Law of the sum of Bernoulli random variables

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1 Law of the sum of Beroulli radom variables Nicolas Chevallier Uiversité de Haute Alsace, 4, rue des frères Lumière Mulhouse December 006 Abstract Let be the set of all possible joit distributios of Beroulli radom variables X,..., X. Suppose that which is a simplex i the -dimesioal space, is edowed with the ormalized Lebesgue measure µ. Suppose also that the iteger is large. The we show that there is subset of whose measure µ ( ) is very close to, such that if the joit distributio of (X,..., X ) is i the the law of the sum X +...+X is close to the biomial law B(, ). This result does t eed ay idepedace assumptio. Next, we show a result of the same kid whe is edowed with a other probability measure ν. Key words: Beroulli radom variable, Biomial law. AMS Subject Classificatio 000: 60-xx, 60Exx. Itroductio The most commo explaatio for the ubiquity of the Gaussia law is the Cetral Limit Theorem. Aother explaatio closely related to the previous oe, is that the Gaussia law is the oly stable law with fiite variace. The proofs of the Cetral Limit Theorem always rest o some idepedece assumptio or at least o some statioarity assumptio. The purpose of our work is to give i a very simplified situatio, aother kid of explaatio for the ubiquity of the Gaussia law. Cosider a sequece X,..., X of Beroulli radom variables. We are iterested i the law of the sum S X X without ay idepedace assumptio about the radom variables X i. Sice the Laplace-Moivre theorem asserts that up to a suitable ormalizatio, for large (ad p ot too small), the biomial distributio B(, p) is close to the Gaussia law, a explaatio for the ubiquity of the Gaussia law may be i our settig: whe is large, the law of S is ofte very close to the symmetric biomial distributio B(, ). We must explai what we mea by very ofte. Fix a positive iteger. Deote by the set all possible joit distributios of Beroulli radom variables X,..., X. is the set of all probability measures o {0, }. To each elemet p (p i ) i {0,} of, oe ca associate the law of the sum S X X. It is a probability measure L (p) o the set {0,..., }. Whe we choose the uiform probability law b (,..., ) o {0, }, we get L (b) B B(, ) the symmetric biomial distributio: B (k) C k

2 for all k i {0,..., }. Let ε > 0, our aim is to estimate the size of the set,ε of probability measures p i such that for all k i {0,..., }, L (p)(k) B (k) ε. To make precise this questio, we have to measure the size of subsets of. This ca be doe with the help of a probability measure o. If µ is a probability measure o, we wish to prove that µ(,ε ) is close to whe is large. I a less formal laguage : whe is large, choosig at radom the joit distributio of (X,..., X ), it is likely that the law of S is very close to the symmetric biomial distributio B(, ). Statemets of results There are may choices for the probability measure µ ad we shall oly cosider two. The first ad the most atural oe is µ µ, the ormalized Lebesgue measure o. Our first result is: Theorem There exists a costat A such that for all positive itegers, ad all positive umbers ε, µ, (,ε ) A ε ad µ, ({p : sup L (p)(i) B (I) ε}) A5/ I {0,...,} ε. Though, the Lebesgue measure is very atural it has a drawback. Oe ca cosider the law of the first Beroulli radom variables (X,..., X ) as a radom variable defied o. This radom variable is the projectio pro : : (p i ) i {0,} (p (j,0) + p (j,) ) j {0,} The poit is that the Lebesgue measure µ, is ot the image of µ, by the map pro ; the family (µ, ) is ot a projective family of probability measures. We would like to fid a projective family (µ, ) of atural probability measures o the sequece of sets ( ). This ca be doe iductively: whe we kow the law (p i ) i {0,} of (X,..., X ) the law (p (i,j) ) i {0,},j {0,} of (X,..., X, X + ) is chose at radom uiformly amog all the possible laws. Let us make it precise. Cosider the atural map which is almost a bijectio, ψ : [0, ] {0,} + : ((p i ) i {0,}, (x i ) i {0,} ) (p (i,j) ) i {0,},,j {0,} where p (i,0) p ix i ad p (i,) p i( x i ). The uiformity meas that the choice of (p (i,j) ) i {0,},,j {0,} give (p i ) i {0,}, is doe at radom with respect to the Lebesgue measure λ o [0, ] {0,}. This eable to trasfer a probability measure from to +. If we have a probability measure µ, o, the product of this measure with the Lebesgue measure λ o [0, ] {0,} gives rise

