Degrees of infinite divisibility, and a relation with p- functions Harn, van, K.

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1 Degrees f infinite divisibility, and a relatin with p- functins Harn, van, K. Published: 01/01/1977 Dcument Versin Publisher s PDF, als knwn as Versin f Recrd (includes final page, issue and vlume numbers) Please check the dcument versin f this publicatin: A submitted manuscript is the authr's versin f the article upn submissin and befre peer-review. There can be imprtant differences between the submitted versin and the fficial published versin f recrd. Peple interested in the research are advised t cntact the authr fr the final versin f the publicatin, r visit the DOI t the publisher's website. The final authr versin and the galley prf are versins f the publicatin after peer review. The final published versin features the final layut f the paper including the vlume, issue and page numbers. Link t publicatin Citatin fr published versin (APA): Harn, van, K. (1977). Degrees f infinite divisibility, and a relatin with p-functins. (Memrandum COSOR; Vl. 7701). Eindhven: Technische Hgeschl Eindhven. General rights Cpyright and mral rights fr the publicatins made accessible in the public prtal are retained by the authrs and/r ther cpyright wners and it is a cnditin f accessing publicatins that users recgnise and abide by the legal requirements assciated with these rights. Users may dwnlad and print ne cpy f any publicatin frm the public prtal fr the purpse f private study r research. Yu may nt further distribute the material r use it fr any prfit-making activity r cmmercial gain Yu may freely distribute the URL identifying the publicatin in the public prtal? Take dwn plicy If yu believe that this dcument breaches cpyright please cntact us prviding details, and we will remve access t the wrk immediately and investigate yur claim. Dwnlad date: 01. Dec. 2017

2 EINDHOVEN UNIVERSITY OF TECHNOLOGY Department f Mathematics PROBABILITY THEORY, STATISTICS AND OPERATIONS RESEARCH GROUP Memrandum COSOR Degrees f infinite divisibility, and a relatin with p-functins by K. van Harn Einhven, January 1977 The Netherlands

3 Dearees f infinite divisibility, and a relatin with p~functins Sunnnary In [2J a classificatin f the infinitely divisible (inf div) lattice distributins was given by means f recurrence relatins. In this paper this classificatin is etended t all inf div distributins n [0,(0); we intrduce a system f increasing classes FA (0 ~ A < (0) f inf div distributin functins F n [0,(0), where F E FA iff F satisfies the functinal equatin fr a KA J a -AY -A (J - e )/(1 - e )df(y) = J F(-y)dKA(y) a that is nndecreasing. Fa is knwn (see [7J) t be the set f all inf div distributins n [a,""), while F"",- lim FA turns ut t be a set f A -l>«> distributins, cntaining all lg-cnve, and s all cmpletely mntne, densities (pssibly with a jump at zer), and which als cntains (save fr nrming) bth the renewal sequences and the standard p-functins f Kingman.

4 I Degrees f infinite divisibility, and a relatin with p-functins by K. van Harn Eindhven University f Technlgy, Eindhven, The Netherlands 1. Intrductin. 00 The class r 1 f infinitely divisible (inf div) lattice distributins {Pn} with P > (i.e. the class f cmpund-pissn lattice distributins) can be characterized by (see [3J r [8J): {p } ~ r iff the quantities r (1), n 1 n fr n = 0,1,2, defined by (1. 1) (n +1)p 1 = n+ n L Pkr -k(l) k=o n (n = 0,1,2, ), are all nnnegative. The class ro f cmpund-gemetric lattice distributins can be characterized similarly (see [7J): {P } r iff the quantities rn(o), n O fr n'" 0,1,2, defined by (1.2) (n = 0,1,2, ), are all nnnegative. In [2J these recursin relatins were generalized t btain classes r f inf div lattice distributins (0 ~ a ~ 1) with the pra perty (1.3) The r a are defined as fllws. Definitin 1.1. A lattice distributin {Pn}~ if the quantities r (a), fr n = 0,1,2, defined by n (1.4) n+l - a 1 - a are all nnnegative. n Pn+l = k~o Pkrn-k(a) with P > 0 is in the class ra (n... 0,1,2, ), Remark. Frm Definitin 1.1 we btain classes f generalized renewal sequences {un}~' if we replace the cnditin that ~~Pn = 1, by the requirement that U :: 1. Especially, taking a... 0, mst prperties f distributins in

5 - 2 - r can be translated int prperties f renewal sequences, and vice versa. A similar crrespndence eists between a class F' f inf div densities (t «> be defined later) and the class f standard p-functins (cf. [4J, Therem 3.1 and ur Therem 5.9). In this paper we give a similar classificatin f al1 inf div distributins n [0,. The starting pint fr this is the abslutely cntinuus analgue f (1.1) (see [7J): a prbability density functin (pdf) f n (0, is inf div iff there is a nndecreasing functin K such that almst everywhere ( 1.5) f() = JXf(X - y)dk(y). 0- Fr pdf's there is n bvius analgue f (1.2); this wuld be (1. 6) f() = (f(x - y)dk(y) 0- with a nndecreasing K. Hwever, (1.6) is satisfied by all pdf's if K is the unit-step functin and by nne fr any ther K. T vercme this difficulty we first generalize the classes r fr a > O. The set f abslutely cntinua us distributins in the intersectin f the resulting classes can then be cnsidered as the analgue f ro Prceeding as in the discrete case we replace the factr in the left-hand side f (1.5) by the functin (0 ~ A < «>; X ~ 0), with c(;o) :=, Cnsidering, mre generally, distributin functins (df's) rather than pdf's, we are led t the fllwing definitin f the classes FA' Definitin 1.2. Fr 0 ~ A < «> a df F n [O,~) is in the class FA if there is a nndecreasing functin KA nr, with KA(X) = 0 fr < 0, such that fr all > 0 (1. 7) J c(yja)df(y) = 0- f F( - y)dka(y) 0- (the representatin is unique),

6 - 3 - Fr A m 0 we btain the class FO f all inf div df's n [O,~) (see [7). It is cnvenient t intrduce the fllwing subclasses f FA fr 0 ~ A < ~ ( 1.8) F + X.- F' := A F(O) > O}, F abslutely cntinuus} Nw we we get ( 1.9) frmally c(;~) let A -+ c in (1. 7) As if >, := lim c(;a)... A+= if..., F() - F(O) = f F( - y)dk (y) <X> 0- ~ If F(O) = 0, the same difficulty arises as in (1.6) fr pdf's, but if F(O) >0, (1.9) makes sense; the df's F with F(O) > 0 and satisfying (1.9) with a nndecreasing K~ crrespnd t the cmpund-gemetric distributins n [O,~) (see [7J), which are knwn t be inf div. This class f distributins we dente by F+ (cf. (1.8», as it turns ut t be the subset f df's F with c F(O) > 0 in a class F c t be defined later. S we have the fllwipg defini- tin. Definitin 1.3. A df F n [O,~) is in the class F+ if F(O) > 0 and if there 00 is a nndecreasing functin K ne, with K () = 0 fr < 0, such that c c (l.9) hlds. In sectin 2 we prve a number f preparatry results, sme f which may be f sme independent interest. In sectin 3 we prve the mntnicity f the FA (cf. (1.3» and cnsider the abslute cntinuity f F in FA in terms f the functin K, Here we als intrduce the class Fc:= A n FA cntaining the A <00 cntinuus analgue f ro In sectin 4 prperties f the classes FA are discussed, and eamples are given. In sectin 5 the classf, which (save fr 00 the prper nrming) cntains bth the renewal sequences and the standard p functins f Kingman (1972), is cnsidered separately, and briefly. Fr general prperties f inf div distributins we refer t [IJ, chapter XIII and [5J, chapter 5.

7 Preliminary results.we first intrduce sme ntatin. The Laplace-Stieltjes-Transfrm (LST) f a nndecreasing functin U n [,~) is dented by 6(1'):= J e -TX du(). 0- The crrespnding small letter u will be used fr the density f U in case f abslute cntinuity. The rdinary Laplace-Transfrm (LT) f u is then A als dented by U(.). If U is a df n [,~), we shall say that U.is a prbability-laplace-stieltjes-transfrm (PLST). Finally, if a df F is in the... class FA' we shall als say that F E FA' and, in case f abslute cntinuity, that f E F~. A We first cnsider the class F 0 in mre detail. Fr ease f reference we state the well-knwn representatin therem given in [IJ. Therem 2.1. A df F is in FO iff the functin is cmpletely mntne (cmp mn), r, equivalently, there is a nndecreasing functin KO n~, with KO() == 0 fr < 0, and (2. 1) ~ f -I dko() < ~, 1... such that F has the frm (2.2) F(.) == ep[ f l) dko(x)] The functin K O ' which is uniquely determined by F, will be called the cannical functin f F. We shall need the fllwing prperties. Lemma 2.2. Let F E FO with cannical functin KO' Then we have i) KO(O) = inf{ > 0 F() > O},. ii) If KO(O) == 0, then

