ON THE RATE Of CONVERGENCE for SOME STRONG APPROXIMATION THEOREMS IN EXTREMAL STATISTICS. Dietma:r Pfe;i.fer

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1 ON THE RATE Of CONVERGENCE for SOME STRONG APPROXIMATION THEOREMS IN EXTREMAL STATISTICS Dietma:r Pfe;i.fer Abstract. Strog approximatio theorems for the logarithms of record ad iter-record times of a i.i.d. sequece are ivestigated with respect to their a.s. rate of covergece which is show to be expoetial. Exact characterizatios of this rate are give by meas of upper ad lower class fuctios for the Wieer process. 1. Itroductio. For a i.i.d. sequece {X i eln}of radom variables with cotiuous distributio fuctio let {~ i>o} - ad {U i >O} deote the iter-record times ad record times, - resp. defied by (1) ~O = 1, ~ +1 = mi {k i Xu +k> Xu } which are a.s. well-defied by the assumptios above. Ivestigatios cocerig the a.s. limitig behaviour of these sequeces were carried out by differet authors (see AMS 1980 subject classificatio: primary: 60 F 15, 60 J 20, sec0dary: 60 J 80, 60 J 85 Key words ad phrases: strog approximatio, record times, iter-record times, Erd~s-type LIL,Wieer process 1

2 D. Pfeife r De h e uvels [ 2 ] ); however, it has tured out that strog app r o ximatio techiques as developed by Deheuvels [1],[2J are most useful i this area. I the preset paper, we shall comp l e te his results by a more detailed ivestigatio of the rate of covergece i his basic represetatio theorems, allowi g at the same time for a simple explaatio for the differet growth behaviour of record ad iter-record times which e.g. becomes apparet from the fact that (2 ) - ~2 2 - E(logll ) =- C+ O(2 ), Var(logll ) = +""6 +O( 2 ) E(logu ) =+ 1- C+ O( 2 ), Var(logU ) =+1 - ~6 + O( 2- ) for + 00 (see Pfei,;fer [5J), where C deotes Euler's costat. 2. Mai results. Followig Deheuvels [2J, the iter-record time sequece ca be represeted as (3) 1I =it{y /-1Dg(1-exp(-T ))} + 1 =y /-log(1-exp(-t )) + R, \ say where {Y ; ein}is a i.i.d. sequece of expoetially distributed radom variables with uit mea, ad {T; ein} is the arrival-time sequece of a uit-rate Poisso process, idepedet of the Y's. The followig result is implicit i his paper [1 J, beig a simple cosequece of (3). LEMMA 1. For +00, we have As will be show i the sequel, the rate of covergece i (4) is maily determied by the limitig behaviour of exp(-t ) which follows from the result below. LEMMA 2. For + 00, (5) log I we have :: -~I = O(log ) a.s. 2

3 Proof. We use a simple Borel-Catelli argumet. I order to prove (5), it suffices to show that i either case (6 ) ( 7) I -2 P (Y < ) < co =l \ L P(a < R/Y < b) <co where a =-2-,b=-2+. =1 - - While the first covergece is obvious, ote that for (7) we have a.s. for t > 0 co Y (8) P (a < R /Y < bit =t) < I P(k+1-b Y < < k+1-a Y ) k=o - -log(1-e-t) - a. We are ow ready to formulate the first mai result. THEOREM 1. For -+ co, we have (9) log ~ = 10gY + T + o(exp(- +H(1/» a.s. whe re t H(1 / t) belogs to the upper class of a Wieer process, i.e. H(t) is a positive fuctio defied i some positive eighbourhood of the origi such that H(t) + ad t-1/ 2H(t) +, ad the itegral (10) I = f t- 3 / 2 H(t) exp(-h 2 (t)/2t) dt 0+ coverges. The above result caot be exteded to lower class fuctios (i.e. H as above with I beig diverget), ot eve whe 0(.) is replaced by 0(. ). Proof. Just as i Theorem 4 i Deheuvels [2J, usig Lemma 2 above ad the Komlos-Major-Tusady Theorem [4 J. 3

4 REMARK. A typical choice for upper ad lower class fuctios is (11) H(1/)= ;r2--{-lo-g--lo-g---+-i--lo-g--(3-)--+-l-o-g-(-4-) l-o-g-(-p-)-+--(-1+-c-)-1-og--(p+--1-)-} for c IR, p ~ 4, givig upper class fuctios for c > 0 ad lower class fuctios for c < O. I fact, this choice is closely related with Erdgs' form of the LIL [3]. We shall ow tur to a correspodig aalysis for the record time sequece. Followig Williams [7} ad Westcott [6 ], this sequece ca be represeted by say where agai {Yi IN}is a LLd. sequece of expoetially distributed radom variables with uit mea, ad (13) 0 < Z+1 < log(1 + U e xp(-y+1» < exp(- T+1), > 0 where T L Yk, > 1 agai defies the arrival-time sequek=1 ce of a uit-rate Poisso process, ad I zk coverges mook=1 toically to some itegrable radom variable Z with mea E(Z) = 1 -C as ca be cocluded from Pfeifer [5J. This gives rise to the followig estimatios. THEOREM 2. For -+ 00, we have (14) log U = Z + T + o(exp(-+h(1 /» ' a.s. where agai t H(1/t) is a upper class fuctio of a Wieer process. 4

5 Proof. Let W L k=+j '" Zk The E(W ) < L E(exp(-T» - k=+1 L E(W ) < "'. By the Borel-Catelli Lemma we therefore have W =0(2 2- ) a.s. =1 2 L for -+"'. But also, W < L exp(-t k ) +W2 -< exp(-t ) +W2 -k=+1 = exp(log-t ) + o(e- ) a.s. for -+"'. The result ow follows as i the proof of Theorem 1. We strogly believe that the rate result i Theorem 2 caot be improved to the case of lower class fuctios either; for a proof of this, however, a better lower boud for Z as i (13) would be ecessary. Refereces. [1J Deheuvels, P.: Strog approximatio i extreme value theory ad applicatios. ColI. Math. Soc. Jaos Bolyai 36, Limit Theorems i Probability ad Statistics, Veszprem (Hugary). North Hollad, Amsterdam (1982). [2] Deheuvels, P.: The complete characterizatio of the upper ad lower class of the record ad iter-record times of a i.i.d. sequece. Z. Wahrscheilichkeitsth. verw. Geb. 62, 1-6 (1983). [3] Erd~s, P.: O the law of the iterated logarithm. Aals of Math. 43, (1942). [4] Komlos, J., Major, P. ad Tusady, G.: A approximatio of partial sums of idepedet rv's, ad the sample df. II. Z. Wahrscheilichkeitsth. verw. Geb. 34, (1976). 5 Pfeifer, D.: A ote o momets of certai record statistics. Z. Wahrscheilichkeitsth. verw. Geb. 66, (1984). 6 Westcott, M.: A ote o record times. J. Appl. Prob. 14, (1977). [7 Williams, D.: O Reyi's 'record' problem ad Egel's series. Bull. Lodo Math. Soc. 5, (1973). Diet ar Pfe ifer, Istitut flir Statistik ud Wirtschaftsmat he a~ik, RWTH Aache, Wlillerstr. 3, D-5100 Aache, West- Germa y 5