3 to a measure o [0, ] {0,} ad its image by ψ is a ew measure µ,+ o +. Sice the map pro + ψ : [0, ] {0,} is the projectio o, the image by pro + of µ,+ is µ,. Takig µ, the ormalized Lebesgue measure o, we get a sequece (µ, ) of atural probability measures o the sequece of simplices ( ) (see sectio 3, for a purely probabilistic poit of view about the measures µ, ). This is our secod choice which also has a drawback for it looses the symmetry betwee the radom variables pr i : R : (p j ) j {0,} p i, i {0, } as well as the symmetry betwee the variables X i, i {,..., }. Our secod result is: Theorem There exists a costat C such that for all positive itegers, ad all positive umbers ε, µ, (,ε ) C l 3/ ε ad µ, ({p : sup L (p)(i) B (I) ε}) C l3/. I {0,...,} ε Remark. I both Theorems ad it is possible to fid a explicit value for the costats A ad C. It is easy to check that the value A works i Theorem whereas it is more difficult to give a explicit value for the costat C ad we do ot give ay. The mai poit of these two results is that they do ot eed ay idepedece assumptio about the variables X i. There must be some other works of the same kid but we have oly fid oe: K. Takeuchi ad A. Takemura ([T,T]) have studied the law of the sum S X X where the X i are Beroulli variables. They oly assume some coditio about cetral biomial momets which are a oe to oe fuctio of the factorial momets (the k th factorial momet of radom variable X is E(X(X )...(X k + ))). This allows them to prove covergece to the ormal law or to the Poisso s law for a triagular array of Beroulli variables X i,. Their hypothesis are oly about the cetral biomial momets of S X, X,. 3 Sketch of proofs The ideas of the proofs of Theorems ad are exactly the same. It is the reaso why, although these proofs are ot difficult, we begi by describig their mai steps. For i,, N, ad k {0,..., }, deote by E i, (k) ad V i, (k) the expectatio ad the variace of the radom variable p L (p)(k) defied o the probability space (, µ i, ). Makig use of the symmetries, we replace the simplex by aother geometrical space where the computatio of expectatios are easier. The we show i both cases, that E i, (k) C k B (k). Sice by the Chebyshev iequality, µ i, ({p : L (p)(k) E i, (k) > ε}) V i,(k) ε, 3

4 Theorems ad ca be deduced from appropriates upper boud o the variaces V i, (k). I the first case, stadard results lead to the iequality V, (k) Ck. I the secod case, computatios are ot as easy as i the first case. Some well kow estimates about biomial coefficiets, eable to show that for all k i {0,..., }, V, (k) C, ad that for all k such that k l, V, (k) C 5, where C is a costat idepedet of ad k. 4 Proof of theorem. The cardial umber of the set is N ad there is a oe to oe correspodece betwee the sets {0, } ad {0,..., N }. Therefore each p i ca be see as a probability measure o {0,,..., N }: {(p 0,..., p N ) R N : p 0, p,..., p N 0, p 0 + p p N }. Furthermore, for all k i {0,..., }, there is a subset E k of {0,..., N } with C k elemets, such that for all p (p 0,..., p N ) i, L (p)(k) i E k p i.. Let σ be a permutatio of the set {0,..., N }. σ iduces the liear map, f σ (x,..., x ) (x σ(0), x σ(),..., x σ(n ) ) which seds oto itself. Therefore the measure µ, is f σ -ivariat. It follows that give a subset E of {0,..., N }, the distributio fuctio of the map p (p 0,..., p N ) L E (p) i E p i depeds oly o the cardial umber of E. This meas that for all k {0,..., }, the map L Ek have the same distributio tha the map L Fk where Hece, F k {0,..., C k }. µ, ({p : L (p)(k) B (k) ε}) µ, ({p : L Fk (p) B (k) ε}). 3. I order to estimate µ, ({p : L Fk (p) B (k) ε}), let us itroduce aother way to see the probability space (, µ, ). Let (Y,..., Y N ) be N idepedet radom variables uiformly distributed i the iterval [0, ]. Arragig them i ascedig order we fid N radom variables Z Z... Z N. The joit distributio of (Z,..., Z N ) is the ormalized Lebesgue measure ν o T {(z,..., z N ) R N : 0 z z... z N }. 4

5 Let φ : R N R N be the map defied by φ(z,..., z N ) (z, z z,..., z N z N, z N ). The image of T by φ is ad sice the map φ is affie, the image of the measure ν by φ is the measure µ,. Now, let F {0,..., m } be a subset of {0,..., N }. For all z (z,..., z N ) T, we have L F (φ(z)) z + (z z ) (z m z m ) z m, hece the distributio of L F is the same as the distributio of the map It follows that ad the same holds for the variaces R m : (z,..., z N ) T z m. E µ, (L F ) E ν (R m ) V µ, (L F ) V ν (R m ). 4. The distributio of R m is well kow, its desity h m is give by the formula h m (t) (see [Da, Du]). Therefore, (N )! (m )!(N m)! tm ( t) N m (N )! E ν (R m ) t 0 (m )!(N m)! tm ( t) N m dt (N )! Γ(m + )Γ(N m) (m )!(N m)! Γ(N + ) (N )! m!(n m )! (m )!(N m)! N! m N, ad E ν (Rm) t (N )! 0 (m )!(N m)! tm ( t) N m dt (N )! Γ(m + )Γ(N m) (m )!(N m)! Γ(N + ) (N )! (m + )!(N m )! (m )!(N m)! (N + )! m (m + ) N (N + ) V ν (R m ) E ν (Rm) E ν (R m ) m(m + ) N(N + ) m Nm(m + ) (N + )m N N (N + ) Nm + Nm Nm m N (N + ) m(n m) N (N + ) m N. Comig back to L Ek, we fid that for all k i {0,..., }, E, (k) E µ, (L Ek ) Ck N Ck B (k), V, (k) V µ, (L Ek ) Ck N. 5

6 Makig use of the Stirlig formula, it is easy to see that C k A where A is a costat idepedet of. It follows that V, (k) Chebyshev iequality we get ad therefore µ, ({p : µ, ({p : L Ek (p) C k ε}) max k0 A. Fially, with A ε L Ek (p) C k ε}) ( + ) A ε, µ, ( ε ) A ε. The secod iequality of Theorem follows from the first i replacig ε by ε/. If we wat to fid a explicit value for A, we ca use the followig iequalities istead of the Stirlig formula π exp( + + )! π exp( + ), (see [Fe], p ). A easy calculatio shows that the value A works. 5 About the defiitio of µ, I this sectio we give two other ways to itroduce the measure µ, : propositio ad. While propositio is ot eeded for the followig, propositio is useful. It replace the simplices, N, by a sigle product space edowed with a product probability. Notatios.. For x i [0, ], we put x (0) x ad x () x.. Deote pro : the map defied by pro ((p i ) i {0,} ) (p i ) i {0,} where p i p i0 + p i for all i {0, }. 3. For each i i {0, }, deote pr i : R {0,} R the map pr defied by pr((p j ) j {0,} ) p i. It iduces a radom variable o ad it is readily see that for all i i {0, }, pr i0 + pr i pr i pro o. 4. For a iteger, the map ψ is defied by: ψ : [0, ] {0,} : ((p i ) i {0,}, (x i ) i {0,} ) (p ij) i {0,},j {0,} where p i0 p ix i ad p i p i( x i ). I the itroductio we preset a rather geometric poit of view about the probability measures µ,. It is possible to give a more probabilistic poit of view about these probability measures. The simplex is the set of all probability laws of a sequece (X,..., X ) of Beroulli radom variables ad the map which associated to each law of (X,..., X ) the law of the first Beroulli radom variables (X,..., X ) is just the projectio pro :. Whe we kow the law of (X,..., X ) what ca we expect about the law of the whole sequece (X,..., X )? This is give by the coditioal distributio give pro of the radom variables P (X i,..., X i, X 0) P (X i,..., X i ) 6 pr (i,0) pr i pro : [0, ],

