A SEMI GROUP SETTING FOR DISTANCE MEASURES IN CONNEXION WITH POISSON APPROXIMATION. D. Pfeifer 1. INTRODUCTION

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1 A SEMI GROUP SETTING FOR DISTANCE MEASURES IN CONNEXION WITH POISSON APPROXIMATION D. Pfeifer Co~uicated by Jerome A. Goldstei 1. INTRODUCTION A classical theorem of probability theory due to Poisso [6J says that if B(,p) are biomial distributios over {0,1,,} with success parameter p e (0,1) ad PO(A) are Poisso distributios over z+ with mea A > 0, the ( 1 ) B (, l) ~ Po (A) ( -+ 00) ~ where -4 meas covergece i distributio. However, relatio (1) does ot give ay impressio o the speed of covergece with respect to some appropriate distace measure, say a metric o the set ~ of all probability measures over z+. The two most importat ad well-ivestigated metrics i coexio with Poisso approximatio are the total variatio distace d defied by (2 ) d(p,q) sup+ Ip(A) -Q(A) I AcZ.; 1: Ip({k}) -Q({k}) I, P,Q e ~ 1

2 ad the cumulative distributio distace do give by ( 3) do (P, Q) = sup + I L P ({ k }) - L Q ( {k}) I, P, Q e?/j ez (see Serflig [8J, [9]). By meas of d ad do' several estimatios for the rate of covergece i (1) have bee worked out i the literature, usig differet approaches such as operator methods (Le Cam [3J, Che [1J, Presma [7J), special probabilistic methods such as couplig techiques (Serflig [8J, [9J), semigroup methods (Pfeifer [4J, Deheuvels ad Pfeifer [2J), ad others, however with some emphasis o the total variatio distace d. For istace, i the light of (1), we have (4) \ d(b('il) {; Po (I.» \2 - (5) 2 \ {; 1 \2 do (B (, il) ; Po (A ) ) for ~ \ (see Serflig [9J). It is the aim of the preset paper to show that i coexio with Poisso approximatio, both metrics ca i a atural way be see withi the same semigroup framework itroduced i Pfeifer [4] by specializig o the uderlyig Baach space as 1 ad 00, resp. This allows for a immediate traslatio of results obtaied for the total variatio distace d to do ad vice versa. 2. THE SEMIGROUP SETTING We begi with a restatemet of a well-kow theorem i the theory of operator semigroups(cf. also Pfeifer [4J). THEOREM 1. Let B be a liear cotractio o some Baach space X. The A B-1 (I stads for the idetity operator) is the (bouded) geerator of a cotractio semigroup T ( U = e ~A, ad 2

3 (6) f ex. For our cosideratios, X = 1 or X = 00 will be a coveiet choice. I either case, the covolutio f * g for 1 f e, g e X is defied by (7) f * g() L f(k)g(-k), ~ 0, where f = (f (0), f (1), ). The agai f * g e X, ad (8) II f * g II ~ X II f II 1 II gil X Further, probability measures P e 9 will be idetified with the probability vector (P({O}),P({1}), ) e 1. Let E k, k e Z+ deote the Dirac measure (or uit mass) at k. The (9) Bf = E1 * f, f e X defies a cotractio o X, ad with A B-1, ( 10) T(~)f e~a f L e-~ ~k IT Ek * f Po (~) * f, E; > 0, f e X Also, for ~ E;, ( 11) (I +.f A) f = B( l)* f, t,; > 0, f ex. ' This gives rise to the followig semigroup represetatio for the metrics d ad do (for simplicity beig restricted to the situatio uder (1». LEMMA. Let f = (1,0,0, ) e 1 ad g = (1, 1,1, ) e 00 be fixed. The for 0 < E;!:., 3

4 ( 1 2 ) d (B (, ~) ; Po (0 ) ~ II T (~) f - (I + i A) f II 1 9, (13) do (B (,~) ; Po (0 ) PROOF. straightforward usig (2), (3),(10) ad (11) Theorem 1 ow allows for a direct estimatio of d ad do by semigroup methods. THEOREM 2. For 0 < ~ {", we have (14 ) d (B (, ~ ) ; Po (~) ) ~ ~ (15) do (B (, ~) ; Po (0) f: 1 ~2 2. ~ROOF. Obvious from the Lemma ad Theorem 1 sice (16) A2 f ( 1, - 2, 1,0,0, ), hece II A 2 f II 1 9, 4, ad (17) A2 g (1,-1,0,0, ) hece IIA2 gil 00 9, 1. It should be poited out that although a 9,oo-approach is also possible for (12) (see Pfeifer [4J, [5J), the preset approach is more atural i that it avoids otherwise ecessary approximatio argumets. CONCLUDING REMARKS. The precedig argumets are ot oly restricted to the simple situatio uder (1) (see for example Deheuvels ad Pfeifer [2J, ad Pfeifer [4J, [5J). I the light of the above Lemma, the results there obtaied for d by semigroup methods easily also carryover to correspodig results ivolvig do by simple chage of the uderlyig Baach 4

5 space. Also,usig a further Taylor expasio i Theorem 1 more sophisticated estimatios for d ad do are possible (see [2] ad [5J). REFERENCES [1 J I)J [4] [5J [7] [8] Che, L.R.Y.: A approximatio theorem for covolutios of probability measures. A. Probe 3(1975), Deheuvels, P. ad Pfeifer, D.: A semi-group approach to Poisso approximatio. (1984). To appear. Le Cam, L.: A approximatio theorem for the Poisso biomial distributio. Pacific J. Math. 10(1960), Pfeifer, D.: A semi-group theoretic proof of Poisso's limit law. Semigroup Forum 26(1983), Pfeifer, D.: Approximatio-theoretic aspects of probabilistic represetatios for operator semigroups. (1984). To appear i: J. Approx. Theory. Poisso, S.D.: Recherches sur la Probabilite des Juge mets e Matiere Crimielle et e Matiere Civile Precedees des Regles Geerales du Calcul des Probabilites. Bachelier, Paris (1837). Presma, E.L.: Approximatio of biomial distributios by ifiitely divisible oes. Theory Probe Appl. 28(1984), Serflig, R.J.: A geeral Poisso approximatio theorem. A. Probe 3(1975), Serflig, R.J.: Some elemetary results o Poisso approximatio i a sequece of Beroulli trials. SIAM Review 20(1978), Istitut fur Statistik ud Wirtschaftsmathematik Techical Uiversity Aache Wullerstr. 3 D-5100 Aache West-Germay 5