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P obabili y in he Enginee ing and In o ma ional Sciences,22, 2008, 389–413. P in ed in he U.S.A.
doi:10.1017/S0269964808000235
CONVEX COMPARISONS FOR
RANDOM SUMS IN RANDOM
ENVIRONMENTS AND APPLICATIONS
JOSