Convergence properties of Kemp s q-binomial distribution. Stefan Gerhold and Martin Zeiner

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1 FoSP Algorithmen & mathematische Modellierung FoSP Forschungsschwerpunkt Algorithmen und mathematische Modellierung Convergence properties of Kemp s -binomial distribution Stefan Gerhold and Martin Zeiner Institut für Analysis und Computational Number Theory Math A Report 008-, May 008

2 CONVERGENCE PROPERTIES OF KEMP S -BINOMIAL DISTRIBUTION STEFAN GERHOLD AND MARTIN ZEINER Abstract. We consider Kemp s -analogue of the binomial distribution. Several convergence results involving the classical binomial, the Heine, the discrete normal, and the Poisson distribution are established. Some of them are -analogues of classical convergence properties. From the results about distributions, we deduce some new convergence results for -Krawtchouk and -Charlier polynomials. Besides elementary estimates, we apply Mellin transform asymptotics.. Introduction Kemp [5] introduced the following -analogue KBn, θ, of the binomial distribution: [ ] n θ / PX KB, 0 n, 0 < θ, θ; n where [ ] n n ; n and z; n z i k ; k ; n k are the -binomial coefficient and the -shifted factorial, respectively. For properties and applications of this distribution, see Johnson, Kemp, and Kotz [4], Kemp [6], and Kemp and Newton [7]. For an introduction to -calculus we refer to the monograph by Gasper and Rahman [3]. As, this distribution tends to a binomial distribution: θ KBn, θ, B n, + θ For n, it tends to the Heine distribution Hθ:. PX H / θ ; e θ, 0, where e z, z C \ { i : i 0,,,... }, z; is a -analogue of the eponential function, since e z e z. The Heine distribution is a -analogue of the Poisson distribution, since H θ P θ. Date: June 4, Mathematics Subject Classification. Primary: 60F05; Secondary: 33D5. Key words and phrases. -binomial distribution, discrete normal distribution, Heine distribution, -Krawtchouk polynomials, -Charlier polynomials, Mellin transform, limit theorems. S. Gerhold received financial support by Christian Doppler Laboratory for Portfolio Risk Management PRisMa Lab.The fruitful collaboration and support by the Bank Austria Creditanstalt BA-CA and the Austrian Federal Financing Agency ÖBFA through CDG is gratefully acknowledged. M. Zeiner was supported by the NAWI-Graz project and the Austrian Science Fund project S96 of the National Research Network S9600 Analytic Combinatorics and Probabilistic Number Theory.

3 STEFAN GERHOLD AND MARTIN ZEINER Moreover, we need a second -analogue of the eponential function, the function E z z;. Note that we have e ze z. The random variable X KB can be written as the sum of independent Bernoulli random variables [7], which leads to the epressions µ n θ i n + θ i and σ θ i + θ i for the mean and variance. We are now interested in seuences of random variables X n with X n KBn, θ n,. Our main results contain analogues to the convergence of the classical binomial distribution to the Poisson distribution and the normal distribution, and show that the limits and n can be echanged. Section deals with two cases of convergent parameter θ n, in particular with the case of constant mean. In Section 3 we show that, if θ n grows sub-eponentially, the distribution of the normalized X n converges to a discrete normal distribution. In Section 4 we eamine the case of an eponentially growing parameter seuence θ n.. Convergent Parameter As noted above we consider seuences of random variables X n with X n X KB n, θ n,. In the present section we will provide convergence results for two different seuences θ n which both tend to a limit as n. We will need the following simple continuity argument. Lemma.. Let θ n be a seuence of real numbers with limit θ 0. Then n lim + θn i E θ. Proof. For small ɛ > 0 and n large enough, we have hence n + θ ɛ i E θ ɛ lim n lim sup n + θn i n + θ ɛ i lim inf n E θ + ɛ. Since E is continuous, the lemma follows. The -number [] is defined as + θn i lim [] : ; n + θ + ɛ i, n + θn i + θ + ɛ i for, we have []. Now we can establish our first convergence result. Theorem.. Let λ be a real number with 0 < λ < n, und put θ n λ/[n λ]. Then we have X KB n, θ n, H λ B n, λ n P λ