8 - 5 - (2.3) J -) dko() < ~- F(O) > 0 0+ Prf. Define y := inf{ > 0 I F() > J. The df G() := F( + y) is again inf div, with cannical functin L O ' say. Accrding t (1.7) fr all > 0 we have G()LO(O) ~ J G( - y)dlo(y) J ydg(y) ~ G(), and as G() > 0 fr all > 0, it fllws that LO(O).... But using (2.2) we then have ~ -yta I -T-1 F(.)... e G(-r) - ep[-y. - (l -e ) dlo(x)] frm which by the uniqueness f the cannical functin it fllws that KO(O) = y. If KO(O)... 0 we can write 0+ -lg F(.)... J (1 0+ and letting. + ~ part ii) f the lemma fllws. Using this lemma we btain the fllwing r~presentatin + FO' Therem 2.3. A df F is in F~ iff F is cmpund-pissn, therem fr the class... i.e. F has the frm (2.4) where ~ > 0 and G is a df with G(O) Prf. Let F E FO' As F(O) > 0, frm the preceding lemma it fllws that KO(O) = 0 and s ~:- J X-JdKO(X) < ~ 0+ Nw if we define G by

9 ( l G() := ~-l f y-idko(y), 0+ then G is a df, and it is easily seen that (2.2) can be rewritten in the frm (2.4). Cnversely, it is well knwn that a PLST F f the frm (2.4) is inf div with F(O) > O. As we nted in sectin 1, fr F+ there is a similar representatin therem. 00 Therem 2.4. A df F is in F+ iff F is cmpund-gemetric, 00 frm... Le. F has the (2.5) where 0 ~ p < 1 and G is a df with G(O) =. Finally, we state a prperty f FO which we need in sectin 3 t prve the mntnicityf the FA' Lemma 2.5. If F E FO and a > 0, then Fa(T) := F(a)F(T)/F(T + a) is a PLST in FO' Prf. Let KO be the cannical functin f F, then ne easily verifies that Fa is f the frm (2.2) with cannical functin K~a) given by K(a) () = I -ay (1 - e )dko(y). Net we cnsider df's in FA fr A > O. Fr < A ~ c -A -I A := lim C(X;A) = (l - e ) define Then, taking LST's in (1. 7) and (1.9), we get (2.6) and (2.7)

10 - 7 - Nw Bernstein's therem immediately yields a characterizatin f the classes FA (0 < A < ~) and F: in terms f LST's (cf. the first part f Therem 2.1). As it is nt clear, that fr an arbitrary df F n [O,~) there is a functin KA f bunded variatin such that (1.7), r (2.6), hlds, we first intrduce the fllwing functins ~A' nting that ~A cincides with KA if F FA' Definitin 2.6. Fr an arbitrary df F n [O,~) we define i) ~A Cr) ii) ;= ca{1 - FCT + A)/P(T)} fr 0 < A < ~, ;= lim ~A(T) = -F'(T)/P(T), HO and, if furthermre F(O) > 0, iii) ~~(T) := lim ~A(T) = 1 - F(O)/F(T) A~ Lemma 2.7. i) Fr 0 ~ A < ~ a df F n [O,~) is in FA iff the functin ~A is cmp mn ii) A df F n [O,~) is in F+ iff F(O) > 0 and the functin ~ is cmp mn ~ ~ We shall need the fllwing lemma, which des nt seem t be generally knwn. Lemma 2.8. If F is an arbitrary df n [O,r) and a > 0, then (2.8) lim F(T + a)/f(t) = ep[-ayfj, T~ where Y F := inf{ > 0 I F() > OJ. Prf. First let Y F = O. Fr each T > 0 there is a e(t) (0,1) such that Hence we can write 5 I - F(T + a)/f(t) = -ap'(, + 8(T)a)/F(T) s -apl(t + 6(T)a)/F(T + 8(T)a), frm which it fllws that fr (2.8) it is sufficient t prve that (2.9) lim -F'(T)/F(T) = 0 T~

11 - 8 - A T d s we use the identity F(,) =, J r "" e-t~()d, and btain fr > 0 -F'(t)/F(t) < Je-t~(X)d~Je-tXF(X)dX < 0 f "" -'['.. Ir fc -'[' -1 ~ + e d/{f( /2) e dl,.. +F(e:/2) ( +1/'r)ep[-'['e:/2J, e; /2 which is less than 2 fr '[' sufficiently large. Hence (2.9) is prved. Finally, using these results fr the df G() := F( + Y F ), we easily btain (2.8) fr a df F with Y F > O. 0 The fllwing prperty f the functins ~A' fllws directly frm (2.8) and (2.9). which we shall use repeatedly, Lemma 2.9. Let F be an arbitrary df n [0,00) and 0 ~ A< "", then (cf. Definitin 2.6) (2.10) lim ~A('[') = C(YF;A) '['~ In the sequel we use withut further cmment the fllwing prperties f the functin KA crrespnding t a df F in FA. Lemma Let 0 < A ~ 00 and F E FA (and F(O) > 0 if A = ~), then (cf. Definitin 1.2 and 1.3) i) If F() > 0 fr all > 0, then KA(O) = 0, c J ii) dk A () = c" {l - F(A)} Prf. Part i) f the lemma immediately fllws frm F()K,,(O) ~ f F( - y)dka(y),.. 0- f c(yj,,)df(y) ~ c(; A)F (), if F FA with < A < c, and, if F E F+, frm c

12 - 9 - F()K~(O) s J F( - y)dk~(y) = F() - F(O) 0- Part ii) is btained frm (2.6) and (2.7) by letting t ~ O. 3. The mntnicity f F",abslute cntinuity Frm nw n we nly cnsider df's F n [O,~) with the prperty (3. 1 ) ">O F() > 0 This is nt an essential restrictin (cf. Therem 4.2i». It fllws that lim ~,,(T) = 0 fr 0 s " < ~ (see (2.10», and that K,,(O) = 0, if F E F" and T-+ Os" :s; "". First we prve that all distributins in the classes FA with 0 < " < "" are inf div. Therem 3.1. Fr all A E (O,~) we have FA C FO' Prf. Let 0 < " < 00 and F E FA' Then (2.6) hlds and can be rewritten as (3.2)... Iterating this equatin and using the fact that F(t + na)/f(n,,) tends t 1 if n ~ 00 (see (2.8», it fllws that (3.3) -1'" 1 - c A KA(kA) II k=o 1 - c A KA (t +ka) -I'" A... If we define 1Tk := c" KA(kA) «1, see Lemma 2.IOH» and Gk(t) := K~(T +ka)/ KA(k,,) (which is a PLST), then (3.3) takes the frm "" F(T) = IT (I - 'IT k )/(1 - 'IT k G k (T» k=o... and (cf. Therem 2.4) F is the limit f a sequence f prducts f cmpundgemetric PLST's, which are inf div. Hence F is inf div. 0

13 Frm (3.3) ne easily btains the fllwing representatin fr df's in FA with a < A < ~ (cf. Therem 2.1, 2.3 and 2.4). - Crllary 3.2. Let a < A < ~, then F E: FA ifff has the frm (3.4)... "" - pg(k).) F(.) "" n k=a - peel + ka)... where a ~ p < I and G is a df with G(a) = a. Net we turn t the general mntnicity prperty. We nte that if F. has this prperty, it als hlds fr F' tified by the bservatin that. '.. (3.5) r = {F E: F+ I F is a lattice df} ct -lg ct F+ and r. The last assertin is jus- Therem 3.3. Fr all A and ~ 1n [a,~) we have (3.6) Prf. In view f Lemma 2.7 we have t shw that, if a < U $ A < ~ and if ~A is cmp mn, then ~~ is cmp man. Accrding t the definitin f ~~ we have (3.7) -1 c {~(1:) ~ ~ ~ (T + A)} = F(T + A + ~) _ FCT + ~) ~ FC. + A) FCL) Dividing by P(T + ~), and ~, and s the right-hand side f (3.7) becmes symmetric in A (3.8) CA{~,,(T) - ~ (T + A)} = F(T + lj) C {~A(T) - ~A(T + lj)} ~ lj P(1: + A) lj... If ~A is camp mn, then, by Bernstein's therem, ~A(1:) - ~AC1: + lj) is camp man t. Further, as F E FA c Fa (Therem 3.1) and lj $ A, frm Therem 2.5 it fllws that the functin Fer + ~)/F(L + A) = F(L' )lfct' + A - ~) (with T' ;= T + lj) is cmp mnt S, frm (3.8) we cnclude that ~ (1:) - ~ (T + }..) is a cmp mn functin. Nw, as lim ~ (T n-+c ~ can write lj ~ + na) = 0 (Lemma 2.9), fr W we ~ {~ (T + ka) - ~ (1: + ka. + A)} lj ~

