TitleA random walk analogue of Levy's th. Studia scientiarum mathematicarum H Citation

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1 TleA random walk analogue of Levy's h Auhor(s) Fuja, Takahko Suda scenarum mahemacarum H Caon 3-33 Issue Dae Type Journal Arcle Tex Verson auhor URL hp://hdlhandlene/10086/15876 Ths s an auhor's verson of an a Rghorgnal publcaon s avalable a hp://dxdoorg/101556/sscmah4 Hosubash Unversy Reposory

2 A random walk analogue of Lévy s heorem 1 Takahko Fuja Graduae School of Commerce and Managemen, Hosubash Unversy, Naka -1, Kunach, Tokyo, , Japan, E-mal:fuja@mahh-uacjp Absrac: In hs paper, we wll gve a smple symmerc random walk analogue of Lévy s Theorem We wll ake a new defnon of a local me of he smple symmerc random walk We apply a dscree Iô formula o some absolue value lke funcon o oban a dscree Tanaka formula Resuls n hs paper rely upon a dscree Skorokhod reflecon argumen Ths random walk analogue of Lévy s heorem was already obaned by GSmons([14]) bu s sll worh nong because we wll use a dscree sochasc analyss o oban and hs mehod s applcable o oher research We noe some connecon wh prevous resuls by Csák, Révész, Csörgő and Szabados Fnally we observe ha he dscree Lévy ransformaon n he presen verson s no ergodc Lasly we gve a Lévy s heorem for smple nonsymmerc random walk usng a dscree bang-bang process Keywords: smple symmerc random walk, local me, dscree Iô s formula, Lévy s Theorem, dscree Tanaka-Meyer formula, dscree Skorokhod equaon, Lévy ransformaon, smple nonsymmerc random walk, dscree bang-bang process 000 AMS subjec classfcaon : 60G50, 60H05, 60J55, 60J65 1 Inroducon The followng celebraed Lévy s Theorem([8], [1]) s well known: (M W, M ) =( W, L ) n law, where W s a Brownan moon, M = max 0 s W s and L =he local me of W a 0=(lm ɛ 0 (1/ɛ) 0 1 ( ɛ,ɛ)(w s )ds) Frs hs paper shows ha here exss a smple symmerc random walk analogue of hs heorem Second we remark ha Dscree Lévy s ransformaon s no ergodc on pah space, whle he queson (he orgnal Lévy s ransformaon s ergodc or no) s sll open Las we gve he Lévy s heorem for a smple nonsymmerc random walk, usng a dscree bang-bang process 1 Ths paper appeared on Suda Scenarum Mahemacarum Hungarca, Vol45, No, (008),

3 Facs Le Z be a smple symmerc random walk, ha s, Z = ξ 1 +ξ + ξ,z 0 =0 where ξ 1,,ξ are d wh P (ξ = 1) = P (ξ = 1) = 1/ We pu M = max 0 s Z s, whle he local me of Z a 0 up o he me s defned as: L = { =0, 1(Z =0 Z +1 =1) (Z =1 Z +1 =0)} Then we oban he followng heorem: Theorem 1 (M Z,M ) =( Z,L ) n law where x := max (x 1, x) (a quas-absolue value funcon) Before provng hs heorem, we prepare a dscree Iô formula for he smple symmerc random walk Ths formula was obaned n [5] when he auhor suded a dervave prcng n a dscree me model Lemma (Dscree Iô formula [5]) f(z +1 ) f(z )= f(z +1) f(z 1) (Z +1 Z )+ f(z +1) f(z )+f(z 1) f(z ) f(0) = f(z +1) f(z 1) (Z +1 Z ) + (Doob-Meyer Decomposon of f(z )) Proof f(z +1) f(z )+f(z 1) f(z +1 ) f(z ) f(z +1) f(z )+f(z 1) = { f(z+1) f(z 1) f ξ +1 =1 f(z 1) f(z +1) f ξ +1 = 1

4 = f(z +1) f(z 1) (Z +1 Z ) (qed) ξ (p) Remark 1 Le Z (p) be a smple nonsymmerc random walk,ha s, Z (p),x 0 = 0 where ξ (p) 1,,ξ are d wh P (ξ (p) = ξ (p) 1 + ξ (p) + =1)=p, P (ξ (p) = 1) = q =1 p In hs case, Iô s formula s he same as n he smple symmerc random walk case, as follows: f(z (p) +1 ) f(z(p) )= f(z(p) +1) f(z (p) 1) (Z (p) +1 Z(p) ) + f(z(p) +1) f(z (p) )+f(z (p) 1) f(z (p) ) f(0) = f(z (p) +1) f(z (p) 1) (Z (p) +1 Z(p) (p 1)) 1 + ( f(z(p) +1) f(z (p) )+f(z (p) 1)) (Doob Meyer Decomposon of f(z (p) )) +(p 1) f(z(p) +1) f(z (p) 1) Remark Szabados ([15]) { obaned he followng dscree verson of Iô formula Le g(k) =ɛ k (1/)f(0) + } k 1 j=1 f(ɛ kj)+(1/)f(k) where 1 (k >0) ɛ k = 0 (k =0) 1 (k <0) He called g a rapezodal sum of f The followng s Szabados s verson of he dscree Iô formula g(z )= f(z )ξ +1 +(1/) 3 f(z +1 ) f(z ) ξ +1

