Gamma Distribution and Gamma Approximation

Transcription

1 Gamma Distributio ad Gamma Approimatio Xiaomig Zeg a Fuhua (Frak Cheg b a Xiame Uiversity, Xiame 365, Chia mzeg@jigia.mu.edu.c b Uiversity of Ketucky, Leigto, Ketucky , USA cheg@cs.uky.edu Abstract The Gamma distributio ad related approimatio properties of this distributio to certai of classes of fuctios are discussed. Two asymptotic estimate formulas are give. Keywords: Gamma distributio, Gamma approimatio, locally bouded fuctios, Lebesgue-Stieltjes itegral, probabilistic methods Itroductio ad Mai Results Probability distributios ad related approimatio properties have played a cetral role i several areas of computer aided geometric desig (CAGD ad umerical aalysis. For istace, the (Berstei basis fuctios of Bézier curves ad surfaces are take from a biomial distributio, ad the basis fuctios of B-splie curves ad surfaces model a simple stochastic process [, ]. Properties of these curves ad surfaces are closely related to the approimatio features of the correspodig basis fuctios. Therefore studyig probability distributios ad related approimatio properties is importat both i theory ad applicatios. The aim of this paper is to do a study i that directio. I this paper we shall cosider the followig Gamma radom variable ξ with probability desity fuctio: p ξ (u { u ep( u, Γ( if u, if u < where is a fied umber i (, ad Γ( is the Gamma fuctio of defied as follows: ( Γ( u e u du, ad study related approimatio process based o the above Gamma distributio. We first prove the followig lemma.

2 Lemma. Let {ξ i } i be a sequece of idepedet radom variables with the same Gamma distributio as ξ, ad let η ξ i. The the probability distributio of the radom variable η is i P (η y u e u du. ( P roof. Let the distributio fuctios of the radom variables ξ i be F i (, i,, 3,..., ad let the distributio fuctio of η be F η (. We have ad where F i (y p ξi (udu u Γ( e u du F η (y (F F... F (y, (F F (y + F (y udf (u, ad (F F... F (y is defied recursively. By covolutio of probability distributios we obtai P (η y F η (y u e u du. Hece, Lemma is proved. I this paper we cosider the followig Gamma approimatio process G : G (f, + f(t/df η (t f(uu e u du (3 where f belogs to the class of fuctio Φ B respectively by or the class of fuctio Φ DB, which are defied ad Φ B {f f is bouded o every fiite subiterval of [,, ad f(t Me βt, Φ DB {f f( f( subiterval of [,, ad f(t Me βt, (M > ; β ; t }. h(tdt; ; h is bouded o every fiite For a fuctio f Φ B, we itroduce the followig metric form: where [, is fied, λ. It is clear that Ω (f, λ Ω (f, λ is o-decreasig with respect to λ; sup f(t f(, t [ λ,+λ] lim λ Ω (f, λ, if f is cotiuous at the poit ; (M > ; β ; t }.

3 If f is of bouded variatio o [a, b], ad b a (f deotes the total variatio of f o [a, b], the +λ Ω (f, λ (f For further properties of Ω (f, λ, refers to Zeg ad Cheg [8]. λ The mai results of this paper are as follows: Theorem. Let f Φ B. If f(+ ad f( eist at a fied poit (,, the for > 3β, we have f(+ + f( f(+ f( G (f, + 3 π + 4 Ω (g, / k + O(, (4 where f(u f(+, < u < ; g (u, u ; f(u f(, u <. (5 We poit out that Theorem subsumes the case of approimatio of fuctios of bouded variatio. From Theorem we get immediately Corollary. Let f be a fuctio of bouded variatio o every subiterval of [, ad let f(t O(e βt for some β as t. the for (, ad > 3β, we have f(+ + f( f(+ f( G (f, + 3 π + 4 Ω (g, / k + O( + 4 +/ k / k (g + O(, (6 Corollary. Uder the coditios of Theorem, if Ω (g, λ o(λ, the G (f, f(+ + f( f(+ f( 3 + o( /. (7 π Theorem. Let f be a fuctio i Φ DB ad let f(t Me βt for some M > ad β as t. If h(+ ad h( eist at a fied poit (,, the for > 3β we have G (f, f( ρ π 4 + [ ] Ω (ψ, /k + ρ 5/ + 7M( + 3/ e 3β, (8 3 3/ 3

