FoSP. FoSP. Convergence properties of the q-deformed binomial distribution. Martin Zeiner

Transcription

1 FoSP Algorthmen & mathematsche Modellerung FoSP Forschungsschwerpunkt Algorthmen und mathematsche Modellerung Convergence propertes of the -deformed bnomal dstrbuton Martn Zener Insttut für Analyss und Computatonal Number Theory Math A Report , February 2009

2 CONVERGENCE PROPERTIES OF THE -DEFORMED BINOMIAL DISTRIBUTION MARTIN ZEINER Abstract We consder the -deformed bnomal dstrbuton ntroduced by Jng 994 and Chung et al 995 and establsh several convergence results nvolvng the Euler and the eponental dstrbuton; some of them are - analogues of classcal results Introducton The -deformed bnomal dstrbuton was ntroduced by Jng [5] n connecton wth the -deformed boson oscllator and by Chung et al [2] Its probabltes are gven by PX QD = = τ τ; n, 0 n, 0 τ, where [ ] n n ; n = and z; n = z ; ; n are the -bnomal coeffcent and the -shfted Pochhammer symbol; an ntroducton to the -calculus and basc hypergeometrc seres can be found n Gasper and Rahman [3] A random varable X whch s -deformed bnomal dstrbuted let us denote by X QDn, τ, Elementary and asymptotc propertes are gven n Jng [5], Jng and Fan [6], Kemp [0, ] and n the encyclopaedc book Johnson, Kemp and Kotz [7] Chung et al [2] and Kupershmdt [2] gave a representaton of the -deformed bnomal dstrbuton as a sum of dependent and not dentcally dstrbuted random varables One mportant property s the convergence for fed paramter τ and n to an Euler dstrbuton wth parameter λ = τ/, e PX E = = λ ; λ; Ths s a -analogue of the Posson dstrbuton snce E λ P λ for For propertes of ths dstrbuton we refer to [7], Kemp [8, 9, ] and Charalambdes and Papadatos [] The -deformed bnomal dstrbuton s a -analogue of the classcal bnomal dstrbuton, snce n the lmt the -deformed bnomal dstrbuton wth paramter n, τ, reduces to the bnomal dstrbuton wth parameters n, τ Ths paper s devoted to study seuences of random varables X n, where the X n are QDn, τ n, -dstrbuted and τ n converges to a lmt c [0, ] Secton 2 deals wth the case c [0, In partcular we show that there s a -analogue of the Date: February 25, Mathematcs Subect Classfcaton Prmary: 60F05; Secondary: 05A30, 33D99, 62E5 Key words and phrases -deformed bnomal dstrbuton, -Posson dstrbuton, Eulerdstrbuton, eponental dstrbuton, lmt theorems M Zener was supported by the NAWI-Graz proect and the Austran Scence Fund proect S96 of the Natonal Research Network S9600 Analytc Combnatorcs and Probablstc Number Theory