5 Usng hs formula and lmng procedure, Szabados proved Iô s formula W for Brownan moon n he followng form: 0 f(s)ds = 0 f(w s)dw s + (1/) 0 f (W s )ds and nvesgaed some consequences We noe ha hs paper s verson of dscree Iô formula also yelds Iô formula f(w ) f(0) = 0 f (W s )dw s +(1/) 0 f (W s )ds by usng lmng procedure by an approprae scale change akng Z δ = δz [/δ] Acually Fuja and Kawansh([6]) prooved he Iô formula usng hs paper verson of dscere Iô s fromula So f we consder he lm case, boh dscree versons of Iô s formula gve he same resul Bu whn he dscree case here exs some dfferences because hs paper s verson gves he Doob Meyer decomposon of f(z ), whle Szabados s verson does no gve n general Proof of Theorem 1 Applyng f(x) = x o dscree Iô formula, we have ha Z = Z +1 Z 1 (Z +1 Z ) We noe ha +(1/) ( Z +1 Z + Z 1 ) x +1 x 1 1 x 1/ x =1 = 1/ x =0 1 x 1 So pung sgn(x) = { 1 x 1 1 x 0, h (x) = 1/ x =1 h 1 (x) = 1/ x =0, 0 oherwse x +1 x + x 1 = { 1/ x =0, 1 0 oherwse, 4

6 Z = sgn(z )(Z +1 Z )+ h (Z ) h 1 (Z )(Z +1 Z ) holds Here we wll show ha L = 1 h (Z ) 1 h 1(Z )(Z +1 Z )by nducon For =1, clearly, L 1 = h (Z 0 ) h 1 (Z 0 )(Z 1 Z 0 )= 1+Z1 holds Assumng, h (Z ) h 1 (Z )(Z +1 Z )=L + h (Z ) h 1 (Z )(Z +1 Z ) So we have ha = L +1 (Z=0 Z +1=1) (Z =1 Z +1=0) = L +1 Z = sgn(z )(Z +1 Z )+L (Dscree Tanaka formula) Pung Ẑ = 1 sgn(z )(Z +1 Z ), we remark ha Ẑ s clearly a smple symmerc random walk On he oher hand, we have ha M Z = Z + M holds So he unqueness of followng dscree Skorokhod Equaon gves a proof of hs heorem (qed) Lemma ( Dscree Skorokhod Lemma) (For a proof of he connuous versons, see [6]) Le us defne he followng hree pah spaces: Ω 1 = {f f :Z + Z, f(0) = 0, and x Z +,f(x +1) f(x) =±1} If we use gamblng ermnology, a smple symmerc random walk Z s he amoun of money whch a gambler A makes afer mes red or black play wh equal probably f hs each be s 1 $ Le us assume ha he gambler A connue o play red connuously Consder anoher gambler B whose -h play s black f Z 1 5 0, red f Z 1 = 1 wh hs each 1 $ be Then Ẑ s he amoun of money whch he gambler B makes afer mes play and clearly s also a smple symmerc random walk 5

7 Ω = {f f :Z + Z + and x Z +,f(x +1) f(x) =0or± 1} Ω 3 = {f f :Z + Z + and x Z +,f(x +1) f(x) =0or1, and,f(0) = 0} where Z + = {x x 0,x Z} Gven f Ω 1 and x Z, here exs unque g Ω and h Ω 3 such ha 1 g() =x + f()+h(), h( +1) h() > 0 only f g() = 0 e h() ncreases only on g()=0 Proof Se g() =x+f() mn 0 s (mn(x+f(s), 0), h() = mn 0 s (mn(x+ f(s), 0)) We can easly see ha g() and h() sasfy boh condons above We wll prove he unqueness Suppose ĝ(),ĥ() sasfy he above condons Then g() ĝ() = h() ĥ() for all 0 If here exss 1 N such ha g( 1 ) ĝ( 1 ) > 0, we se = max{ < 1 g() ĝ() =0, N {0}} Then g() > ĝ() 0 for all < 1 and hence by he above condons h( 1 )=h( ) = 0 Snce ĥ s ncreasng, we have ha 0 <g( 1) ĝ( 1 )= h( 1 ) ĥ( 1) h( ) ĥ( )=g( ) ĝ( )=0 Ths s a conradcon Here ĝ g and ĥ h (qed) Remark 3 Ths precse random walk analogue of Lévy s heorem was already obaned by G Smons([14]) He gave a proof of hs heorem by smlar dscussons bu whou a dscree sochasc calculus Remark 4 Sasho and Tanemura([13]) dsplayed smlar dscree Skorokhod equaons hrough her research abou Pman ype heorem for one dmensonal dffusons and Smlarly, we have he followng facs We pu L = { =0, 1,Z =0 Z +1 =1} L + = { =0, 1,Z =1 Z +1 =0} 6