7 i (i,...i ) {0, }. The probability measures µ, are the oly such that these variables are all uiformly distributed i the iterval [0, ] ad idepedet coditioally to the law of (X,..., X ). This the meaig of the ext propositio which we state without proof. Propositio The sequece (µ, ) is the uique sequece of probability measures such that : i. µ, is the ormalized Lebesgue measure o, ii. for all iteger, µ, is a probability o, iii. for all iteger, the image of µ, by pro is µ,, iv. for all iteger ad all family (B i ) i {0,} of Borel subsets of [0, ], pr (i,0) µ, ( B i, i {0, } pro pr i pro ) where λ is the Lebesgue measure o [0, ]. i {0,} λ(b i ) Remark. iv ca be replace as well by: for all itegers ad all family (B i ) i {0,} of Borel subsets of [0, ], µ, (pr (i,0) B i, i {0, } λ(b i ]0, pr pro ) i [) pr i {0,} i where λ is the Lebesgue measure o [0, ]. By defiitio, the probability µ, is the image by ψ of the probability µ, λ where λ is the Lebesgue measure o [0, ] {0,}. We ca iterate this process from dow to ad we see that the probability measure µ, is the image of the Lebesgue measure o Ω [0, ] { } [0, ] {0,}... [0, ] {0,} by a map φ : Ω. It will be more efficiet to defie φ o a uique probability space Ω which does ot deped o. Notatios. Deote by J the set { } ( k {0, }k ) ad Ω the set [0, ] J. For j (j,..., j k ) i J, deote by Z j : Ω [0, ] J [0, ] the radom variable defied by Z j ((ω i ) i J ) ω j. Deote by Q the ifiite product of Lebesgue measures o Ω. Propositio For all itegers, cosider the map φ : Ω R {0,} defied by pr (i,...,i ) φ Z (i ) Z (i ) (i ) Z(i 3) (i,i )...Z(i ) (i,...,i ). The φ (Ω) ad the image by φ of the probability measure Q is µ,. thus Proof. It is easy to check that φ (ω) is i for all ω i Ω. Ideed, pr (i,...,i ) φ pr (i,...,i ) φ Z (i ) (i,...,i ), pr (i,...i,0)(φ (ω)) + pr (i,...,i,)(φ (ω)) pr (i,...,i )(φ (ω)) Z (i,..,i ) + pr (i,...,i )(φ (ω)) ( Z (i,..,i )) pr (i,...,i )(φ (ω)) 7

8 ad it follows by iductio that pr i (φ (ω)) pr i (φ (ω)) pr 0 (φ (ω))+pr (φ (ω)) Z (0) +Z(). i {0,} i {0,} Next we prove by iductio that the image by φ of the probability measure Q, is µ,. Suppose the image by φ of the probability measure Q is µ,. Usig the sequece of maps (ψ ), it is easy to fid a iductio relatio satisfied by the sequece of maps (φ ), we have φ (ω) ψ (φ (ω), (Z i (ω)) i {0,} )). Now φ ad (Z i ) i {0,} are idepedet radom variables, therefore the image by ω Ω (φ (ω), (Z i (ω)) i {0,} ) of the probability measure Q is the product of the image by φ of Q ad of the Lebesgue measure o [0, ] {0,} which is λ. By iductio hypothesis we get µ, λ ad by defiitio, the image of µ, λ by ψ is µ,. 6 Calculatio of the first two momets of p L (p)(k) Notatios. Let be a positive iteger.. For each subset F of {0, }, we shall deote by L F the map defied by p (p i ) i {0,} L F (p) p i. i F. For all itegers k i {0,...}, F,k deote the subset of elemet i (i l ) l {,...,} {0, } such that l i l k. Let be a iteger ad let k be a iteger i {0,..., }. We would like to estimate E, (k) E µ, (L (.)(k)) E µ, (L F,k ) p i dµ, ((p i ) i {0,} ) i F k pr i (φ (ω)) dq(ω). Ω i F k ad E µ, (L (.)(k)) E µ, (L F,k ) ( p i ) dµ, ((p i ) i {0,} ) i F k ( pr i (φ (ω))) dq(ω). Ω i F k Set f,k i F k pr i φ.. By propositio, E Q (f,k ) E Q ( pr i φ ) E Q ( pr (i,...,i ) φ Z i i F,k (i,...,i ) F,k E Q (pr (i,...,i ) φ Z (i) (i )),...,i ) (i,...,i ) F,k (i,...,i ) ) 8

9 ad sice φ ad Z i (i,...,i ) are idepedet, E Q (f,k ) E Q (pr (i,...,i ) φ )E Q (Z (i) (i,...,i ) )) (i,...,i ) F,k E Q (pr (i,...,i ) φ ). (i,...,i ) F,k Furthermore, F,k F,k {} F,k {0}, thus E Q (f,k ) ( We have also (i,...,i ) F,k {} (E Q(f,k ) + E Q (f,k )). E Q (pr (i,...,i ) φ ) + E Q (f,0 ) E Q (Z (0) ) E Q(Z () ) E Q(f, ), therefore, by iductio, we get Hece E Q (f,k ) C k. E, (k) C k. (i,...,i ) F,k {0}. The quadratic mea E Q (f,k ) is a little more difficult to estimate. The mai idea is to decompose F,k i the two sets F 0,k {(i,..., i ) F,k : i 0}, F,k {(i,..., i ) F,k : i } ad to observe that for each i i F,k 0 ad each j i F,k, the two variables Z (i ) (i ) Z(i 3) (i,i )...Z(i ) (i,...,i ), Z(j ) (j ) Z(j 3) (j,j )...Z(j ) (j,...,j ) E Q (pr (i,...,i ) φ )) are idepedet. We have E Q (f,k) E Q (( pr i φ ) ) + E Q (( pr i φ ) ) i F 0,k + E Q (( pr i φ )( i F,k pr i φ )) i F 0,k T + T + T 3. j F,k The first term gives T E Q (( Z (0) Z(i) (0) Z(i3) (0,i )...Z(i) (0,i...,i ) ) ) i F 0,k E Q (Z ( Z (i ) (0) Z(i 3) (0,i )...Z(i ) (0,i...,i ) ) ), i F 0,k sice Z is idepedet of the others Z i, we get E Q (( pr i φ ) ) E Q (Z )E Q( Z (i ) (0) Z(i 3) (0,i )...Z(i ) (0,i,...,i ) ) ). i F 0,k i F 0,k 9