3 2 MARTIN ZEINER convergence of the classcal bnomal dstrbuton wth constant mean to a Posson dstrbuton In Secton 3 we eamne the case c = Here the lmt dstrbuton can be - dependng on the growth rate of the parameters τ n - degenerate, truncated eponental-lke or eponental 2 Parameter seuences wth lmt < In the present secton we study seuences of random varables X n whch are QDn, τ n, -dstrbuted, where the parameters τ n converge to a c [0, In partcular we prove a -analogue of the convergence of the classcal bnomal dstrbuton wth constant mean to a Posson dstrbuton As noted above the seuence converges n the case of constant parameters τ n = τ to an Euler dstrbuton wth parameter τ/ The followng theorem shows that the Euler dstrbuton s the lmt dstrbuton for every convergent parameter seuence τ n wth lmt n [0, Theorem 2 Let X n QDn, τ n, Then, for n, f τ n τ and 0 τ < X n Eτ, Proof Observe that [4, Lemma 2] stll remans true for θ > Then the theorem follows mmedately from the facts e = /E and ; for n We are now nterested n specal choces of the parameters τ n such that the lmt X of the seuence X n converges to a Posson dstrbuton for From the prevous theorem we conclude mmedately Theorem 22 Let X n QDn, τ n, wth τ n λ n for and τ n τ for n wth the addtonal property τ λ n the lmt Then the followng dagram s commutatve: n QDn, τ n, Eτ, B n, λ n P λ n One very natural way to choose the parameters s to set τ n = λ [n] Our net goal s to establsh a convergence result, whch s analogous to the convergence of the classcal bnomal dstrbuton wth constant mean to a Posson dstrbuton and reduces n the lmt to that theorem For ths purpose we start wth an elementary fact Lemma 23 Let f n, n N, be a seuence of contnuous functons whch converge pontwse to a contnuous lmt f Assume that for each n the functon f n has a sngle root ˆ n, and f has a sngle root ˆ Then ˆ n ˆ Proof For gven ε > 0 choose a δε < mnfˆ + ε, fˆ ε Then there ests a N = Nδε such that for all n N we have f n ˆ + ε fˆ + ε < δε Therefore f n ˆ + ε > 0 Moreover there ests a M = Mδε such that for all n M we have f n ˆ ε fˆ ε < δε Therefore f n ˆ ε < 0 Hence, by contnuty, for all n man, M we have ˆ ˆ n < 2ε

4 CONVERGENCE PROPERTIES OF THE -DEFORMED BINOMIAL DISTRIBUTION 3 The essental key to apply ths lemma s the followng representaton of the means µ n τ,, whch allows us to etract mportant propertes of the means easly Lemma 24 The means µ n τ, have the alternatve representaton n µ n τ, = ; τ = Proof By defnton, the means are gven by n µ n τ, = τ τ; n = Usng the well-known formula eg see [3, p 20] z; k = we obtan µ n τ = k [ ] k /2 z n τ = n Etractng coeffcents n ths formula we get 2 [τ ]µ n τ = [ n [ ] [ ] n n = = [n]! [n ]! Now we show by nducton that S := = = = ] /2 τ /2 /2 []![ ]! /2 []! []![ ]! = ; For = ths s true Now suppose that ths formula s shown for Usng the recurson formula for the bnomal coeffcents and splttng sums we get [ ] [ ] S = + /2 + [ = + 2/2+ + = [ ] + /2 + =2 ] [ ] /2 Notce that the frst sum euals S Shftng ndces n the other sums we obtan 2 [ ] S = ; /2 = 2 [ ] + + 2/2 =0 =

5 4 MARTIN ZEINER Now combne the second and thrd term as well as the fourth and the ffth term to [ ] S = ; 2 + 2/2 =0 = ; + ; = ;, where we used formula Remark 25 An alternatve way to prove ths lemma s to use Kemp s [0, p 300] representaton of the probablty generatng functon, to dfferentate and to manpulate the sum Usng the monotoncty of the -bnomal coeffcents n n we mmedately get Theorem 26 The means µ n τ, are strctly ncreasng n n for τ > 0 and τ Now we turn to the convergence result: Theorem 27 F µ > 0 Then the followng commutatve dagram s vald: n QDn, τ n, Eτ, B n, µ n P µ n wth τ n determned by µ n τ, = µ and τ = lm n τ n Proof Frst we check, that for gven µ > 0 and fed there ests a unue seuence τ n n N wth µ n τ n, = µ The functon µ n τ, s contnuous and strctly ncreasng n n and τ by the prevous theorem Moreover, we have lm τ 0 µ n τ, = 0 Choosng τ n n such a way that τn n, then µ n τ n, becomes arbtrary large Conseuently there s a unue soluton of µ n τ, = µ By Lemma 23 the seuence τ n converges to a lmt τ where τ s the unue soluton of µ E τ, = µ, where µ E τ, s the mean of an Euler-dstrbuton mt parameters τ and Ths mean can be wrtten as τ µ E τ, = τ, see [8] or take the lmt n n Lemma 24 and manpulate the sum Agan be Lemma 23 we get that τ µ/n It remans to check that τ n / converges to µ n the lmt But ths s agan a conseuence of Lemma 23 snce τ/ s the unue soluton of µ E τ, = µ and µ E τ, tends to τ for 3 Parameter seuences wth lmt In ths secton we nvestgate n seuences X n of random varables, where X n s QDn, τ n, -dstrbuted and the parameters τ n converge to The behavour of the seuences X n depends on the growth rate of τ n Therefor we wll dstngush three cases: Frstly we eamne the case τn n, where t wll turn out that the lmt dstrbuton s degenerate Then we study the case τn n c wth 0 < c < Here the lmt dstrbuton wll depend only on c and be lke a truncated eponental dstrbuton Fnally we turn to the case τn fn c where 0 < c < and fn = on; ths wll lead to an eponental dstrbuton Consder seuences of random varables X n QDn, τ n, wth τ n and addtonally τn n frst Then we have the followng theorem:

6 CONVERGENCE PROPERTIES OF THE -DEFORMED BINOMIAL DISTRIBUTION 5 Theorem 3 Let X n QDn, τ n, wth τ n and τn n Then n X n converges to the pont measure at 0 Proof The probablty that Y n = n X n s eual to 0 s gven by whch converges to by assumpton PY n = 0 = τ n n Now let us nvestgate n seuences X n QDn, τ n,, where τ n and τn n c for a c 0, Before we can establsh the dstrbuton of the lmt of such a seuence, we start wth several lemmas, whch allow us to compute the asymptotc behavour of certan sums of probabltes of QDn, τ n, -dstrbuted random varables and ts means and varances The frst lemma s an analogue to Lemma 24 and gves an alternatve representaton of the varance: Lemma 32 The second moment of X n τ, can be wrtten as n n 2 τ ; n = a τ wth a = = ; + 2 = = Proof Startng as n the proof of Lemma 24 we obtan for the coeffcents of τ n EX 2 the formula [ ] [τ ]EX 2 = 2 /2 = It remans to show that [ ] T := 2 /2 = = ; + 2 For ths purpose we use nducton For = ths formula s true Now suppose that t has been shown for We use smlar deas as before and wrte [ ] [ ] T = /2 + = [ ] = 2 T + 2 /2 =2 ] [ + 2 /2 = Shftng nde n the frst sum and addng 2 we obtan T = T /2 = = 2 [ ] T /2 =0 ] 2 [ + + 2/2 =0 =

7 6 MARTIN ZEINER Now we use the proof of Lemma 24 agan and nducton hypothess to get T = T + 2S = 2 ; whch completes the proof = 2 = ; = = ; + 2 =, + 2; 2 The net three lemmas are devoted to the asymptotc behavour of sums of powers of θ n, where 0 < θ n < and θ n Lemma 33 If fn for n and θ n 0 < c <, then θ n fn log c, n Proof We use Euler-MacLaurn s formula and obtan θn = 0 θ nd log θ n = log θ n log θ n 0 = fn + log θn fn 2 + log θ n snce the term wth the ntegral s bounded 0 such that θ fn n {} θ 2 nd {} 2 0 θ n {} θn fn 2 log c, c wth Lemma 34 For θ n and θ n, θ fn n c for a c 0, and gn/fn β, gn n we have and as n gn gn θ n cβ log c fn θn e cβ log c fn Proof For the frst sum, we rewrte ths sum as gn θn = θ gn + n θ n The growth of the denomnator s gven n Lemma 33, and the numerator tends to c β, snce θn gn = θn gn {gn} c β because of θ n

8 CONVERGENCE PROPERTIES OF THE -DEFORMED BINOMIAL DISTRIBUTION 7 gn To get the asymptotc of the second sum we wrte θn = gn θn + gn gn gn + θn + gn gn gn θ n The frst and the thrd sum on the rght-hand sde are O gn and therefore asymptotcally neglgble The second sum st bounded by ; n ; gn + ; n gn gn gn gn + ; n ; 2 n/2 θ n gn gn gn + gn gn gn + θ n θn By the frst part of ths lemma the lower and the upper bound has the asserted asymptotc Lemma 35 If θ n and θ n wth θn n c for 0 < c <, then n θn c + c log c log 2 n 2 c and as n n θn e c + c log c log 2 n 2 c Proof To estmate ths sum we use Lemma 33 agan and the dentty n t = t tn nt n t t 2 Hence, settng t = θ n, n θn c nθn n θ n n2 log 2 c + c log c n2 c log 2 c Here we used that under the assumpton τ n n c we have τ n n log c To see ths, wrte τ n = c /n + fn Then τ n n = c + n = n fn c n n =: c + s n Now suppose for a moment fn 0 The sum s n must tend to 0 and we have s n = s n fnnc Therefore fnn 0 Smlarly, fnn 0 for negatve f Conseuently fnn 0 for arbtrary f Now consder τ n n = n c /n + nfn log c The asymptotc for the sum wth the -bnomal coeffcent s obtaned as n Lemma 34 Now we are ready to establsh the essental key n provng the convergence result: we gve the asymptotc behavour of sums of probabltes and the means and varances of QDn, τ n, -dstrbuted random varables

9 8 MARTIN ZEINER Lemma 36 Let X n be QDn, τ n, -dstrbuted and denote wth µ n τ n, and σnτ 2 n, the correspondng mean and varance If τ n and τn n c wth 0 < c < and fn βn, fn < n, then as n fn =0 τ n τ n ; n c β µ n τ n, c log c n σ 2 nτ n, + 2c log c c2 log c 2 n 2 Proof We start the frst asserton Snce fn < n we can wrte S n := fn =0 τ n [ ] fn n τ n ; n = τ n The summands are bounded by e 2, hence fn n S n τ n = n τ n =0 τ n n τ n = n τ n =: Ŝn Estmatng the product and usng agan the boundedness of the summands yelds = Ŝ n τ n τ n ; n fn + n fn τ n τ n ; n fn + n n τn = n n τn =: Ŝ n Snce [4, Lemma 2] remans true for θ n > ε > and wth use of Lemma 34 wth gn := fn and fn := n we obtan Ŝ n τ n e e cβ log c n cβ In an analogous way we fnd a lower bound of Ŝn that s asymptotc euvalent to c β Now we prove the second proposton of the lemma: Use Lemma 24, easy estmates of the -Pochhammer symbol and the asymptotcs gven n Lemma 34 to obtan and µ n τ n, n = ; n n ; 2 + ; n n/2 µ n τ n, ; n n = = n τn e e c log c n τn c log c n Smlarly we proceed for the second moments of X n τ n, and estmate wth use of Lemma 35 n n EXn 2 ; + 2 τn 2; n τn 2 c + c log c log c 2 n 2 = =

10 CONVERGENCE PROPERTIES OF THE -DEFORMED BINOMIAL DISTRIBUTION 9 To bound the second moment from above we do the followng: n EXn 2 ; n ; 2 + 2n n [n + ; + 2 n/2 Thus Hence = = on n = n n = n on 2 + 2; n ; + 2 ; + 2 n = n = on 2 + 2; n n n 2 c + c log c log c 2 n 2 = n [ ] n τn = [ ] n τ n n τn = EX 2 nτ n, 2 c + c log c log c 2 n 2 σ 2 nτ n, = EX 2 nτ n, µ n τ, 2 + 2c log c c2 log c 2 n 2 2 c + c log c log c 2 = τn 2 c n 2 log c After ths anayss of the means and varances t s now easy to obtan the lmtng dstrbuton of the seuence X n Theorem 37 Let Y n QDn,, τ n wth τ n and τn n c wth 0 < c < Then the seuence of the normalsed random varables X n = Y n µ n /σ n converges to a lmt X wth PX = e c e +2c log c c 2 for [ c c log c + 2c log c c 2 + 2c log c c 2 ] τn and P = for = c log c + 2c log c c 2 Proof The support of X s gven by [ lm µ nτ n, n σ n τ n,, lm n 0 y σ n+µ n τ y n ] n µ n τ n, σ n τ n, Usng Lemma 36 the stated support follows mmedately Computng the dstrbuton functon of X yelds wth use of Lemma 36 PX n = τ n ; n y c α y