4 CONVERGENCE PROPERTIES OF KEMP S -BINOMIAL DISTRIBUTION 3 Proof. We only have to show that X KB n, θ n, H λ. Note that [ ] n λ / PX n [n λ] n + λ [n λ] i [ n where the last line follows from Lemma.. ] λ / n λ n + λ i n λ / λ e λ, ; Theorem. yields limit relations for orthogonal polynomials. For these we use the same parametrization as the encyclopaedic report by Koekoek and Swarttouw [8]. Corollary.3. Let θ n λ/[n λ] be as in the preceding Theorem. i The -Krawtchouk polynomial K k ; n θ n, n; converges for n to the -Charlier polynomial C k ; λ;. ii For, the -Krawtchouk polynomial K k ; n θ n, n; converges to the Krawtchouk polynomial K k ; λ/n, n. Proof. The -Krawtchouk polynomials are orthogonal w.r.t. Kemp s -binomial distribution, while the -Charlier polynomials are orthogonal w.r.t. the Heine distribution. Therefore, part i follows from Theorem.. It is also easy to verify directly, e.g. from the representation of the polynomials by basic hypergeometric series [8]. As for part ii, the Krawtchouk polynomials are orthogonal w.r.t. the binomial distribution. Again, direct verification would be straightforward, too. For our net result, we note the following elementary fact. Lemma.4. Let f n, n N, be a seuence of functions that are increasing in, and suppose that for each n there is a uniue solution n of f n 0. Moreover, assume that f n < f m for n < m and that f n converges pointwise to a limit f with a uniue solution ˆ of f 0. Then n n N converges to ˆ. Proof. Since f n < f m for n < m, the seuence n is decreasing and n ˆ. Therefore n converges to a limit ˆ. Moreover, thus ˆ. 0 lim f n n lim f n f, Our second convergence property is an analogue of the classical convergence of the binomial distribution with constant mean to the Poisson distribution. Theorem.5. Fi µ > 0. Then we have X KB n, θ n, Hθ B n, µ n P µ where the parameter seuence θ n is chosen such that µ n µ, and θ lim θ n.

5 4 STEFAN GERHOLD AND MARTIN ZEINER Proof. First we check that for given µ > 0 and fied there is a uniue seuence θ n n N, such that µ n θ n, µ. The function µ n θ, is strictly increasing in θ and µ n 0, 0. Since µ n n+, n i n+ i n+ n and µ n θ, is continuous in θ, there eists a uniue solution θ n of µ n θ, µ for each n µ. As µ n θ, is increasing in n, we can apply Lemma.4 to obtain lim θ n θ, with θ the uniue solution of µ θ, µ. Thus X KB n, θ n, Hθ by Lemma.. The function µ n θ, is also increasing in, so we get θ n µ or euivalently θn +θ µ n n for by Lemma.4. So X KBn, θ n, B n, n µ. It remains to check that θ/ converges to µ for then Hθ P µ. The value θ/ is the uniue solution of µ θ, µ. Moreover, µ θ, is increasing in and θ and lim µ θ, θ because H θ P θ. Thus we can again apply Lemma.4. Analogously to Corollary.3, Theorem.5 implies the following result about -Krawtchouk and -Charlier polynomials. Corollary.6. Let θ n and θ be as in Theorem.5. i The -Krawtchouk polynomial K k ; n θ n, n; converges for n to the -Charlier polynomial C k ; θ;. ii For, the -Krawtchouk polynomial K k ; n θ n, n; converges to the Krawtchouk polynomial K k ; µ/n, n. 3. Sub-Eponentially Increasing Parameter Now we consider parameter seuences θ n fn with fn and n fn for n. These assumptions on fn will be in force throughout the section. Theorems 3. and 3.3 and Lemmas are devoted to the asymptotic behavior of the seuence µ n of means. As they tend to infinity, we will normalize our seuence of random variables to X n µ n /σ n. Still, this seuence does not converge in distribution without further assumptions on fn. A fruitful way to proceed is to pick subseuences along which the fractional part {fn} is constant. Theorem 3.7 shows that this induces convergence to discrete normal distributions. To investigate the seuence of means, we begin by providing an elementary estimate for the variance. Lemma 3.. If θ n fn with the above assumptions on fn, then the seuence of variances satisfies σ n /. Proof. By, the variance σ n euals n i fn fn + i fn i fn + i fn + n < fn i fn + {fn} i n fn + {fn} i + + i+ {fn} {fn} i i + i. n µ i fn + i fn i+ {fn} + i+ {fn}

