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

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1 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: Departmet of Statistics Faculty of Mathematical Scieces, Ferdowsi Uiversity, Mashhad, Ira Bozorg@math.um.ac.ir, Fax: Abstract: I this paper, we study some P P strog it theorems for the sequece f fi X g, for each fi > 0 ad weighted sums a kx k where fx ; g is a sequece of egative depedece Sub-Gaussia radom variables ad a k is a array of oegative real umbers. Key Words: Negative Depedet,Sub-Gaussia, Strog Law of Large Numbers, Weighted Sums,Martigale..Itroductio Some P covergece theorems for weighted sums a k X k has bee studied by Chow [4] for the case where fx ; g is a sequece of idepedet, geeralized Gaussia radom variables. The case of m-depedet geeralized Gaussia r.v.'s has bee discussed by Ouy [6], ad the strog law of large umbers for sequeces of idepedet Sub-Gaussia radom variables has bee obtaied by Taylor ad Chug Hu [9]. I this paper, we exted some of these results ad prove somep strog it theorems for the sequece of f fi X g, for each fi > 0, ad weighted sums P a k X k where fx ; g is a sequece of egative depedet Sub-Gaussia radom variables ad a k is a array of oegative real umbers. Also P by sub-gaussia techiques we prove that a k X k coverge with probability oe for each, where E[X jf ] = 0, F = ff(x ; ;X ) ad P j=k a j = O(k fi ) for every fi > 0.To prove these results we eed to the followig defiitios, lemmas ad theorems. Defiitio. A symmetric radom variable X is said to be Sub- Gaussia (SG) r.v. if there exist a oegative real umber ff such that for Λ This research supported by cetral research of statistics each real umber t Ee tx» exp[ ff t (:) The umber, fi(x) = iffff 0:E(e tx )» exp[ ff t ]; t Rg, will be called the Gaussia stadard of the radom variable X. It is evidet that X will be a Sub-Gaussia radom variable if ad oly if fi(x) <. Moreover fi(x) = sup t6=0 " l(e(e tx )) t # = ; ad iequality (.) holds for ff = fi(x). Defiitio.A symmetric radom variable X is strictly Sub-Gaussia if E(X )=fi (X): Defiitio 3.The radom variables X ; ;X are said to be ND if we have P[ P [ ad (X j» x j )]» (X j >x j )]» P [X j» x j ]; (:) P [X j >x j ]; (:3)

2 for all x ; ;x R. A ifiite sequece fx ; g is said to be ND if every fiite subset X ; ;X is ND. The coditios (.) ad (.3) are equivalet for =, but these do ot agree for 3 (see [3],Pages 3-4). The followig Lemmas ad Theorems which our work is based o ca be foud i Taylor, Chug Hu (987),Buldygi, Kozacheko (980) ad Bozorgia, Taylor (996). Theorem. ( [0]) Let X be Sub-Gaussia radom variable ad a is a real umber (a 6= 0), the ax is sub-gaussia radom variable with fi(x) =jajfi(x). Lemma. ( [9]) If X is a Sub-Gaussia radom variable with fi(x)» ff, the i) For every t R, ad E[e tjxj ]» exp[ ff t (:4) For every ">0, we have P[X>"]»exp[ " ff ] (:5) P [jxj >"]»exp[ " (:6) ff Lemma. ( [9]) If X is bouded (jxj»m) ad has zero mea (E(X) = 0), the X is Sub- Gaussia radom variable with fi(x)» p M. Lemma 3.( [3]) Let X ; ;X be ND radom variables ad f ;f be a sequece of Borel fuctios which all are mootoe icreasig (or all are mootoe decreasig), the f (X ); ;f (X ) are ND radom variables. Lemmas 4.( [3]) Let X ; ;X be ND oegative radom variables. The Examples. E[ X j ]» E[X j ) Let X has uiform distributio i (a; b) iterval, the Y = X a+b is Sub-Gaussia radom variable with fi(y )» p (b a). ) Let X be a cotiuous radom variable with d.f. F (x), the Y = F (X) is Sub- Gaussia with fi(y )» p. 3) Let X be a radom variable with d.f. Normal ad E(X) =0; Var(X)=ff, the X is strictly Sub-Gaussia with fi (X) =ff.. Strog Limit Theorems I this sectio we obtai some strog it theorems for sequece f fi P X k g for each fi>0, where fx ; g is a sequece of egative depedet Sub-Gaussia radom variables with fi(x )» ff, for every, uder the coditios o P ff k. Theorem. Let fx ; g be a sequece of ND Sub-Gaussia radom variables with fi(x )» ff. i) S = P X k is a Sub-Gaussia r.v. with ff = P i= ff i. i P If i= ff i = O( fi ) for every fi>0, the! X k =0 W:P:: If ff = ff = = ff = ff, the for some fi>,! fi X k =0 W:P:: i) By Lemmas,3,4 ad Theorems we have E[e ts ]» E[e tx k ]» exp[ ff t ]; hece S is a Sub-Gaussia r.v. with ff = P i= ff i. For each " > 0 by part i ad Lemma (., we have P [j X k j >"]» exp[ " ( fi) ] < ; c