10 The last thig to see for the computatio of the first term is that E Q (( Z (i ) (0) Z(i 3) (0,i )...Z(i ) (0,i,...,i ) ) ) E Q (( ( ) Z(i ) (i )...Z(i ) (i,...,i ) ) ) E Q (f,k), i F 0,k i F,k Z (i) thus T 3 E Q(f,k). Exactly the same argumets show that T 3 E Q(f,k ). By idepedece, the last term gives T 3 E Q (( Z (0) Z(i ) (0) Z(i 3) (0,i )...Z(i ) (0,i,...,i ) )( Z () Z(i ) () Z(i 3) (,i )...Z(i ) (,i,...,i ) )) i F 0,k E Q (Z (0) i F,k Z() )E Q(( Z (i) (0) Z(i3) (0,i )...Z(i) (0,i,...,i ) ))E Q(( Z (i) () Z(i3) (,i )...Z(i) (0,i,...,i ) )) i F,k 0 i F,k 6 E Q(f,k )E Q (f,k ). Fially, we get the relatio E Q (f,k) 3 [E Q(f,k) + E Q (f,k ) + E Q (f,k )E Q (f,k )]. 3. This recursio relatio ad the equality E Q (f m,l ) m C l m, eable to fid a recursio relatio betwee V Q (f,k ), V Q (f,k ) ad V Q (f,k ): V Q (f,k ) E Q (f,k) E Q (f,k ) 3 [E Q(f,k) + E Q (f,k ) + E Q (f,k )E Q (f,k )] 4 [E Q(f,k ) + E Q (f,k )] Hece, 3 (V Q(f,k ) + V Q (f,k ) + E Q (f,k )E Q (f,k )) + [E Q(f,k ) + E Q (f,k ) ] E Q(f,k )E Q (f,k ) 3 (V Q(f,k ) + V Q (f,k )) + (E Q(f,k ) E Q (f,k )) 3 (V Q(f,k ) + V Q (f,k )) + [ + (C k C k )]. V µ, (L F,k ) 3 [V µ, (L F,k )+V µ, (L F,k )]+ [ + (C k Ck )]. 6. A upper boud for V, (k) V µ, (L F,k ) We shall eed the followig lemma. Lemma There exists a costat C such that for all itegers we have:. for all k i {0,..., }, C k C k c. 0

11 . for all k i {0,..., } such that k l, C k C k c 5/. Proof. I what follows, C deotes a costat whose value may chage at each lie. Sice C k C k! k ( k)!k! k + C k + k + k, we ca use the classical Laplace-Moivre estimate about the biomial law: Let (a ) be a sequece of o egative real umbers which go to 0 as goes to ifiity. The for all positive itegers ad all itegers k such that k a /3 we have C k + δ ( (k) π exp (k ) ) where lim sup δ (k) 0 k: k / a /3 (actually, it is a slight extesio of the Laplace-Moivre theorem which deals oly with itegers k such that k a where a is a fix real umber; see [Fe] p. 85 theorem, or [Le] p. 36 propositio 8.). It follows that for all positive itegers ad all itegers k such that k a /3, C k C exp ( (k ) ) where the costat C does ot deped o. Makig use of the mootoicity of the biomial coefficiets, we get the followig iequality C k C 5/ for all itegers ad all itegers k such that k l. This last iequality implies. Now let us prove. For all positive itegers ad all itegers k such that k l, we have C k C k C exp ( (k ) ) k +. Put t k. We get C k C k C t exp ( t ) + C 3/ ad sice the fuctio t R t e t is bouded, C k C k C. Propositio 3 There exists a costat C such that or all positive itegers, v : sup V µ, (L F,k ) C k {0,...,}

12 Proof. Let be a iteger. Sice for all k i {0,..., }, V µ, (L F,k ) 3 (V µ, (L F,k )+V µ, (L F,k ))+ [ + (C k Ck )], we have v 3 v + [ + (C k Ck )]. With the previous lemma we get v 3 v + C. By iductio, we get that for all itegers, The sum v ( 3 ) v + C ( i) ( 3 )i. ( i) ( 3 )k is easy to estimate: ( i) ( 3 )i 0 i / + /<i, sice ad sice we have 0 i / /<i ( i) ( 3 )i 4 0 i / ( 3 )i ( i) ( 3 )i ( 3 )/ C, v ( 3 ) v + C C. Propositio 4 There exists a costat C such that or all positive itegers u : sup V k {0,...,}: k µ, (L F,k ) C l 5. Dem. Let be a iteger. First ote that the two variables L F,k ad L F, k have the same law, so it suffices to prove the propositio for k /. By lemma, for all k i {0,..., } such that k l, we have V µ, (L F,k ) 3 [V µ, (L F,k ) + V µ, (L F,k )] + [ + (C k Ck )] 3 [V µ, (L F,k ) + V µ, (L F,k )] + C 5, Fix a iteger k + l. We prove by iductio o l that for all iteger l l, V µ, (L F,k ) 3 l l l Cl i V µ, l (L F l,k i ) + C ( i) 5 ( 3 )i.