6 CONVERGENCE PROPERTIES OF KEMP S -BINOMIAL DISTRIBUTION 5 The following theorem is our first result about the seuence of means in the case of a sub-eponentially increasing parameter. It does not reveal the behavior of the O-term as clearly as Theorem 3.3, but will be useful later on Lemma 3.5. Theorem 3.. Let X n KBn, θ n, with θ n fn. Then, for n, 3 µ n fn + c{fn}, + o, where c{fn}, : + {fn} {fn} + + O. l {fn} + l+{fn} l 0 l 0 Proof. We start from 4 µ n n i fn n + i fn and split the sum into two parts w.l.o.g. fn < n: fn + fn i l 0 fn + fn i l lfn i l 0 fn l lfn fn + l fn l fn + l li l lfn l fn l l l{fn} l + O fn l l l{fn} l + O fn fn fn + l l l{fn} lj + O l j 0 fn + j++{fn} l + O fn j 0 l fn + j++{fn} + + O fn j++{fn} j 0 fn + + O fn j {fn} j 0 For the upper portion of the sum, we find n i fn + + fn i i fn O n fn fn i + {fn} i + O n fn,

7 6 STEFAN GERHOLD AND MARTIN ZEINER since in + fn i + fn n i fn n i n fn. In the limit, the term c{fn}, tends to. To see this, apply the Euler-Maclaurin formula to f + b with b > 0, which yields 5 with Since fl l 0 0 R does not change sign, we have R The first integral in 5 is 0 So we have fd + f0 0 + f 0 + R {} {} + f d. 6 f log +b +b + +b 3 fd log + b log l 0 0 f d 0 f d log b + b o,. log log + b + O,. fl log log + b + + o,. 4 Application to the sums appearing in c{fn}, gives c{fn}, {fn} + {fn} + o,. Note that for the error term in the representation 3 for µ n increases. This is why the limits for and n can t be echanged. Indeed, it is clear from 4 that µ n tends to n/ for. The following theorem represents the O-term from Theorem 3. as a Fourier series, which shows that it is a -periodic function of fn. Theorem 3.3. Let X n KBn, θ n, with θ n fn. Then, as n, 6 minfn/,n fn. µ n fn + + π sinkfnπ + O k>0 log sinh kπ log

8 CONVERGENCE PROPERTIES OF KEMP S -BINOMIAL DISTRIBUTION 7 Proof. We write µ n n + fn i + + O n fn fn i and apply the Mellin transformation [] to ht + t i. By the linearity of the Mellin transformation M and the properties M π sin πs and Mhαts α s Mhs, we see that Mhs i s π sin πs π s sin πs. +t Echanging M and the sum is permitted by the monotone convergence theorem. From the inverse transformation formula we get 7 h fn πi c+i c i fns s π sin πs ds for c 0,. To evaluate this integral, we choose the integration contour γ k γ k, γ k, γ k,3 γ k,4 with γ k, {s s } + iv : T k v T k { γ k, s s u + it k : u } γ k,3 {s s } + iv : T k v T k { γ k,4 s s u it k : u } where T k π log k + h fn lim k πi 4. Then γ k, lim k πi γ k, + πi γ k,3 + πi γ k,4 + residues, since the integral on the left side eists. Now we estimate the integrals on the right side. fns π T k s sin πs ds +iv π γ k,3 +iv sinπ + ivdv T k fn π fn π fn fn +iv sinπ + iv dv π dv sin π + sinh πv

9 8 STEFAN GERHOLD AND MARTIN ZEINER fns π s sin πs ds γ k, fnu+it k u+it k π sinπu + it k du π π fn π fn sinh πt k fnu u+it k sin πu + sinh πt k du u du sint k log sinh πt k k du 0 u The integral over γ k,4 is treated similarly. Now let us compute the residues: s has simple poles at z k : πik log, and sin πs has a simple pole at 0. First we consider the residue at z k for k 0: lim z z k fnz π z z k z sin πz πik fn log e fnπik π i sinh π z z k lim πik sin log π z z k z π k log log The sum etended over the residues at the poles z k, k 0, therefore euals log sinh k 0 iπe ifnkπ Putting together the summands k and k, kπ log e ifnkπ e ifnkπ cos ifnkπ + i sin fnkπ cosfnkπ i sinfnkπ we obtain Finally, by the epansions i sinfnkπ,. π sinkfnπ. log sinh k>0 the residue at z 0 0 is fn +. kπ log fns fn log s + Os s log s + + Os π sin πs s + Os, It is worthwhile to evaluate the sum in 6 in the limit 0. First note that, if n is fied and 0, then 4 easily yields { fn + {fn} if {fn} > 0 µ n fn + if {fn} 0.