3 i ad also P [j fi X k j >"]» exp[ " fi ff ] < : which these complete the proof.. Theorem 3. Let fx ; g be a sequece of ND radom variables satisfyig P [a» X i» b] =, for each i where a<b, the for every fi>,! fi (X k E(X k )) = 0; W:P:: Defie Y k = X k E(X k ), k =;; ;, the, E(Y k ) = 0 ad jy k j» (b a), W.P., hece by Lemma fy k ; k g be a sequece of Sub-Gaussia radom variables with fi(y k )» p (b a). Thus by Theorem (3.i, for every fi>,wehave! fi (X k E(X k )) = 0; W:P:: Corollary. Let fx ; g be a sequece of ND idetically distributed radom variables with E(X )=0, Var(X )= ad E[X k ] < k, the p S is a asymptotically Sub-Gaussia radom variable, whe! with fi( p S )». For t R p E[e t S ]» we have p E[e t X k ]= [ + t + ffi( )]! e t ; whe!..some Strog Limit Theorems for weighted sums I this sectio, we obtai some strog it theorems for weighted sums T = P a k X k ad P a k X k, where fx ; g is a sequece of egative depedece Sub-Gaussia radom variables ad a k is a array of oegative real umbers.also we prove T = P a k X k covergece W.P.. uder the coditio that E[X jf ]=0,F = ff(x ; ;X ) ad P j=k a j = O(k fi ) for every fi>0 ad. Lemma 5. Let fx ; g be a sequece of ND Sub-Gaussia radom variables with fi(x k )» ff. The i) T is a Sub-Gaussia radom variable with fi(t )» ff p A for all. For every ">0 P[jT j>"]» exp[ " ff A Where A = P a k. i) By Lemmas,3,4 ad Theorems, for every h R we have E[e htm ]» my exp[ h ff P m a k Hece, by Fatou's Lemma E[e ha kx k ]» ]» exp[ h ff A E[e ht ]» exp[ h ff A This follows by part i ad Lemma. Corollary. i) If P exp[ " ff A ] <, the! a k X k =0 W:P:: (:) I particular if A = ffi(l ()), the (.) holds. If S = P X k ad fi>0 the 3

4 ! = (l (+fi)= ())S =0 W:P:: Theorem 4. Let fx ; g be a sequece of ND Sub-Gaussia r.v.'s P i) If! a k = l 6= 0<, the for every fi>0! fi a k X k =0 W:P:: (:) If a k = O( fi ) for some k» ad fi>, the! a k X k =0 W:P:: (:3) i If P a k = O( fi ) for some fi > 0, the! a k X k =0 W:P:: (:4) By Lemma 5 for some 0 <B<, ad ">0 ad P [j P [j fi X» a k X k j >"]» fi " exp[ P ff ] < ; a k a k X k j >"]» exp[ " fi ff B ] < : " exp[ P ff ] a k Now (.) ad (.3) follow from the Borel Catelli Lemma, ad (.4) follows from part (. Theorem 5. Let fx ; g be a sequece of ND Sub-Gaussia r.v.'s. The for every x R x P [max jt jj x]»exp[ j»m ff A By Lemmas,3,4 ad Theorems for every h R we have Ee hjtmj» Ee htm +Ee htm» exp[ h ff A Sice ft m ; F m ; m g is a martigale ad fjt m j; F m ; m g is submartigale ad '(t) =e th for each h 0 is icreasig ad covex fuctio, the by the submartigale iequality P [max j»m jt jj x]=p[max j»m '(jt jj) '(x)]» For h = E['(jT m j)] '(x) x ff A» exp[ hx + h ff A wehave P[max jt jj x]»exp[ x j»m ff A Theorem 6. Uder the assumptios of Theorem 5 i) If ft m ; m g coverges i probability for every, the it coverges W.P.. T = P a k X k coverges W.P. for each. i) Let T m! l i probability for every, the there exist a subsequece fm k ; k g such that T mk! l W.P.. We defie S k = By Theorem 5 max m k <m»m k+ jt m T mk j: P [S k >"]»exp[ P ff j=mk + a j " Hece by the Borel Catelli Lemma S k! 0 a.e., whe k!.thus jt m l j»s k + jt mk l j!0 For every N >mby Lemma 4 W:P:: 4

5 P [jt N T m j >"]»exp[ P ff j=m+ a j " [0] Buldygi,V.V. ad Kozacheko Yu.V. (980) Sub-Gaussia radom variables. Ukraiia Math.J.3, 980, If m!, the left had side of above iequality teds to zero. Hece, ft m ; m g coverges i probabilityby the Cauchy criterio. Now part i shows that T coverges W.P.. Let fx ; g be a sequece of idepedet Sub-Gaussia r.v.'s with fi(x )» ff, for every, the the assumptio E[X jf ]=0 ca be replaced by E(X ) = 0, ( i.e.e(x )=E[X jf ] = 0). Thus all the above Theorems, Lemmas, ad Corollaries are true i this case. Refereces [] Amii, M ad Bozorgia, A. Negatively depedet bouded radom variables probability iequalities ad the strog law of large umbers. Joural of Applied Mathematics ad Stochastic Aalysis, 3:3 (000), [] Amii, M. Azaroosh, H.A. ad Bozorgia, A. The almost sure covergece of weighted sums of egatively depedet radom variables.j. of Scieces I.R.I. Vol.0 No., 999, - 6. [3] Bozorgia, A., Patterso, R.F ad Taylor, R.L. Limit theorems for depedet r.v.'s.world Cogress Noliear Aalysts'9, 996, [4] Chow, Y.S. Some covergece theorems for idepedet r.v.'s.a.mat. Statist. 37, 966, [5] Chow, Y.S. Probability Theory.Spriger Verlag, 978, [6] Kim, Chheag.Ouy. Some covergece theorems for depedet geeralized Gaussia r.v.'s.j.natioal.chiao.tug.uiversity, 976, [7] Lehma, E. Some cocepts of depedece. A.Math.Statist.37, 966, [8] Stout, W.F. Some results o the complete ad almost sure covergece of liear combiatios of idepedet radom variables ad martigale differeces.a.math.statist. 39, 968, [9] Taylor, R.L. ad Tie-Chug Hu. Sub-Gaussia techiques i provig strog law of large umbers.the Teachig of Mathematics, 987,