theoret Probab 20 4, MR M Gordi A cetral limit theorem for statioary processes Soviet Math Dol 0, MR P Hall ad CC Heyde Martigale limit theory ad its applicatio Academic Press, New Yor 980 J Haa, E Cetral limit theorem for time series regressio Z Wahrscheilicheitstheorie verw Geb 26, MR O Kalleberg Foudatios of Moder Probability Spriger, New Yor 997 MR J Klicarová ad D Volý O the exactess of the Wu-Woodroofe approximatio Stochastic Proccesses ad their Applicatios 9, MR D Volý Approximatig martigales ad the cetral limit theorem for strictly statioary processes Stochastic Processes ad their Applicatios 44, MR98662 D Volý Martigale approximatios of o adapted stochastic processes with oliear growth of variace Pe Bertail, P Douha ad P Soulier, editors, Depedece i Probability ad Statistics, volume 87 of Lecture otes i Statistics, pages 4 56 Spriger 2006 MR WB Wu ad M Woodroofe Martigale approximatios for sums of statioary processes The Aals of Probability 32 2, O Zhao ad M Woodroofe O martigale approximatios The Aals of Applied Probability 8 5, MR
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 informationOn the optimality of McLeish s conditions for the central limit theorem
O the optimality of McLeish s coditios for the cetral limit theorem Jérôme Dedecker a a Laboratoire MAP5, CNRS UMR 845, Uiversité Paris-Descartes, Sorboe Paris Cité, 45 rue des Saits Pères, 7570 Paris
More informationRegression with an Evaporating Logarithmic Trend
Regressio with a Evaporatig Logarithmic Tred Peter C. B. Phillips Cowles Foudatio, Yale Uiversity, Uiversity of Aucklad & Uiversity of York ad Yixiao Su Departmet of Ecoomics Yale Uiversity October 5,
More information62. 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 informationProduct 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 informationErratum 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 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 informationEntropy 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 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 informationSolutions to Tutorial 5 (Week 6)
The Uiversity of Sydey School of Mathematics ad Statistics Solutios to Tutorial 5 (Wee 6 MATH2962: Real ad Complex Aalysis (Advaced Semester, 207 Web Page: http://www.maths.usyd.edu.au/u/ug/im/math2962/
More informationOn the weak invariance principle for non adapted sequences under projective criteria.
O the weak ivariace priciple for o adapted sequeces uder projective criteria. Jérôme Dedecker, Florece Merlevède, Dalibor Voly To cite this versio: Jérôme Dedecker, Florece Merlevède, Dalibor Voly. O the
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 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 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 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 informationNotes 27 : Brownian motion: path properties
Notes 27 : Browia motio: path properties Math 733-734: Theory of Probability Lecturer: Sebastie Roch Refereces:[Dur10, Sectio 8.1], [MP10, Sectio 1.1, 1.2, 1.3]. Recall: DEF 27.1 (Covariace) Let X = (X
More informationA Proof of Birkhoff s Ergodic Theorem
A Proof of Birkhoff s Ergodic Theorem Joseph Hora September 2, 205 Itroductio I Fall 203, I was learig the basics of ergodic theory, ad I came across this theorem. Oe of my supervisors, Athoy Quas, showed
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 information1+x 1 + α+x. x = 2(α x2 ) 1+x
Math 2030 Homework 6 Solutios # [Problem 5] For coveiece we let α lim sup a ad β lim sup b. Without loss of geerality let us assume that α β. If α the by assumptio β < so i this case α + β. By Theorem
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall 2013 Lecture 21 11/27/2013
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall 2013 Lecture 21 11/27/2013 Fuctioal Law of Large Numbers. Costructio of the Wieer Measure Cotet. 1. Additioal techical results o weak covergece
More informationLecture 19. sup y 1,..., yn B d n
STAT 06A: Polyomials of adom Variables Lecture date: Nov Lecture 19 Grothedieck s Iequality Scribe: Be Hough The scribes are based o a guest lecture by ya O Doell. I this lecture we prove Grothedieck s
More informationOn the behavior at infinity of an integrable function
O the behavior at ifiity of a itegrable fuctio Emmauel Lesige To cite this versio: Emmauel Lesige. O the behavior at ifiity of a itegrable fuctio. The America Mathematical Mothly, 200, 7 (2), pp.75-8.
