Some Probability Inequalities for Quadratic Forms of Negatively Dependent Subgaussian Random Variables

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1 Joural of Sceces Islamc epublc of Ira 6(: (005 Uvers of ehra ISSN hp://scecesuacr Some Probabl Iequales for Quadrac Forms of Negavel Depede Subgaussa adom Varables M Am A ozorga ad H Zare 3 Deparme of Mahemacs Facul of Scece Ssa ad aluchesa Uvers Zaheda Islamc epublc of Ira Deparme of Sascs Facul of Mahemacal Scece Ferdows Uvers Mashhad Islamc epublc of Ira 3 Deparme of Mahemacs Facul of Scece Zabol Uvers Zabol Islamc epublc of Ira Absrac I hs paper we oba he upper expoeal bouds for he al probables of he quadrac forms for egavel depede subgaussa radom varables I parcular he law of eraed logarhm for quadrac forms of depede subgaussa radom varables s geeralzed o he case of egavel depede subgaussa radom varables Kewords: Negavel depede; Quadrac forms; Subgaussa radom varable; Ieraed logarhm Iroduco Le { } be a sequece of subgaussa radom varables ad le A = ( a = be a arra of real umbers ad A = ( a = We defe he quadrac forms (QF Q = A ( = Whou loss of geeral we ma assume ha A s smmerc For a marx = ( b = = b ad μ ( s he larges egevalue of Moreover r( ad r ( sad for race of ad ra of respecvel ad dag ( α α deoes he dagoal marx wh dagoal elemes α α Some expoeal bouds for he al probables of QF s have bee suded b Mosch [7] for he case = { } s a depede subgaussa radom varables he lm behavors of QF s have bee suded b ma auhors such as Kresberg [6] Goze ad homrov [5] Gras ad aqqu [4] I hs paper we exed some expoeal bouds for he al probables of QF s for egavel depede (ND subgaussa radom varables he b usg hese equales we oba he law of he eraed logarhm (LIL ad some probabl equales for QF s Defo he radom varables are sad o be ND f we have P[ ( x ] P( x I = = ad P[ I ( > x ] P( > x = = E-mal: bozorg@mahumacr 63

2 Vol 6 No Wer 005 Am e al J Sc I Ira for all x x A fe sequece { } s sad o be ND f ever fe subse s ND he followg lemmas are lsed for referece obag he ma resuls he ex secos Dealed proofs ca be foud alor ad Chug Hu [9] uldg ad Kozacheco [] ozorga Paerso ad alor [] Mosch [7] Chug [3] Lemma ([] Le { } be a sequece of ND radom varables ad { f } be a sequece of orel fucos all of whch are moooe creasg (or all are moooe decreasg he { f ( } s a sequece of ND radom varables Lemma ([] Le be a fe sequece of ND radom varables ad be all oegave (or oposve he [ = ] [ ] = Defo A smmerc radom varable s sad o be subgaussa (SG radom varable f here exss a oegave real umber α such ha for each real umber α exp[ ] ( α he umber τ( = f{ α 0: E( e exp[ ] } wll be called he Gaussa sadard of he radom varable I s evde ha wll be a subgaussa radom varable f ad ol f τ ( < Moreover l( ( τ ( sup[ ] / = 0 ad equal ( hold for α = τ ( A subgaussa radom varable alwas sasfes he relaos E( = 0 ad E( τ ( If E( = τ ( he s called srcl subgaussa Lemma 3 ([9] If s a subgaussa radom varable wh τ ( α he α [ ] exp[ ] Lemma 4 ([7] For a posve marx dmeso hz Z = e q d Z h > 0 hz exp( ( q ( s he des of a -dmesoal Gaussa vecor wh mea zero ad varace marx Lemma 5 ([3] If { } he P( E o lm P( E of E s a sequece of eves Expoeal ouds for al Probables I hs seco we oba upper expoeal bouds for he probables > x] ad P[ Q > x] for ever x > 0 Pu α = τ ( V = dag( α α = 4 V AV μ = μ( V AV ad Q = Q r( V AV hroughou he secos ad 3 we suppose ha { } s a sequece of ND subgaussa radom varables Sce some proofs are he same as Mosch [7] we abbrevae hem I fac we oba upper bouds ha are greaer ha upper bouds Mosch [7] bu hese equales mpl our ma resuls for ND subgaussa radom varables he followg lemmas pla a esseal role obag our resuls Lemma Le { } be a sequece of ND subgaussa radom varables wh τ ( for all Le A be a posve semdefe marx ad Q A he = E[exp( hq] exp( l( 4 hλ = 0 h /4μ ad λ λ are he egevalues of A Proof Smlar o he oao Lemma of Mosch [7] le ra ( = > 0 here exss a orhogoal marx U such ha A = U LU ad L = dag( λ λ 00 λ λ are posve egevalues of A he A = V LV 0 64