10 CONVERGENCE PROPERTIES OF KEMP S -BINOMIAL DISTRIBUTION 9 Moreover, the O-term in 6 is o for 0, as follows readily from the estimates in the proof of Theorem 3.3. These two facts combined imply π sinkfnπ lim 0 log sinh k>0 kπ log { {fn} if {fn} > 0 0 if {fn} 0. Note that in the special case fn αn with positive α, the summands tend to the summands of the Fourier series of {fn}, if {αn} > 0. The following three lemmas complete our analysis of the means µ n and prepare for the main result of this section, Theorem 3.7. Lemma 3.4. If we choose a subseuence n k such that {fn k } β constant, then: a For k µ nk fn k + cβ, + o, where cβ, is a constant depending on β and. b i c0, c/, / ii cβ, + c β, Proof. Use 6 and simple properties of sin. Lemma 3.5. Set β {fn}. Then Proof. We define cβ, + β { 0 0 β < / / β <. ĉ{fn}, : c{fn}, + {fn}. By Theorem 3., ĉβ, is strictly increasing in β. Therefore we have for 0 β < / ĉ0, ĉβ, < ĉ/, + 0. / / Thus 8 β cβ, < β and cβ, + β <. Similarly, we get for / β < 9 β cβ, < and cβ, + β < + β < 3. Lemma 3.6. ii For β, i If β, then fn + cβ, Z. Thus µ n fn + cβ, fn + β + cβ,. µ n > fn + Thus µ n fn + if fn n and µ n < fn + if fn n and µ n fn + if fn n. if fn n.

11 0 STEFAN GERHOLD AND MARTIN ZEINER Proof. i: From 8 we get for 0 β < / fn + β < fn + cβ, < fn + β and therefore fn + < fn + cβ, < fn +. Similarly, from 9 we get for / < β < fn + < fn + cβ, < fn + 3. ii: Assume fn n. Then n i fn + i fn fn i fn + i fn + fn ifn+ i fn n + + i fn fn+ i fn + i fn fn i + i + fn i+ + i+ + o fn + + o. We used ; the o-term is non-negative and vanishes only for fn n. If fn n, then the third sum vanishes and the second sum just runs up to n < fn, so µ n < fn +. Note that similarly to the proof of ii we can prove the properties of cβ, in Lemma 3.4 b. We can now state and prove the main result of this section. It shows that the limit distribution of the normalized X n is discrete normal. Theorem 3.7. Let n k k N be an increasing seuence of natural numbers and X nk KBn k, θ nk, with θ nk fnk and {fn k } β constant. Recall that we assume fn and n fn throughout the present section. Then X nk µ nk /σ nk converges for k to a limit X, with 0 P X β + c σ + σ e e β e β β/, Z, where c cβ, is the constant from Lemma 3.4 and σ lim k σ nk. distribution of X is symmetric iff β 0 or β /. The Proof. For simplicity we write in the following n instead of n k. By, we see that the σ n converge. First we consider the case β /: [ ] n PX n µ n + µ n + [ ] n µ n + θn µn + µn + µn + / n + θ n i µn +fn+ µn + µn + / n + i fn.

12 CONVERGENCE PROPERTIES OF KEMP S -BINOMIAL DISTRIBUTION The product in the denominator euals n + i fn + i n fn fn i fn + + i fn fn fn fn ++ fn + fn / n fn + i+ fn fn+ fn fn fn ++ fn + fn / n fn + i fn fn+ fn i + fn fn fn i + fn fn ++ fn + fn / β ; fn + β+ ; n fn. The second euality uses the easy relation 3 n n + z i nn / z n + z i. The last two terms in tend to e β and e β+. The -binomial coefficient in tends to e. The eponent of resulting from and is µ n + fn + µ n + µ n + / + fn fn + fn + fn / fn + β + c + fn + β + c + / fn + β + c + fn + fn fn + fn + fn / + δδ fn + fn + + δδ β +, where c cβ, and { 0 β < / δ β + c β > / by Lemma 3.5. Putting things together, we obtain PX n µ n + e e β e β+ δ+ δ+ β. By normalizing X n we get 0.The distribution of X is symmetric iff β + c β + c β + c β + c + β + c β + c. This is true for β 0 by Lemma 3.4 b i. For 0 < β < we have β + c 0 by Lemma 3.5. But then we must have β + c, which would contradict 8 since euality only holds for β 0. For β > we must have β + c 3 by Lemma 3.5, but this would be a contradiction to 9. For β / define { µn if fn n Hµ n : µ n if fn n

