Société de Calcul Mathématique SA Mathematical Modelling Company, Corp.

Transcription

1 oiété de Calul Mathéatique A Matheatial Modellig Copay, Corp. Deisio-aig tools, sie 995 iple Rado Wals Part V Khihi's Law of the Iterated Logarith: Quatitative versios by Berard Beauzay August 8 I this Fifth Part, we first give a quatitative versio of Khihi's law of the iterated logarith (94) ; we the explai the oetios ad differees with our preset wor. The opariso betwee Khihi's ethods ad ours lead to the followig olusios: Khihi's ethods ay hadle the ase of a sigle urve, whereas ours ay hadle oly the ase of two syetri barriers; Quatitative estiates obtaied by eas of Khihi's ethods are of probabilisti type, ad are uh weaer tha the results derived by eas of operator theory; BB RW, Part V, July 8

2 To say that Khihi's urve ( x) xlog Log ( x) the urve b( x) xlog ( x) = is a "seurity urve" is iorret. Tae = whih is above Khihi's urve, ad tae two istats. The the probability to hit bx betwee these two istats, ad thus to go above ( x ), is stritly positive. I. Khihi's Law of the Iterated Logarith: quatitative versio Let us oe ba to the origial settig: the P( X = ) =. We set = X = tie ad, origially, both fortues are equal). X 's are idepedet variables with sae law, (i other words, we do ot osider oly eve values of the We itrodue Khihi's urve, or barrier, defied by the equatio: ( x) xlog Log ( x) =, whih is a real futio, defied for all real x e. The lassial stateet of the law of the iterated logarith is: alost surely, lisup = + ( ) (see The explaatio give by Wiipedia, rather obsure, is as follows: "Thus, although the quatity ( ) is less tha ay predefied with probability approahig oe, the quatity will evertheless be greater tha ifiitely ofte; i fat, the quatity will be visitig the eighborhoods of ay poit i the iterval (-,) alost surely." The diffiulty is i the uderstadig of the words "alost surely", both i theory ad i pratie. There is a atural probability o the ifiite produt =,, whih is siply the produt of the eleetary probabilities o eah layer. With respet to this "global" probability, the words "alost surely" are well-defied. But, for this probability, every eleetary path has probability, ad so does a fiite uber of paths. Moreover, it is quite hard to obtai quatitative results for this probability, whih is well-suited oly for probabilisti arguets.

3 O the otrary, if we stop at tie, we have a preise ad ituitive defiitio of the probability of a evet: up to tie, we have paths, ad the probability of ay evet is : uber of paths satisfyig the evet, divided by. For istae, the probability of the evet ; is perfetly lear: we out the uber of paths for whih, at soe poit, the rado wal is above the urve, ad we divide this uber by. ie the paths divide ito two at eah step, a estiate obtaied at a give step will reai valid at later stages. For istae, the stateet X = has the sae probability (/), o atter whether we osider it at stage or at ay later stage. Let us observe that the stateet fro Wiipedia ay be quite isleadig. Ideed, if oe reads: "The quatity ( ) will evertheless be greater tha ifiitely ofte", this is true for ost urves, ad ot oly for Khihi's urve. Ideed, we reeber (Part I) that the rado wal oes ba ifiitely ay ties to the x axis, so for istae the value = ay be expeted at tie = 5. But the, osider the situatio where ireases liearly fro this poit; it will evetually ross the urve ( ) ; i fat, this is true for ay urve suh that ( ) whe +. We ostrut this way a ifiite uber of situatios i whih exeeds 4 ( ) or ay ultiple, as oe wishes. Therefore, we thi that i suh stateets, a preise defiitio of the probabilities ust be give. This is what we do ow. We give Khihi's results i quatitative settigs, whih are ew, as far as we ow. Theore. Quatitative stateet of LIL, - Let ad. We set: (, ) = + ( Log ( + )) ( ) ad: (, ) = ( + ), ( + ) B The, for all : ( (, )) (, ) P B The set B (, ) is ade of the paths whih are above the "safety strip" after the tie ( ) + at least oe +. The Theore says that the probability of suh a evet teds to whe 3