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 informationMath Solutions to homework 6
Math 175 - Solutios to homework 6 Cédric De Groote November 16, 2017 Problem 1 (8.11 i the book): Let K be a compact Hermitia operator o a Hilbert space H ad let the kerel of K be {0}. Show that there
More informationA Weak Law of Large Numbers Under Weak Mixing
A Weak Law of Large Numbers Uder Weak Mixig Bruce E. Hase Uiversity of Wiscosi Jauary 209 Abstract This paper presets a ew weak law of large umbers (WLLN) for heterogeous depedet processes ad arrays. The
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 informationIt is often useful to approximate complicated functions using simpler ones. We consider the task of approximating a function by a polynomial.
Taylor Polyomials ad Taylor Series It is ofte useful to approximate complicated fuctios usig simpler oes We cosider the task of approximatig a fuctio by a polyomial If f is at least -times differetiable
More informationSequence A sequence is a function whose domain of definition is the set of natural numbers.
Chapter Sequeces Course Title: Real Aalysis Course Code: MTH3 Course istructor: Dr Atiq ur Rehma Class: MSc-I Course URL: wwwmathcityorg/atiq/fa8-mth3 Sequeces form a importat compoet of Mathematical Aalysis
More informationBIRKHOFF 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 informationDistribution 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 informationNotes 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 informationMath 155 (Lecture 3)
Math 55 (Lecture 3) September 8, I this lecture, we ll cosider the aswer to oe of the most basic coutig problems i combiatorics Questio How may ways are there to choose a -elemet subset of the set {,,,
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 informationECE 901 Lecture 12: Complexity Regularization and the Squared Loss
ECE 90 Lecture : Complexity Regularizatio ad the Squared Loss R. Nowak 5/7/009 I the previous lectures we made use of the Cheroff/Hoeffdig bouds for our aalysis of classifier errors. Hoeffdig s iequality
More informationSolution to Chapter 2 Analytical Exercises
Nov. 25, 23, Revised Dec. 27, 23 Hayashi Ecoometrics Solutio to Chapter 2 Aalytical Exercises. For ay ε >, So, plim z =. O the other had, which meas that lim E(z =. 2. As show i the hit, Prob( z > ε =
More informationLECTURE 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 informationSome 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 informationRiesz-Fischer Sequences and Lower Frame Bounds
Zeitschrift für Aalysis ud ihre Aweduge Joural for Aalysis ad its Applicatios Volume 1 (00), No., 305 314 Riesz-Fischer Sequeces ad Lower Frame Bouds P. Casazza, O. Christese, S. Li ad A. Lider Abstract.