14 It fllws that ~ is the limit f a sequence f sums f cmp mn functins. II Hence ~ is cmp mn, and the therem is prved. 0 II By letting A ~ ~ in (3.8) we easily see that if ~ is cmp mn, then ~ll is cmp mn fr all II (0,00), s we have F: c n F~. On the ther hand X< ~ = lim ~A is cmp mn, if all ~A are, and s F+.. n F~. This allws us 00 A~ A<OO t define the classes F and F' as fllws (cf. (1.8» Definitin 3.4. (3.9) F:= n FA (= lim FA' because f Therem 3.3), A <00 X~ F' := {F F F is abslutely cntinuus} If F FA' then Therem 3.3 ensures the eistence f the nndecreasing functins K (ll ~ X), crrespnding t F. Frm (3.8) we btain the fllwing il prperties f K II Crllary 3.5. If X E (O,J and if the functin K A, crrespnding t F E FA' is abslutely cntinuus (density k A ), then fr all II [O,AJ the functin K is als 'abslutely cntinuus (density k ), and the fllwing inequality II hlds (3. 10) II Prf. First take lle (O,A). As we saw in the prf f Therem 3.3, F(r + ll)/f(t + A) is cmp mn, s there is a nndecreasing functin KA zer fr negative arguments, such that F(, + lj)/f(, + A) = KX (f). Writing A A,il ~A = KA and <PlJ = K, frm (3.8) it fllws that fr all > 0 lj (3.11) J c(y;a)dk].i(y) = -y A A 4,ll f f c(u;].i)dka(u)dka,j.l(y) 0- If K).. has a density k A, then, changing the rder f integratin in (3.11) and nting that K, 1\,].1 (0) = lim F(, + ill/fer + A) = 1, we btain the abslute,~ cntinuity f K, with density k J lj lj given by '

15 (3. 12) k () l.j -1 r = C(X;A) {c(;l.j)ka() + J C(X-y;l.J)kA(-y)dKA,l.J(y)} Fr the case l.j = 0, we let l.j ~ 0+ in (3.8) t btain 0+ (3. 13) which, using (2.6), is equivalent t (3. 14) J c(y;a)dko(y) = f Y dk A (y) + J KA(-y)(l-e-AY)dK(y) It fllws that KO is abslutely cntinuus with density ko' given by (3. 15) f ka(-y)o-e-ay)dk(y)} Finally, (3.10) is btained frm (3.12) and (3.15). 0 Frm the definitin f FA it fllws that the abslute cntinuity f KA is sufficient fr the abslute cntinuity f Fe) - F(O). This bservatin generalizes a therem by Tucker (1962), if restricted t the half-line; in ur case the prf is very simple. Therem 3.6. If A EO [O,J and if the functin K A, crrespnding t F EO FA' is abslutely cntinuus, then F() - FeO) is abslutely cntinuus. Prf. If KA has a density k A, then the right-hand side f (1.7) is abslutely cntinuus with density u given by u() = I ka ( - y)df(y) 0- As C(X;A) > 0 fr all > 0, frm (1.7) it nw fllws that F() - F(O) = J df(y) = C(y;A) u(y)dy J

16 S, F() - F(O) is abslutely cntinuus with density fo given by (3. 16) fo() '" C(X;A) -\ f ka ( - y)df(y) 0- Crllary 3.7. If 0 then F F~. + ~ A < ~ and F FA\FA with abslutely cntinuus K A, Finally, we give a characterizatin f F; (cf. (1.5», which easily fllws frm the definitin f FA' Therem 3.8. Let 0 ~ A < r, then a pdf f is in F~ iff there is a nndecreasing functin KA such that almst everywhere (3. 17) c(,a)f() ~ J f( - y)dk A (y) 4. Prperties f FA' eamples In this sectin we shall frequently use the characterizatin f FA given by Lemma 2.7. Fr ntatinal cnvenience we dente the functins ~A' crrespnding t a df Fv (see Definitin 2.6), by ~;v). We start with sme prperties f the classes FA' The first f them is well knwn fr FO (see [5J). Therem 4.1. Fr 0 ~ A ~ ~ the class FA is clsed under weak cnvergence, i.e. a df F, fr which there are Fn in FA' FA such that F(T) '" lim Fn(T), is again n-+<><> Prf. In view f (3.9) it is sufficient t cnsider the case 0 < A < 00. The functins ~~n), crrespnding t Fn FA' are camp mn, and as ~in) + ~A' if Fn + P, it fllws that ~A is cmp man t. S F E FA' 0 It turns ut that every FA (0 ~ A ~ ~) is clsed under translatins t but nly FO and Fm are clsed under scale transfrmatins.