4 ireases. I other words, if we fix a width for the safety strip, that is fixed, it beoes less ad less probable to pass above the urve ( + ) whe ireases. Let us tae for exaple =. Theore gives the estiate: P, ( Log ( ) ) For istae, if we wat the right-had side to be.5, we fid = 85. Therefore, the probability to have 85 for whih ( ) is <.5. For a better uderstadig, this stateet ay be overted ito a proportio of paths, as follows: Fix ay 85. The proportio of paths, whih satisfy ( ) at least i oe plae, betwee 85 ad, is saller tha 5%. Proof of Theore Our proof is a quatitative versio of the origial "Law of the Iterated Logarith", by A Khihi. We adapt the presetatio give by [Velei]. I all that follows, (width of the strip) is fixed, so we oit it fro ost otatio. For easy referee, the reader ay tae =. We eed several steps. We reall fro Part I, Lea, that, for all : P( ) () We also reall fro Part I, Corollary 5, that for ay real x ad ay, we have: (, ) P x P x () ad, fro Part I, Lea IV., that for ay ad ay x, x, we have: x P x e (3) We defie = +, ad, for all, iteger, we set =. The ext Lea gives a estiate o the uber of paths whih are above the strip at least oe, i the iterval of tie, + : Lea. Let C,, ( ) =. The: + 4

5 Log ( ) P C Proof of Lea Usig (), we have: P C P + ad, usig (3): ie =, + =, ad we get: P Log Log + + ( ) exp ( ) exp( ) ( ) ( ( ) ) P C Log Log = Log = Log This proves Lea. We set D,, ( ) =. We have: + Lea 3. For ay, D C. Proof of Lea 3 Ideed, if there exists a suh that the iequality ( ) ( ) holds, we have a fortiori, sie the futio is ireasig. This proves Lea 3. We dedue: Log ( ) P D (4) Let B = D ; the the sets B are dereasig whe ireases. The set Therefore,,, +, B = = B is the set of all paths whih are above the seurity strip ( ) tie. We ow estiate its probability. B is by defiitio: at least oe after Lea 4. For all, we have: 5

6 P B ( Log ) ( ) Proof of Lea 4 By defiitio of the sets: P( D ) P B + = ad Lea 4 follows fro (4) ad the iequality: + + dx = + + = x ( ) The proof of Theore is oplete. We oe ba here o what we said about the defiitio of probabilities. I all the stateets above, up to Lea 5, all probabilities refer to a bouded iterval for (for istae + ). This is ot the ase for Lea 6 ( ), but we iediately have a upper boud fro bouded itervals, dedued fro (3). We ow tur to the opposite theore: ay paths eter the safety strip. More preisely, let (sall). We will show that there is (, ) suh that if ( ),, the:,... ; P =. ( ) The stateet is as follows: 6 Theore 5. Let,. et =, = 4., Log = + Log. For ay, we have: ( ) Log = + + Log ( ) ( ) ad,, P suh that ( ) 6

7 What this stateet says, i siple words, is that it is ore ad ore uliely to stay ostatly below ( ) ( x) strip ( ) ( x), ( x). For a fixed width, the probability that eters, at soe tie, the teds to whe +. Proof of Theore 5 =. Let : it refers to the width of the safety strip, We set as before ( ) Log Log ( ) 6 is fixed ad is ost of the tie oitted fro the otatio. We itrodue = ad, for ay iteger, =. We write the quotiet uder the for of the su of two ters, whih will be treated separately: et Y = ad Z = + () ( ) large with large probability, whereas ( ) ( ) =. The geeral idea of the proof is to show that Y is Z is sall. We start with the study of Z. Lea 6. For all, we have: ( ) ( ) = 4 Proof of Lea 6 ( ) Log Log This is lear, sie = ad by the hoie of. ( ) Log Log Lea 7. Let 4 Log ( ) = + +. The: ( Log ( ) ) Log ( ), P ( ) Proof of Lea 7 7

8 Let us hoose = i Theore. We have, for ay : (, ) P ( Log ( ) ) We hoose so that 4, that is +. o we get: Log ( Log ) P(, ( ) ) that is: P, ( ) Here, = ad the oditio is satisfied as soo as: Log ( ) 4 Log ( ) ( ) ( Log ( ) ) ( ) Log Log This proves Lea 7. Lea 8. - Let be as before. If, we have:, P Z Proof of Lea 8 Ideed, for ay, we have both This proves Lea 8. ( ) ad ( ) ( ) 4 o a set of probability. We ow tur to the ter Y have, with = : 4 =. We reall fro Part I, Propositio 9 that, if, we 8