3 J Sc I Ira Am e al Vol 6 No Wer 005 V = (( U ( U L0 = dag( λ λ = 4 λ = ad Lemma 4 for suffcel small h > 0 hq hv E [ e ] q( d( = hv = E exp( h u ( = = + γ γ = = = E{exp( h h } γ = u γ + = max{0 γ } ad γ = = max{0 γ } Now b Cauch Schwarz equal we have hv γ + γ = = { E exp( h E exp( h } + h γ h γ / [ ( ( ] = = + 4hγ 4hγ / h [( e ( e ] = e ( = = he secod equal holds b Lemma ad hrd equal s rue b Lemma 3 Now b ( ad ( smlar o he proof of Lemma [7] we have hq h e q ( d = 4hλ / / (de L0 ( π exp( ( d = λ exp( l( 4 hλ (3 = 0< 4h maxλ < Lemma Le { } be a sequece of ND subgaussa radom varables ad A be a posve semdefe marx he for 0 h /4μ hq 4hμ exp( h ( + (4 3 4hμ / Proof rasformao / α we assume ha α = for all Lemma we have exp( ( l( 4 (5 hq r A = hλ I he oher had exp( hλ l( 4 hλ = hλ + hμ + hμ 3 exp(4 ( (4 (4 + 4hμ = exp(4 h λ ( + (6 3 4hμ for 0 h /4μ Now (4 oba of (5 ad (6 heorem Le { A } be a sequece of posve semdefe smmerc marces he for ever 0< δ < he followg equales are rue for all : (A μ μ > ] exp( ( ( 4 3 for 0 (( δ / μ ( ( δ ( δ > ] exp( ( 8μ 3δ for (( δ / μ (C ( δ > ] C( δ exp( for some cosa C ( δ > 0 ad all > 0 Proof Mehod of he proof s he same as proposo [7] he case of depede Corollar Uder he assumpos of heorem If { } s a sequece of posve real umbers such ha μ / 0 he for ever 0< δ < ad suffcel large 65

4 Vol 6 No Wer 005 Am e al J Sc I Ira ( δ > ] exp( 4 For ever > 0 > 48 [ ] exp( m( μ PQ heorem Uder he assumpos of heorem he followg equales are rue: EQ > ] μ μ exp( ( ( for 0 (( δ / μ ad EQ > ] ( δ ( δ ( δ exp( ( + 8μ 3δ 4μ for (( δ / μ acov( = = = aα = Proof Marov equal ad (4 for ever ad > 0 we have h EQ > ] e hq ( EQ = + h e E exp( h( Q r( V AV EQ 4hμ exp( h + h ( + + h (7 3 4hμ Hece b pug h = / ad δ h = for μ 4 0 (( δ / μ ad (( δ / μ respecvel he proof s compleed 3 A Applcao o he LIL I hs seco b usg he oaos of seco ad heorems ad we prove a law of he eraed logarhm for quadrac forms he case ha { } s a sequece of ND subgaussa radom varables heorem 3 Le { } be a sequece of ND subgaussa radom varables ad { A } be a sequece of posve semdefe smmerc marces ad Q = A he Q lm w p χ ( If ad he μ = (3 / o(( / log = o (3 / (( / log Q EQ lm w p χ ( χ = ( log = (33 / / Wh log x for a x > 0 = log log x ad log x = max{l x } Proof heorem for = χ ( ad δ (0 we have ad ( δ > χ( ] C( δexp( log ( δ χ( ] C( δexp( log Sce we have lm χ( ] = Hece b Lemma 5 Q P[ o] = χ ( Le = χ ( ad δ (0 Sce μ = / o(( he here exss N > 0 such ha for log ever N ad ε > 0 66

5 J Sc I Ira Am e al Vol 6 No Wer 005 μ (log / ε Sce ε s arbrar b (33 we ca assume 0 (( δ / μ Now wh subsug he value of (33 o heorem we have EQ > χ ( ] log 8 3 / exp( ( μ (log (log + 4 / Hece b (3 ad (3 EQ > χ ( ] log + 8 μ (log / ( exp( ( o(( o( o( Sce as he rgh sde of he las equal eds o zero ad so lm EQ χ ( ] = Hece b Lemma 5 Q EQ P[ o ] = χ ( hs complees he proof For subgaussa sequece of ND radom varables we have he followg example Example Le { } be a sequece of sadard ormal dsrbuos such ha Cov ( 0 for each he s a sequece of ND srcl subgaussa radom varables wh τ ( = hece = / lm (8 log 4 w p efereces ozorga A Paerso F ad alor Lm heorems for depede radom varables World Cogress Nolear Aalss 9: (996 uldg VV ad Kozacheo YuV Sub_Gaussa radom varables Uraa Mah J 3: (980 3 Chug K A Course Probabl heor hrd Edo Academc Press (00 4 Gras L ad aqqu MS Ceral lm heorems for quadrac forms wh me-doma codos A Probab 6(: (998 5 Goze F ad homrov A NA asmpoc dsrbuo of quadrac forms Ibd 7(: (999 6 Kresberg A Khch ad Marcewcz-Zgmudpe equales for quadrac forms of depede radom varables Sascs ad Probab Le 60(3: 3-37 (00 7 Mosch Esmaes for al probables of quadrac ad blear forms subgaussa radom varables wh applcaos o he low eraed logarhm Probabl ad Mahemacal Sascs : (99 8 Morcz F Serflg J ad Sou WF Mome ad probabl bouds wh quas-superaddve srucure for he maxmum paral sum A Probab 0: (98 9 alor L ad e-chug Hu Sub_Gaussa echques provg srog law of large umbers Amr Mah Mohl 94: (987 67