17 Therem 4.2. Let ::;; A ::;; QQ and a > 0, then i) A df F is 1n FA iff the df F () := F( - a) is in FA' a ii) A df F is in FA iff the df F () := F(a) is in FaA' a Prf. The therem is knwn fr A = 0 and fllws fr A = QQ, as sn as it has been prved fr finite A's. S let 0 < A < 00 and a > O. In case i) we... -at... have F (T) = e F(T), frm which ne easily btains the fllwing relatin a. (4. 1 ) As ~A and ~~a) are ~th nnnegative, it fllws that ~A is cmp mn iff ~~a) is cmp mn Hence i) is prved. In case ii), using F (T) = F(T/a), ne easily verifies that a (4.2) It fllws that ~A is cmp mn iff ~~~) is cmp mn, and ii) fllws. 0 In view f Therem 4.2ii) fr many purpses, such as asympttic behaviur and prperties f mments, it is sufficient t cnsider nly the class Fl in stead f all classes FA fr 0 < A < QQ. Still, the mntnicity f the FA (Therem 3.3) is an interesting prperty, and the separate FA are needed t define the class F' Als, in the discrete case, where the lattice is kept fied, the classes r are essentially distinct. a In the fllwing therem we state sme simple prperties f the FA fr < A < QQ, which are well knwn r trivial fr A = O. Therem 4.3. Fr 0 < A < (lq we have i) if F FA' then F(T + v)/f(v) e FA (0::;; v < (0),... v ii) if FE FA' thenf e FA (0 ~ v ::;; I), n-l iii) if F E FA' then F (T):= 11 F(-r+kv)/F(kv) ef (O::;;v::;;A/n; n=i;2, ), n,v k=o v iv) if FE FA' v) a df F is then Fv(T) := F(v)F(T)/F(T + v) e FA (0 ::;; v < (0), in FA iff the functin F(A)F(T)/F(T + A) is a PLST in F:. Prf. In all five cases Lemma 2.7 is applied. We give the prf fr the cases ii), iv) and v); the ther cases are easily verified. il). Frm

18 we btain the relatin (4.3) A A 1-v d v[f(,)/f(, + A)J {- dt ~A (t)} If F E FA' then the last factr in the right-hand side f (4.3) is cmp mn Further, by Therem 2.5 we knw that F(A)F(t)/F(T + A) E F O ' and hence A A A I-v (v) [F(A)F(T)/F(T + A)J. E FO fr 0 $; v $; 1. Nw, as CPA (T) ~ 0, frm (4.3) it fllws that cp~v) is cmp mn iv). We nte that, if F E FA' then Fv is indeed a PLST (Therem 2.5). S we may calculate the functin cp~v) fr Fv and btain (4.4 ) If F E FA' then CPA (,) - CPA (T + v) is cmp mn, and using again Therem 2.5, frm (4.4) we cnclude that cp~v) is cmp mn v). If F FA' then the functin F(A)F(,)/F(T + A) is a PLST FA' say. As FA (0) = F(A) lim F(t)/F(, + A) = F(A) (see (2.8», we have t-l>«! (4.5) == 1 It fllws that CPA is cmp mn iff cp~a) is cmp mn, s q) is prved. 0 We nte that nw part i), ii) and iv) f the preceding therem als fllw fr A = ~, while part iii) takes the frm n-l (4.6) F E F.. F (T): = IT F (t + kv) /F (kv) E F (0 $; v < ~; n = 1,2, ) ~ n,v k=o v Finally, we prve that U FA is dense in FO in the sense f weak cnvergence. A>O Therem 4.4. If F F O ' then there is a. decreasing sequence {A n }7 with An-+O and there are Fn FA (n = 1,2, ), such that fr every t ~ 0 n (4.7) FCT) - lim F (,) n-l>«! n + Prf. On accunt f Therem 2.3, fr F E FO we can give a prf alng the same lines as in [2J fr the discrete case. Fr an arbitrary F E FO this prf and the prf f De Finetti's therem (see [5J, p. 112) suggest the -2 fllwing chice: take An = n,and Fn such that

19 n-1 (4.8) F ('r) == II n k==o _L"n-i 2 - n 2F (T + kin )... which n accunt f (4.6) is a PLST in F -2' We can rewrite Fn as n _1 n-]... -~... n 2 n F (1)., {I + n (F (T) - I)} IT [I + k(n)j, n k-o where the k satisfy k(n)., (l/n) (n + ~) unifrmly in k, as can be prved withut t much difficulty. Nw it easily fllws that F (1) + every T ~ 0 as n + ~. n F(T) fr Net we mentin sme simple eamples f distributins in FA' nting that in the fllwing sectin sme mre eamples fr F are given. As the prfs f e the fllwing statements are ften rather technical, but nt very difficult, they are mitted; we nly give very brief indicatins. J) By taking F F+ in (4.6) we get e n-] II k=o _ pg(ka) - pg(t + ka) FA (n.. 1,2,,c; 0 S P < 1; G.. df; 0 S A < e) 2) Take G(T) = ~(~ + T)-l in eample 1, then it fllws that n-l ~l + ka ~2 + ka II k I F (n-i,2,,e; 0<]JlS1l2; OSA<CO). k=o ]J 1 + A + T ]J 2 + ka + T A 3) Calculating the functin <P"" we btain fr psitive ]J. l ]Jl ]J3 ll2 2 + I { } E F""!(]Jl + ]J3) :::;; ~1 + T ]J3 + T ]J2 + 1 (i '" 1,2,3) ]J2 S ma(]jt,]j3) ) By chsing F(T).. 1l(]J + T) in (4.6) we get n-i II II + kv c F' (n = ) k=o ]J + kv + T " J t"'; II > ; S v < c 5) Take n = 2 in eample 4 and calculate <P A, then V ]J ~ + v F ) ]J + T ]J + V + T A A S v (]J > 0; v ~ 0. 6) Immediately frm the preceding eample U A>O FA (ll > 0)