13 Ideed, if l l, the for all i {0,..., l}, therefore k i l l ( l) l( l), V µ, l (L F l,k i ) 3 (V µ, l (L F l,k i )+V µ, l (L F l,k i ))+ C ( l) 5. Together with the iductio hypothesis, this imply that V µ, (L F,k ) 3 l l Cl i [ 3 (V C µ, l (L F l,k i ) + V µ, l (L F l,k i )) + ( l) 5 ] l + C ( i) 5 ( 3 )i 3 l+ { 3 (V µ, (l+) (L F (l+),k (l+) ) + V µ, (l+) (L F l,k ) l + (C j l + C j l )V µ, (l+) (L F (l+),k j )} + C j l j0 ( j) 5 ( 3 )j l+ 3 l+ Cl+V i µ, (l+) (L F (l+),k i ) + C As before, it is easy to prove that l j0 ( j) 5 ( 3 )j C 5. Furthermore (remember that 0 L F ), l j0 ( j) 5 ( 3 )j. 3 l l Cl i V µ, l (L F l,k i ) 3 l l Cl i ( 3 )l. Thus, with l l we fid that V µ, (L F,k ) 3 l l l Cl i V µ, l (L F l,k i ) + C ( i) 5 ( 3 )i ( 3 ) l + C 5 C Ed of proof of theorem Let k be a iteger betwee l ad + l. By propositio 3 ad Chebyshev iequality, for all positive umbers ε, () µ, ({p : L F,k (p) C k ε}) C ε, 3

14 thus µ, ({p : max L F,k (p) C k ε}) 4 l C l k + l ε. Let k be a iteger i {0,..., } such that k l. By propositio 4 ad Chebyshev iequality, for all positive umbers ε, thus () µ, ({p : L F,k (p) C k ε}) C ε 5, µ, ({p : max It follows that k: l L F,k (p) C k ε}) C ε 5. µ, ({p : max k {0,...,} L (p)(k) L (B )(p) ε}) C l ε 3/. Let δ be a positive umber. We shall use iequality () with ε iequality () with ε δ. For each subset I of {0,..., }, the set k: k l {p : L (p)(i) L (B )(I) δ} is icluded i the uio of δ {p : L (p)(k) L (B )(k) 4 l } ad of k: k l {p : L (p)(k) L (B )(k) δ } δ 4 l ad which does ot deped o I, therefore (remember that L (p)(k) L F,k (p)) + k: k l µ, ({p : sup L (p)(i) L (B )(I) δ)} I {,...,} k: k l µ, ({p : L (p)(k) L (B )(k) δ }) µ, ({p : L (p)(k) L (B )(k) C l ε + C ε C l 3/ 4 δ (6 + } C l3/. δ Refereces [Da, Du]: D. Dacuha-Castelle, M. Duflo, Exercices de probabilités et statistiques. Problèmes à temps fixe, Masso, Collectio Mathématiques appliquées pour la Maîtrise, Paris p , 98 [Fe]: W. Feller, a itroductio to probability theory ad its applicatios, vol, 3 th ed., Wiley, 968. [Le]: E. Lesige, Pile ou Face, Ue Itroductio au Calcul des Probabilités, ellipse, 00. [T,T]: K. Takeuchi, A. Takemura, O sum of 0 radom variables. Uivariate case, A. Ist. Statist. Math. 39 (987), Part A, δ 4 l }) 4

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