11 0 MARTIN ZEINER wth for α = + 2c log c c 2 < log c + c log c c log c + 2c log c c 2 Smplfyng c α yelds the theorem Now we turn to the thrd case, whch treats wth seuences of random varables X n QDn, τ n, where τ n and τn fn c for a c 0, and fn = on Ths case s very smlar to the prevous one, and so we start wth an analogue of Lemma 35 Lemma 38 Let fn, fn = on, θn fn c wth 0 < c < Then n n [ ] θn fn2 n log 2 and θn e fn2 c log 2 c as n Proof Follow the proof of Lemma 35 and observe that nθ n n θ n tends to zero Followng the proof of Lemma 36 and usng Lemma 38 nstead of Lemma 35 we obtan Lemma 39 If τ n and τn fn βfn, then as n gn =0 τ n c wth 0 < c < and fn = on, gn τ n ; n c β µ n τ n, fn log c σ 2 nτ n, fn2 log c 2 As an mmedate conseuence we get the dstrbuton of the lmt of X n, whch s an eponental dstrbuton and s agan ndependent of Theorem 30 Let Y n QDn,, τ n wth τ n and τ fn n c wth 0 < c < and fn = on Then the seuence of the normalsed random varables X n = Y n µ n /σ n converges to a normalsed eponental dstrbuton wth parameter, e PX = e, Proof Lemma 39 yelds mmedately that the support of the lmt dstrbuton s [, Computng the dstrbuton functon gves PX = τ n ; n y c + log c = e y 0 y σ n+µ n τ y n Comparng ths result wth Theorem 37 we see that ths corresponds to takng the lmt c 0

12 CONVERGENCE PROPERTIES OF THE -DEFORMED BINOMIAL DISTRIBUTION References [] C A Charalambdes and N Papadatos, The -factoral moments of dscrete - dstrbutons and a characterzaton of the Euler dstrbuton, n Advances on models, characterzatons and applcatons, vol 80 of Stat Tetb Monogr, Chapman & Hall/CRC, Boca Raton, FL, 2005, pp 57 7 [2] W-S Chung, K-S Chung, H-J Kang, and N-Y Cho, -deformed probablty and bnomal dstrbuton, Internat J Theoret Phys, , pp [3] G Gasper and M Rahman, Basc hypergeometrc seres, vol 35 of Encyclopeda of Mathematcs and ts Applcatons, Cambrdge Unversty Press, Cambrdge, 990 Wth a foreword by Rchard Askey [4] S Gerhold and M Zener, Convergence propertes of Kemp s -bnomal dstrbuton preprnt [5] S C Jng, The -deformed bnomal dstrbuton and ts asymptotc behavour, J Phys A, , pp [6] S C Jng and H Y Fan, -deformed bnomal state, Phys Rev A 3, , pp [7] N L Johnson, A W Kemp, and S Kotz, Unvarate dscrete dstrbutons, Wley Seres n Probablty and Statstcs, Wley-Interscence [John Wley & Sons], Hoboken, NJ, thrd ed, 2005 [8] A W Kemp, Hene-Euler etensons of the Posson dstrbuton, Comm Statst Theory Methods, 2 992, pp [9], Steady-state Markov chan models for the Hene and Euler dstrbutons, J Appl Probab, , pp [0], Certan -analogues of the bnomal dstrbuton, Sankhyā Ser A, , pp Selected artcles from San Antono Conference n honour of C R Rao San Antono, TX, 2000 [], Characterzatons nvolvng U U + V = m for certan dscrete dstrbutons, J Statst Plann Inference, , pp 3 4 [2] B A Kupershmdt, -probablty I Basc dscrete dstrbutons, J Nonlnear Math Phys, , pp Martn Zener Graz Unversty of Technology, Department for Analyss and Computatonal Number Theory, Steyrergasse 30/II, 800 Graz, Austra E-mal address: zener at fnanzmathtu-grazacat