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 informationEFFECTIVE WLLN, SLLN, AND CLT IN STATISTICAL MODELS
EFFECTIVE WLLN, SLLN, AND CLT IN STATISTICAL MODELS Ryszard Zieliński Ist Math Polish Acad Sc POBox 21, 00-956 Warszawa 10, Polad e-mail: rziel@impagovpl ABSTRACT Weak laws of large umbers (W LLN), strog
More information2.2. 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 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 informationSOME 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 informationSequences and Limits
Chapter Sequeces ad Limits Let { a } be a sequece of real or complex umbers A ecessary ad sufficiet coditio for the sequece to coverge is that for ay ɛ > 0 there exists a iteger N > 0 such that a p a q
More informationThe picture in figure 1.1 helps us to see that the area represents the distance traveled. Figure 1: Area represents distance travelled
1 Lecture : Area Area ad distace traveled Approximatig area by rectagles Summatio The area uder a parabola 1.1 Area ad distace Suppose we have the followig iformatio about the velocity of a particle, how
More informationsin(n) + 2 cos(2n) n 3/2 3 sin(n) 2cos(2n) n 3/2 a n =
60. Ratio ad root tests 60.1. Absolutely coverget series. Defiitio 13. (Absolute covergece) A series a is called absolutely coverget if the series of absolute values a is coverget. The absolute covergece
More information4. Partial Sums and the Central Limit Theorem
1 of 10 7/16/2009 6:05 AM Virtual Laboratories > 6. Radom Samples > 1 2 3 4 5 6 7 4. Partial Sums ad the Cetral Limit Theorem The cetral limit theorem ad the law of large umbers are the two fudametal theorems
More informationMa 530 Infinite Series I
Ma 50 Ifiite Series I Please ote that i additio to the material below this lecture icorporated material from the Visual Calculus web site. The material o sequeces is at Visual Sequeces. (To use this li
More informationSlide Set 13 Linear Model with Endogenous Regressors and the GMM estimator
Slide Set 13 Liear Model with Edogeous Regressors ad the GMM estimator Pietro Coretto pcoretto@uisa.it Ecoometrics Master i Ecoomics ad Fiace (MEF) Uiversità degli Studi di Napoli Federico II Versio: Friday
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 informationAlmost Sure Invariance Principles via Martingale Approximation
Almost Sure Ivariace Priciples via Martigale Approximatio Florece Merlevède a, Costel Peligrad b ad Magda Peligrad c a Uiversité Paris Est, Laboratoire de mathématiques, UMR 8050 CNRS, Bâtimet Coperic,
More informationExact convergence rates in the central limit theorem for a class of martingales
xact covergece rates i the cetral limit theorem for a class of martigales M. l Machkouri L. Ouchti Laboratoire de Mathématiques Paul Pailevé, UMR 854 CNRS-Uiversité des Scieces et Techologies de Lille,
More informationREVIEW 1, MATH n=1 is convergent. (b) Determine whether a n is convergent.
REVIEW, MATH 00. Let a = +. a) Determie whether the sequece a ) is coverget. b) Determie whether a is coverget.. Determie whether the series is coverget or diverget. If it is coverget, fid its sum. a)
More informationTheorem 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 informationki, 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 informationLecture Chapter 6: Convergence of Random Sequences
ECE5: Aalysis of Radom Sigals Fall 6 Lecture Chapter 6: Covergece of Radom Sequeces Dr Salim El Rouayheb Scribe: Abhay Ashutosh Doel, Qibo Zhag, Peiwe Tia, Pegzhe Wag, Lu Liu Radom sequece Defiitio A ifiite
More informationON WELLPOSEDNESS QUADRATIC FUNCTION MINIMIZATION PROBLEM ON INTERSECTION OF TWO ELLIPSOIDS * M. JA]IMOVI], I. KRNI] 1.