9 P( ) exp () We set: D = Propositio 9. For all, we have: P D ( ) Log / where = as before. 4 Proof of Propositio 9 ie the defiitio of, the evets D are idepedet. We have: D relies upo oseutive differees P( D ) = P P = sie ad have the sae law. I the estiate () above, we replae by ad by ( ) ; we obtai: ( ( )) Log Log ( ) exp exp = P But: = =, fro the hoie of. Therefore: P exp ( ) ( ) Log Log Log = = ( Log ( )) This proves Propositio 9. 9

10 We eed a quatitative versio of the seod Borel-Catelli Lea: Lea. Let D be a sequee of idepedet evets; let D be their opleets. The : P D exp P( D) = = Proof of Lea We have, for all : P D = ( P( D )) exp P( D ) = = = usig the iequality x e x, x. This proves Lea. et, for all, u =. We dedue fro Propositio 9 ad Lea : / / ( Log ( )) P D exp u = = ie the series of geeral ter hoosig large eough. The u is diverget, the su = u a be ade arbitrarily large, exp u = is lose to. o the itersetio D = sall probability. But this itersetio is the set of all paths for whih ay =,...,. More preisely, we have: Lea. Let as i Lea 7. We have, if = + Log has a very : for P,, Proof of Lea

11 We hoose large eough so that, if, for, we have u, ad therefore: exp = u, that is u Log. But, + u o we hoose large eough for : Log that is: + Log This proves Lea. Let us fiish the proof of Theore 5. We have: = + (3) Let E be the set: E Y =, =,..., The, Lea says that P( E ). Let E be the set: The P( E ). We have:, E = Z

12 ad therefore: ( ) P E E = P E E P E + P E = P E + P E + = P E E But E E is the set for whih there is a,, with Y ad. We de- due: Y + Z Y Z Z This fiishes the proof of Theore 5. II. Copariso with our results Let b( x ) be the barrier b( x) xlog ( x) =. Let be ay istat. The fat that the eergy left after the istat teds to whe + ay at first sight loo otraditory with Khihie's result, aordig to whih ( x) xlog Log ( x) = is a seurity urve, sie our barrier b is above Khihie's barrier. But i fat, there is o otraditio. Let us explai the situatio ore i detail. We distiguish betwee 4 types of paths (see piture): A : all paths whih ever touh b or b before the istat (suh a path is draw i bla); the proportio of suh paths at tie is what we all E (the eergy left at that tie). B + : all paths whih touh b but do ot touh b before the istat (suh a path is draw i red).

13 B : all paths whih touh b but do ot touh b before the istat (suh a path is draw i blue). C : all paths whih touh both b ad b before the istat (suh a path is draw i gree). Of ourse, these four sets are disjoit, ad their uio represets all possible paths. What we saw, for b( x) xlog ( x) + =, is that: P B B C whe +. I other words, it beoes ore ad ore uliely that a path ever touhes the barrier or its opposite. This barrier is above Khihi's urve, whih eas that the probability to touh b after the tie teds to whe +. I other words, alost every path returs ear Khihi's urve ifiitely ay ties, but this is ot so for the barrier b. This is ot otraditory with our result. It eas siply that, for istae : P( B + ). whe + P( B ). whe + PC.8 whe + We do ot ow the probability of eah situatio, beause our ethods aot hadle the separately. o the total probability of hittig b teds to whe + (our result), but the probability to hit either b or b after tie teds to whe +. The opariso betwee Khihi's ethods ad ours lead to the followig olusios: Khihi's ethods ay hadle the ase of a sigle urve, whereas ours ay hadle oly the ase of two syetri barriers; Quatitative estiates obtaied by eas of Khihi's ethods are of probabilisti type, ad uh weaer tha the results derived by eas of operator theory; To say that Khihi's urve ( x) xlog Log ( x) the urve b( x) xlog ( x) = is a "seurity urve" is iorret. Tae = whih is above Khihi's urve, ad tae two istats. 3

14 The the probability to hit bx betwee these two istats (ad thus to go above ( x ) ) is strilty positive. Referees [Velei] Y. Velei, Uiversité de Geève "Chapitres hoisis de Théorie des Probabilités" : 4