20 \! The class F OIl In [7] Steutel prved the infinite divisibility f the cmp mn pdf's n (0, ). Le. f mitures f epnential pdf' and. slightly mre gener.l, f the lg-cnve pdf's n (0. ). We nw prve that these are in the class. F!. Therem 5.1. If a pdf f n (0,00) is lg-cnve, then. f e F!. Prf. Fr h > 0 define p!h) :_ y h hf(6(2n + l)h) (n ) where Y is such that h I~ p~h).. 1. As f is a lg-cnve functin,' {p!h)}~ is a lg-cnve sequence, and hence satisfies the recursin relatin (1.2), with nnnegative rtl. (0) (see [7]). Cnsidering {p~h)}~ as a prbability disttibutin n {0,h,2h.3h,. }. it fllws: (ef. (3.5» that fr all h > 0 (h) OIl.' +.'... {Pn 10 E F Nw,fr the df Fj crrespnding t. f,.we have As F GO [lh] (h) F() lim 2 hi( H2n + 1 )h)- limr Pn h+o n-o h+o nh~ is clsed under, weak cnvergence (Therem 4.1), we cnclude thatf E F c.d Crllary 5.2. If f is a cmp mn pc.if n (0,00), then f E F!; r, in terms f PLST's: fr all df's G n (0,00) we have (5.1) :- f A simple eample f the situatin abve is. prvided by the gamma-distributin... (5.2) F(T)... + We knw that if f is a cmp mn pdf n (0,00), then p + (I - p)f(t) ef 0 fr < psi. Nw the questin arises fr which df's F we have p + (1 - p)f(t) E: ~. It will turn '. ut that this hlds fr all F F 00 T this end we use the f01- lwing characterizatin f F which can be cnsidered as an etensin f GO Lemma 2.7ii) frm F: t F

21 Therem 5.3. A df F is ~n F iff ~ F(T)-1 is camp man. "" dt Prf. In view f Definitin 3.4 and Lemma 2.7i) we have t prve that ~A is d camp man fr all A < "" iff ~(T) := r.r F(T) is camp man. If ~ is camp mn, then d... -]...-1 W(T) - WeT + A) = - r.r{f(t + A) - F(T) } is camp man. As F(T + A) ~ F(T), it fllws that is a camp man functin, and hence ~A is camp man. Cnversely, if all ~A are camp man, then s are the functins - ~T ~}" Cr), which can be written as Dividing by F(A), and using the fact that if A ~ "", then c}" ~ 1, F(T + },,)/F(A) ~ 1 (see (2.8» and ~O(T + A) ~ (see (2.10», it fllws that (5.3) and hence is a camp man functin. We nte that in the preceding therem it is essential that F has prperty... -at (3.1); fr eample, the PLST G(T) := e ~/(~ + T), with a > a and ~ > 0, is d in F"" «5.2) and Therem 4.2i», while r.r G(T) is nt camp man. As Therem 5.3 is used fr the fllwing therems, (3.1) is als essential there. Therem 5.4. If F E F"" and p ;::: 0, then F(T)/{p + (1 -p)f(t)} is a PLST in F"". Prf. Defining q>(t) := F(T)/{p + (l-p)f(-c)}, we can write -I q>(t) = {I + w(-c)}, with WeT) := p{f(t) - I}. On accunt f Therem 5.3