8 Then max(z 1, 0) = 1 (Z 1)(Z +1 Z )+L + max(0, Z )= 1 (Z 0)(Z +1 Z )+L = 1 (Z 0)(Z +1 Z )+(1/) 1 {0} (Z ) (1/) 1 {0} (Z )(Z +1 Z ) max(z, 0) = So we have ha 1 (Z 0)(Z +1 Z )+(1/) 1 {0} (Z )+(1/) 1 {0} (Z )(Z +1 Z ) Z = (1 (Z 0) 1 (Z 0))(Z +1 Z )+ 1 {0} (Z ) = (sgn(z )(Z +1 Z )+ 1 {0} (Z )+ 1 {0} (Z )(Z +1 Z ) = sgn(z )(Z +1 Z )+L Then we have also he followng heorem Theorem Z L = Z n law Remark 4 Csák, Csörgő and Révész ([1],[],[11])consdered a local me of random walk n he followng way: ζ(x, ) = {, Z = x} 7

9 and hen hey showed ha ζ(x, ) = Z x x sgn(z ˆ x)ξ +1 where sgn(x) ˆ = 1 x 1 0 x =0 1 x 1 Csák and Révész([1]) obaned a nearly rue Lévy s heorem for a smple symmerc random walk usng ζ(x, ) We remark ha her verson of a dscree Tanaka Meyer formula s also dfferen from hs paper s verson bu we pon ou ha applyng hs paper s verson of dscree Iô formula o f(y) = y x, hs verson of dscree Tanaka Meyer formula s very easly obaned Here we also noe ha Myazak and Tanaka ([10],[16]) also researched a random walk analogue of Pman s heorem Remark 5 Ths knd of problem s also relaed o he so called Lévy s ransformaon: W Ŵ = 0 sgn(w s )dw s, whch s measure-preservng on pah space Wheher hs ransformaon s ergodc or no, a queson rased by DRevuz and MYor([1]), s sll open Dubns and Smorodnsky([3]) gave a proof of ergodcy n he modfed, dscree and nfne me horzon case Roughly speakng, her defnon of Lévy ransform s Z = he one ha skppng he fla pah from 1 sgn(z ˆ )ξ +1 Also Dubns, Emery and Yor([4]) made some consderaons abou Lévy ransformaon on some connuous marngales We noe ha for orgnal Lévy s ransformaon problem, Malrc([9]) obaned recenly ha he unon of zero pons of eraed Lévy s ransforms s as dense n R + Ths paper s verson of Lévy s ransformaon s he followng naural generalzaon of he connuous Lévy s ransformaon: Defnon 1 T (Z ) =Ẑ = sgn(z )(Z +1 Z ) s called dscree Lévy ransformaon 8