Yugoslav Joural of Operatios Research 1 (00), Number 1, 49-60 ON WELLPOSEDNESS QUADRATIC FUNCTION MINIMIZATION PROBLEM ON INTERSECTION OF TWO ELLIPSOIDS M. JA]IMOVI], I. KRNI] Departmet of Mathematics
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 informationStudy the bias (due to the nite dimensional approximation) and variance of the estimators
2 Series Methods 2. Geeral Approach A model has parameters (; ) where is ite-dimesioal ad is oparametric. (Sometimes, there is o :) We will focus o regressio. The fuctio is approximated by a series a ite
More informationHyun-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 informationThe 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 informationChapter 6 Infinite Series
Chapter 6 Ifiite Series I the previous chapter we cosidered itegrals which were improper i the sese that the iterval of itegratio was ubouded. I this chapter we are goig to discuss a topic which is somewhat
More informationWeighted Approximation by Videnskii and Lupas Operators
Weighted Approximatio by Videsii ad Lupas Operators Aif Barbaros Dime İstabul Uiversity Departmet of Egieerig Sciece April 5, 013 Aif Barbaros Dime İstabul Uiversity Departmet Weightedof Approximatio Egieerig
More informationLecture 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 informationOn the convergence rates of Gladyshev s Hurst index estimator
Noliear Aalysis: Modellig ad Cotrol, 2010, Vol 15, No 4, 445 450 O the covergece rates of Gladyshev s Hurst idex estimator K Kubilius 1, D Melichov 2 1 Istitute of Mathematics ad Iformatics, Vilius Uiversity
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 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
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 information1 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 informationMath 525: Lecture 5. January 18, 2018
Math 525: Lecture 5 Jauary 18, 2018 1 Series (review) Defiitio 1.1. A sequece (a ) R coverges to a poit L R (writte a L or lim a = L) if for each ǫ > 0, we ca fid N such that a L < ǫ for all N. If the
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 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 informationTechnical Proofs for Homogeneity Pursuit
Techical Proofs for Homogeeity Pursuit bstract This is the supplemetal material for the article Homogeeity Pursuit, submitted for publicatio i Joural of the merica Statistical ssociatio. B Proofs B. Proof
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 informationGeneralized 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 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 information5.1. The Rayleigh s quotient. Definition 49. Let A = A be a self-adjoint matrix. quotient is the function. R(x) = x,ax, for x = 0.
40 RODICA D. COSTIN 5. The Rayleigh s priciple ad the i priciple for the eigevalues of a self-adjoit matrix Eigevalues of self-adjoit matrices are easy to calculate. This sectio shows how this is doe usig
More informationON CONVERGENCE OF BASIC HYPERGEOMETRIC SERIES. 1. Introduction Basic hypergeometric series (cf. [GR]) with the base q is defined by
ON CONVERGENCE OF BASIC HYPERGEOMETRIC SERIES TOSHIO OSHIMA Abstract. We examie the covergece of q-hypergeometric series whe q =. We give a coditio so that the radius of the covergece is positive ad get
More informationMATH4822E FOURIER ANALYSIS AND ITS APPLICATIONS
MATH48E FOURIER ANALYSIS AND ITS APPLICATIONS 7.. Cesàro summability. 7. Summability methods Arithmetic meas. The followig idea is due to the Italia geometer Eresto Cesàro (859-96). He shows that eve if
More informationCouncil for Innovative Research
ABSTRACT ON ABEL CONVERGENT SERIES OF FUNCTIONS ERDAL GÜL AND MEHMET ALBAYRAK Yildiz Techical Uiversity, Departmet of Mathematics, 34210 Eseler, Istabul egul34@gmail.com mehmetalbayrak12@gmail.com I this
More informationINFINITE SEQUENCES AND SERIES
INFINITE SEQUENCES AND SERIES INFINITE SEQUENCES AND SERIES I geeral, it is difficult to fid the exact sum of a series. We were able to accomplish this for geometric series ad the series /[(+)]. This is
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 information6.3 Testing Series With Positive Terms
6.3. TESTING SERIES WITH POSITIVE TERMS 307 6.3 Testig Series With Positive Terms 6.3. Review of what is kow up to ow I theory, testig a series a i for covergece amouts to fidig the i= sequece of partial
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 informationSOME GENERALIZATIONS OF OLIVIER S THEOREM
SOME GENERALIZATIONS OF OLIVIER S THEOREM Alai Faisat, Sait-Étiee, Georges Grekos, Sait-Étiee, Ladislav Mišík Ostrava (Received Jauary 27, 2006) Abstract. Let a be a coverget series of positive real umbers.
More informationAssignment 5: Solutions
McGill Uiversity Departmet of Mathematics ad Statistics MATH 54 Aalysis, Fall 05 Assigmet 5: Solutios. Let y be a ubouded sequece of positive umbers satisfyig y + > y for all N. Let x be aother sequece
More informationLecture 4. We also define the set of possible values for the random walk as the set of all x R d such that P(S n = x) > 0 for some n.