22 is a cmp mn functin, and as als $(1) ~ 0, frm a therem f Feller (see [IJ, p. 441) it fllws that ~ is cmp mnt Nw, as ~(O) = 1, there is a df A d A -1 G such that ~ = G, while --d G(~) is cmp mnt S (Therem 5.3) G E F [ ~ 00 Therem 5.5. If F ~ F~ and 0 < p ~ 1, then G(~) ;= p +... F+ (l -p)f(~). ~ ~. Prf. We calculate the functin ~~ crrespnding t G, and btain cp(~). 1 - p+(l-p)f(o);:: (l-p){l-:(o)} ~('r)... p + (I -p)f(~) F('r) p + (1 -p)f(r) Using the preceding therem (and Lemma 2.7ii), if F(O) > 0), it nw fllws that cp~ is cmp mn, and, as G(O) ~ P > 0, by Lemma 2.7ii) we may cnclude that G F+ 0 ~. Crllary 5.6. i) If a df F n [,~) has a lg-cnve density and if 0 < P ~ 1, then (5.4 ) p + (1 - p)i(r) ~ F+ ~ ii) Fr all p (0,1] and all df's G n (O,~) (5.5) p + (1 - p) f ~(~ + T)-ldG(~) ~ F:. As an eample we have (5.6)... ( ) v / ~ e F+ ( ) F T :- v + T ~ + T ~ 0 < v ~ ~, as F can be put in the frm (5.5). Frm the crrespndence between r 0 and the r.enewal sequences (see the remark in sectin 1) and Therem 5.5, we easily btain the fllwing prperty f renewal sequences Crllary 5.7. If {un}o ~s a renewal sequence, then s is {vn}o' where (5.7) (n = 0,1,2, ; 0 < P ~ 1)

23 Net we mentin anther subset f F This subset was suggested by a special "" class f p-functins (see [4]). Distributins f this type als ccur as passage-time distributins in [6]. Therem 5.8. If p E [0,1), ~ > and v > 0, with v S ~, and if G is a df n (0,""), then (5.8)... II 1 - P F (1') : = -~ E F ~ ~ + l' 1 - p{v/(v + T)}G(1') Prf. In view f Therem 5.3 we calculate d d v A \l (I - p)d't F ( T ) = F (ll + T) (1 - p v + -r G ( T ) ) J = which, lking at (5.6), fr v S ~ is camp mn Hence F E F"", while F is abslutely cntinuus because f the epnential cmpnent. By taking G(T) - in (5.8) we get the fllwing eample (5.9) The p-functins are clsely related t the pdf's in F",,; in fact, a represen- A tatin therem can be derived fr F E F"", very similar t that fr LT's f p-functins (cf. [4J, p. 55)1. We shall reprt n this in detail later, and restrict urselves here t a representatin therem fr F, which can easily be btained frm Therem 5.3. Frm this representatin therem it fllws withut much difficulty that (save fr nrming) F cntains bth the renewal sequences and the standard p-functins. Therem 5.9. If F is a df in F, then F has the frm (5. 10) where h is such that (5. I I ) ep[-h(-r)j is a PLST in FO Cnversely, every functin h, satisfying (5.1]), defines by (5.10) a PLST F in F "" A similar representatin therem has recently been prved by J. Hawkes fr ptential kernels in Levy-prcesses (ral cmmunicatin).

24 Prf. Let F be in F, and define the functin h by I her) := F(,) - I Then (5.10) hlds, and as h(o) = 0, he,) ~ 0 and ~, he,) is cmp mn (Therem 5.3), it fllws (see [IJ, p. 441) that ep[-h(,)j is a PLST G, say. Clearly, the functin ~O' crrespnding t G, is cmp mn, and s G E F O '... Cnversely, let h be a functin such that ep[-h(,)j is a PLST G in FO' Then... d d... he,) = -lg G(,) ~ 0 and dt he,) = - dt lg G(,) is cmp mn, s (see [IJ, d d p. 441) {I + he,)} is a PLST F. As dt F(,) = d, he,) is cmp mn, we have F E F 0 00 We cnclude this sectin with a simple applicatin f Therem 5.9; fr all a > 0 and v > 0 we have (5. 12) F(,) := {I + a 10g(1 + VT)}-I E F 00 Acknwledgement. I wuld like t thank F.W. Steutel fr suggesting the prblem and fr many helpful discussins.

25 References.[IJ Feller, W. (1971). An intrductin t prbability thery and its applicatins 2, 2nd. ed. Wiley, New Yrk. [2J Harn, K. van and Steutel, F.W. (1976). Generalized renewal sequences and infinitely divisible lattice distributins. Stchastic Prcesses and their Applicatins [3J Katti, S.K. (1967). Infinite divisibility f integer valued randm variables. Ann. Math. Statist [4J Kingman, J.F.C. (1972). Regenerative phenmena. Wiley, Lndn. [5J Lukacs, E. (1970). Characteristic functins, 2nd. ed. Griffin, Lndn. [6J Miller, H.D. (1967). A nte n passage times and infinitely divisible distributins. J. Appl. Prbe [7J Steutel, F.W. (1970). Preservatin f infinite divisihility under miing, and related tpics. Math. Centre Tracts 33. Math. Centre, Amsterdam. [8J Steutel, F.W. (1971). On the zers f infinitely divisible densities. Ann. Math. Statist [9J Tucker, H.G. (1962). Abslute cntinuity f infinitely divisible distributins. Pacific J. Math