13 STEFAN GERHOLD AND MARTIN ZEINER Then [ ] n P X n Hµ n + Hµ n + Hµn+fn+Hµn+Hµn+ / n + i fn. The -binomial-coefficient tends to e, and the product can be transformed as above. This time the eponent of euals Hµ n + fn + Hµ n + Hµ n + / + fn fn + fn + fn / fn + + fn + fn + + fn + / + fn fn + fn + fn / So we have. P X n Hµ n + e e. Normalizing X n yields 0. The discrete normal distribution is defined by PX α / k kα k / α R, Z. So the limit distributions in the preceding theorem are normalized discrete normal distributions with parameters α + β if β < α + β if β > α 0 if β. For, they converge to the standard normal distribution [9]. Therefore, as in Theorems. and.5, the limits and n can be echanged. Indeed, for, the distribution of X n in Theorem 3.7 tends to the binomial distribution Bn,. The latter converges to the standard normal distribution after normalization. Again, the convergence of the distributions in Theorem 3.7 yields a convergence property of the corresponding orthogonal polynomials. The orthogonal polynomials for the discrete normal distribution are the Stieltjes-Wigert polynomials S k ; [, 8]. Corollary 3.8. Let be a real number, and fn as usual. Then the -Krawtchouk polynomial K k fn+o ; fn n, n; tends to ; k S k ; as n. As above, a direct proof of the corollary easily follows from the series representations of the polynomials. 4. Eponentially Growing Parameter Theorem 4.. For θ > 0 and X n KBn, θ n,, the distribution of n X n tends to the Heine distribution H/θ as n.

14 CONVERGENCE PROPERTIES OF KEMP S -BINOMIAL DISTRIBUTION 3 Proof. Define Y n n X n. Then, by 3 with z θ n, [ ] n nn n n / θ n PY n Therefore n + θ i n [ ] n n +n θ n nn / n θ n n + n i θ [ ] n +/ θ [ ] n n i + i θ / θ PY n / ; θ e. θ /θ; n. It follows from the preceding theorem that the -Krawtchouk polynomials converge to the alternative -Charlier polynomials, which is a known result [8, 4.5.]. If the parameter grows only slightly faster than in Theorem 4., then the limit distribution is degenerate. Theorem 4.. If X n KBn, n fn, with fn for n, then the distribution of n X n tends to the point measure δ 0 as n. Proof. Again we put Y n n X n. Then [ ] n PY n [ ] n It suffices to prove which is euivalent to By 3, we have lim n fnn n n / n + i n fn n +nfn+n fn / n +. i n fn n +nfn+n/ n + i n fn, n +nfn+n/ lim n + i n fn. n n +nfn+n/ n + i n fn + n+fn i This tends to as n by Lemma.. n + fn++i. Acknowledgement. We thank Stephan Wagner for the idea of using Mellin transform asymptotics for the seuence of means in Section 3, and Tom Koornwinder for the hint to think about our convergence results in terms of orthogonal polynomials.

15 4 STEFAN GERHOLD AND MARTIN ZEINER References [] J. S. Christiansen and E. Koelink, Self-adjoint difference operators and classical solutions to the Stieltjes-Wigert moment problem, J. Appro. Theory, , pp. 6. [] P. Flajolet, X. Gourdon, and P. Dumas, Mellin transforms and asymptotics: harmonic sums, Theoret. Comput. Sci., , pp Special volume on mathematical analysis of algorithms. [3] G. Gasper and M. Rahman, Basic hypergeometric series, vol. 35 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 990. With a foreword by Richard Askey. [4] N. L. Johnson, A. W. Kemp, and S. Kotz, Univariate discrete distributions, Wiley Series in Probability and Statistics, Wiley-Interscience [John Wiley & Sons], Hoboken, NJ, third ed., 005. [5] A. Kemp and C. Kemp, Weldon s dice data revisited, The American Statistician, 45 99, pp. 6. [6] A. W. Kemp, Certain -analogues of the binomial distribution, Sankhyā Ser. A, 64 00, pp Selected articles from San Antonio Conference in honour of C. R. Rao San Antonio, TX, 000. [7] A. W. Kemp and J. Newton, Certain state-dependent processes for dichotomised parasite populations, J. Appl. Probab., 7 990, pp [8] R. Koekoek and R. F. Swarttouw, The Askey-scheme of hypergeometric orthogonal polynomials and its -analogue, Tech. Rep. 98-7, Delft University of Technology, 998. [9] P. Szab lowski, Discrete normal distribution and its relationship with Jacobi theta functions, Statist. Probab. Lett., 5 00, pp Stefan Gerhold Vienna University of Technology, Wiedner Hauptstraße 8 0, 040 Vienna, Austria Martin Zeiner Graz University of Technology, Steyrergasse 30, 800 Graz, Austria address: sgerhold at fam.tuwien.ac.at address: zeiner at finanz.math.tu-graz.ac.at