Radom Walks ad Browia Motio Tel Aviv Uiversity Sprig 20 Lecture date: Mar 2, 20 Lecture 4 Istructor: Ro Peled Scribe: Lira Rotem This lecture deals primarily with recurrece for geeral radom walks. We preset
More informationMIDTERM 3 CALCULUS 2. Monday, December 3, :15 PM to 6:45 PM. Name PRACTICE EXAM SOLUTIONS
MIDTERM 3 CALCULUS MATH 300 FALL 08 Moday, December 3, 08 5:5 PM to 6:45 PM Name PRACTICE EXAM S Please aswer all of the questios, ad show your work. You must explai your aswers to get credit. You will
More informationSequences 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 informationECONOMETRIC THEORY. MODULE XIII Lecture - 34 Asymptotic Theory and Stochastic Regressors
ECONOMETRIC THEORY MODULE XIII Lecture - 34 Asymptotic Theory ad Stochastic Regressors Dr. Shalabh Departmet of Mathematics ad Statistics Idia Istitute of Techology Kapur Asymptotic theory The asymptotic
More informationy X F n (y), To see this, let y Y and apply property (ii) to find a sequence {y n } X such that y n y and lim sup F n (y n ) F (y).
Modica Mortola Fuctioal 2 Γ-Covergece Let X, d) be a metric space ad cosider a sequece {F } of fuctioals F : X [, ]. We say that {F } Γ-coverges to a fuctioal F : X [, ] if the followig properties hold:
More information16 Riemann Sums and Integrals
16 Riema Sums ad Itegrals Defiitio: A partitio P of a closed iterval [a, b], (b >a)isasetof 1 distict poits x i (a, b) togetherwitha = x 0 ad b = x, together with the covetio that i>j x i >x j. Defiitio:
More informationAdvanced Analysis. Min Yan Department of Mathematics Hong Kong University of Science and Technology
Advaced Aalysis Mi Ya Departmet of Mathematics Hog Kog Uiversity of Sciece ad Techology September 3, 009 Cotets Limit ad Cotiuity 7 Limit of Sequece 8 Defiitio 8 Property 3 3 Ifiity ad Ifiitesimal 8 4
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 informationarxiv: v1 [math.fa] 3 Apr 2016
Aticommutator Norm Formula for Proectio Operators arxiv:164.699v1 math.fa] 3 Apr 16 Sam Walters Uiversity of Norther British Columbia ABSTRACT. We prove that for ay two proectio operators f, g o Hilbert
More informationSingular Continuous Measures by Michael Pejic 5/14/10
Sigular Cotiuous Measures by Michael Peic 5/4/0 Prelimiaries Give a set X, a σ-algebra o X is a collectio of subsets of X that cotais X ad ad is closed uder complemetatio ad coutable uios hece, coutable
More informationECE 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 informationMath 2784 (or 2794W) University of Connecticut
ORDERS OF GROWTH PAT SMITH Math 2784 (or 2794W) Uiversity of Coecticut Date: Mar. 2, 22. ORDERS OF GROWTH. Itroductio Gaiig a ituitive feel for the relative growth of fuctios is importat if you really
More information1 6 = 1 6 = + Factorials and Euler s Gamma function
Royal Holloway Uiversity of Lodo Departmet of Physics Factorials ad Euler s Gamma fuctio Itroductio The is a self-cotaied part of the course dealig, essetially, with the factorial fuctio ad its geeralizatio
More informationMa 4121: Introduction to Lebesgue Integration Solutions to Homework Assignment 5
Ma 42: Itroductio to Lebesgue Itegratio Solutios to Homework Assigmet 5 Prof. Wickerhauser Due Thursday, April th, 23 Please retur your solutios to the istructor by the ed of class o the due date. You
More informationLinear Elliptic PDE s Elliptic partial differential equations frequently arise out of conservation statements of the form
Liear Elliptic PDE s Elliptic partial differetial equatios frequetly arise out of coservatio statemets of the form B F d B Sdx B cotaied i bouded ope set U R. Here F, S deote respectively, the flux desity
More information