10 Observaon Ths Lévy s ransformaon T :Ω M 1 Ω M 1 s no ergodc f M 4 where Ω M 1 = {f f : {0, 1,M} Z, f(0) = 0, and x N {0},f(x +1) f(x) =±1} If we ake M = 4, we observe very explcly ha T 8 = d Also we can observe ha when M =5,T 16 = d, when M =6, T 3 = d, when M =7, T 3 = d, We denoe he pah Z 0 =0,Z 1 =1,Z =,Z 3 =3,Z 4 = 4 as , he pah Z 0 =0,Z 1 = 1,Z =,Z 3 = 1,Z 4 = as + Then we observe ha T (+ + ++) = +++, T( + ++) = + +, T(+ +) = +, T( + ) =++ +, T(+ + +) = + +, T( + +) = + +, T(+ + ) =, T( )=++++ and T (+ + + ) = ++, T( ++ ) =+, T(+ )= ++, T( ++) = + +, T(+ + ) = +, T( + ) =+ ++, T(+ ++) = +, T( +) = +++ So pung Ω 4 1 = {++++, +++, + +, +, ++ +, + +, + +, } and Ω 4 1 = {+++, ++, +, ++, ++, +, + ++,, +}, Ω 4 1 =Ω4 1 Ω4 1 holds, e Ω 4 1 has wo ergodc componens Also denong Z 0 =0,Z 1 =1,Z =,Z 3 =3,Z 4 =4, as++++, pung Ω 4 1 = {++++, +++, + +, +, ++ +, + +, + +, } and Ω 4 1 = {+++, ++, +, ++, ++, +, + ++, + }, we oban Ω 1 =Ω 4 1 Ω4 1 holds e Ω 1 has a leas wo ergodc componens Defnng φ(m) = nf{ T :Ω M 1 ΩM 1, T = d}, we noe ha generally φ(m) s so far no known Las we gve he Lévy s heorem for a smple nonsymmerc random walk Z (p) Theorem 3(Lévy s heorem for Z (p) ) where J (p) (Z (p) mn Z s (p), mn 0 s 0 s (Z s (p) )=( J (p),l J (p) ) n law (J (p) 0 = 0) s a dscree bang-bang process whch s defned as follows: he ranson probably { p(x, y) ofj (p) s he followng: p y = x +1 For x 1, p(x, y)= 1 p y = x 1 { 1 p y = x +1 For x 0, p(x, y)= p y = x 1 We noe ha L J (p) 0, 1, (J (p) =0 J (p) +1 s he local me of J (p) a 0 up o he me := { = (p) =1) (J =1 J (p) +1 =0)} 9

11 Proof of Theorem 3 We consder he followng sochasc dfference equaon: X +1 X = sgn(x )(Z (p) +1 Z(p) )(X 0 =0) By he defnon of J (p), X =J (p) For J (p), applyng he dscree Iô formula, we ge ha Here we noe ha 1 Z (p) )=Z (p) ha J (p) = sgn(j (p) )(J (p) +1 J (p) )+L J (p) (p) sgn(j )(J (p) Then by Dscree Skorokhod Lemma and J (p) +1 J (p) )= 1 (p) (sgn(j )) (Z (p) = Z (p) +1 + L J (p), we ge References L J (p) = mn 0 s Z (p) s, J (p) = Z (p) mn 0 s Z (p) s [1] Csák,E and Révész,P: A combnaoral proof of a heorem of P Lévy on he local me, Aca Sc Mah, 45, (1983), [] Csörgő, M and Révész,P:On Srong Invarance for Local Tme of Paral Sums, Soc Proc Appl 0, (1985), [3] Dubns, L and Smorodnsky, M : The modfed dscree Lévy s ransformaon s Bernoull, Sem Prob XXVI Lecure Noes n Mahemacs, Vol 156 Sprnger, (199), [4] Dubns, L, Emery, M, and Yor M : On he Lévy ransformaon of Brownan moons and connuous marngales, Sem Prob XXVII Lecure Noe n Mahemacs, Vol 1557, Sprnger, (1993), 1-13 [5] Fuja,T, : Marngale mehods n prcng dervaves (n Japanese), The Hosubash Revew, Vol 15, No10,(001), 1-6 [6] Fuja, T and Kawansh,Y : A proof of Iô s formula usng dscree Iô s formula, n hs Volume [7] Ikeda N and Waanabe S : Sochasc dfferenal equaons and dffuson processes, Second Ed, Norh-Holland Publ Co, Amserdam Oxford New York; Kodansya Ld, Tokyo,(1989) 10

12 [8] Lévy, P : Processus sochasques e mouvemen Brownen, Gauher- Vllars, Pars,(1948) [9] Malrc, M : Densé des zéros des ransformés de Lévy érés d un mouvemen brownen, Compes Rendus Acad Sc Pars, Ser I 336 (003), [10] HMyazak and HTanaka : A heorem of Pman ype for smple random walks on Z d, Tokyo J Mah, Vol 1, No1 (1989), [11] Révész, P : Random Walk n Random and Non-random Envronmens, World Scenfc,nd ed (005) [1] Revuz, D, and Yor, M Connuous Marngales and Brownan Moon, Thrd Ed, correced prnng, Sprnger, (005) [13] Sasho, Y and Tanemura, H :Pman ype heorem for one-dmensonal dffuson Processes, Tokyo J Mah Vol 13, No, (1990), [14] Smons G : A dscree analogue and elemenary dervaon of Lévy s equvalence for Brownan moon, Sa and Prob Le 1, (1983), [15] Szabados, T: An elemenary nroducon o he Wener process and sochasc negrals, Suda Scenarum Mahemacarum Hungarca 31 (1996), [16] Tanaka,H : Tme reversal of random walks n one-dmenson, Tokyo J Mah, Vol1, No 1, 1989),