4 where ρ h(+ h(, ad h(t h(+, < t < ; ψ (t, t ; h(t h(, t <. (9 Remark. If f is a fuctio with derivative of bouded variatio, the f Φ DB. Thus the approimatio of fuctios with derivatives of bouded variatio is a special case of Theorem. Auiliary Results To prove Theorem ad Theorem we eed some auiliary results. Lemma. For a fied (,, we have P roof. G (,, G (u k ( + k ( + k...( +,, k k > ( Direct computatios give G (,, ad G (u k, u +k e u du t +k e t dt +k Γ( + k k ( + k ( + k...( +. k Lemma 3. For (, there hold G ((u, ; ( G ((u 4 3( +, ; ( G ((u 6 49( + 3, ; 3 (3 G (e βu, e 6β, for > 3β. (4 P roof. By Lemma ad direct computatios we get G ((t, ; G ((t 4, ( + ; 3 G ((t 6, ( ;

5 I additio, if > 3β, puttig u t, we have β G (e βu, ( β ( β e 6β. e βu u e u du t e t dt Lemma 4. [3, Chapter ] Let {ξ k } be a sequece of idepedet ad idetically distributed radom variables with the epectatio Eξ, the variace E(ξ Eξ σ >, E(ξ Eξ 4 <, ad let F stad for the distributio fuctio of (ξ k Eξ/σ. If F is ot a lattice distributio, the the followig equatio holds for all t (, + F (t π t e u / du E(ξ ξ3 6σ 3 ( t e t / + O(. (5 π Lemma 5. Let If v u <, the K (, u K (, u u v e v dv. ( u. (6 P roof. Let ξ be a Gamma radom variable with probability desity fuctio defied as i (. Let {ξ i } i be a sequece of idepedet radom variables with the same Gamma distributio as ξ. The by direct computatio we have E(ξ, E(ξ Eξ σ, (7 E(ξ Eξ 3, E(ξ Eξ <. (8 Let η ξ i. By the additio operatio of radom variables, we obtai i E(η, E(η Eη through simple computatio. Thus by Chebyshev iequality it follows that K (, u P ( η u ( u. 5

6 3 Proof of Theorem Let f satisfy the coditios of Theorem, the f ca be epressed as f(u f(+ + f( + g (u + [ f(+ f( sg(u + δ (u f( where g (u is defied i (5, sg(u is sig fuctio ad δ (u Obviously, {, u, u. ] f(+ + f(, (9 G (δ,. ( Let {ξ i } i be a sequece of idepedet radom variables with the same Gamma distributio ad their probability desity fuctios are { u ep( u, if u p ξi (u Γ(, if u <. where (, is a parameter. Let η ξ i ad F stad for the distributio fuctio of i (ξ i Eξ i /σ. The by Lemma, the probability distributio of the radom variable η is i Thus G (sg(u, P (η y u e u du. u e u du u e u du P (η F (. ( By Lemma 4, (7, (8 ad straightforward computatio, we have F ( E(ξ a 3 6σ 3 π + O( 3 π + O(. ( It follows from (9 ( that f(+ + f( f(+ f( G (f, + 3 π G (g, + O(. (3 We eed to estimate G (g,. Let K (, u u v e v dv. The by Lebesgue-Stieltjes itegral represetatio, we have G (g, 6 g (ud u K (, u (4

7 Decompose the itegral of (4 ito four parts, as where g (ud u K (, u, (g +, (g + 3, (g + 4, (g,, (g 3, (g / +/ g (ud u K (, u,, (g g (ud u K (, u, 4, (g +/ / g (ud u K (, u, g (ud u K (, u. We will evaluate, (g,, (g, 3, (g ad 4, (g separately. First, for, (g, ote that g (. Hece, (g +/ / g (u g ( d u K (, u Ω (g, /. (5 To estimate, (g, ote that Ω (g, λ is o-decreasig with respect to λ, thus it follows that / /, (g g (ud u K (, u Ω (g, ud u K (, u. Usig itegratio by parts with y /, we have Ω (g, ud u K (, u Ω (g, yk (, y + From (6 ad Lemma 5 we get, (g Ω (g, y ( y + Usig itegratio by parts oce agai, from (6, (7 it follows that, (g Ω (g, + Puttig u / t for the last itegral we get Cosequetly / Ω (g, u ( u 3 du, (g ( Ω (g, + Usig a similar method to estimate 3, (g, 3, (g ( Ω (g, + 7 K (, ud u ( Ω (g, u. (6 ( u d u( Ω (g, u. (7 / we get Ω (g, u ( u 3 dt. Ω (g, / tdt. Ω (g, / tdt. (8 Ω (g, / tdt. (9

8 Fially, by assumptio we kow that g (t M(e βt as t. Usig Hölder iequality ad Lemma 3, we have, for 3β, 4, (g M eβu d u K (, u M (u e βu d u K (, u ( M (u 4 d u K (, u / ( e βu d u K (, u / M(+e3β. (3 Equatios (5 ad (8 (3 lead to G (g,, (g +, (g + 3, (g + 4, (g Ω (g, / ( + Ω (g, + Ω (g, / udu + M(+e3β +4 Ω (g, / k + M(+e3β. (3 Theorem ow follows from (3 ad (3. 4 Proof of Theorem By direct computatio we fid that where G (f, f( Itegratio by parts derives h(+ h( G ( u, A, (ψ + B, (ψ + C, (ψ, (3 A, (ψ A, (ψ B, (ψ C, (ψ t + ( t ( t ( t ( t ψ (udu d t K (, t, ψ (udu d t K (, t, ψ (udu d t K (, t, ψ (udu d t K (, t + K (, tψ (tdt + K / (, vψ (vdv ψ (uduk (, t ( / Note that K (, v ad ψ (, it follows that / K (, vψ (vdv Ω (ψ, 8 [ ] Ω (ψ, /k.

9 O the other had, by Lemma 5 ad usig chage of variable v /u, we have / K (, vψ (vdv / Ω (ψ, v dv ( v Thus, it follows that A similar evalutatio gives A, (ψ + B, (ψ + Net we estimate C, (ψ, by the assumptio that f(t Me βt Lemma 3 we have C, (ψ M + e βt d t K (, t Ω (ψ, /udu [ ] Ω (ψ, /k. [ ] Ω (ψ, /k. (33 [ ] Ω (ψ, /k. (34 M + (t 3 e βt d 3 t K (, t M 3 ( + / ( + (t 6 d t K (, t (M >, β, ad usig / e βt d t K (, t 7M( + 3/ e 3β 3 3/. (35 Fially, we estimate the first order absolute momet of the Gamma approimatio process: G ( u,. By direct calculatio, we have Note that thus G ( u, u u e u du ( ( uu e u du + ( uu e u du t e t dt t e t dt ( e + G ( u, 9 (u u e u du t e t dt. t e t dt, ( e. (36

10 Now usig Stirlig s formula (cf. [4]: Γ(z + πz(z/e z e cz, (z + < c z < (z, from (36 we have π G ( u, π ( e c, (37 ad a simple calculatio derives 8 3/ π ( e c. (38 3/ Theorem ow follows from (3 (38 combiig with some simple calculatios. Ackowledgemet The first author was supported by Natioal ad Fujia provicial Sciece Foudatio of Chia. The secod author is supported by USA NSF uder grats DMI-46 ad DMS Refereces [] W. Dahme ad C. Micchelli, Statistical ecouters with B-splie, AMS Series o Cotemporary Mathematics, 59 (986, [] R. N. Goldma, Ur models ad B-splies, Costructive Approimatio, 4 (988, [3] W. Feller, A Itroductio to Probability Theory ad Its Applicatios, Joh Wiley & Sos, Ic. New York, Lodo, Toroto, 97. [4] V. Namias, A simple derivatio of Stirig s asymptotic series, Amer. Math. Mothly, 93 (986, 5-9. [5] E. Omey, Operators of probabilistic type, Theory Prob. Appl., 4 (996, 9-5. [6] A. N. Shiryayev, Probability, Spriger-Verlag, New York [7] C. Zhag ad F. Cheg, Triagular patch modelig usig combiatio method, Computer Aided Geometric Desig, 9 (, [8] X. M. Zeg ad F. Cheg, O the rates of approimatio of Berstei type operators, J. Appro. Theory 9 (, 4-56.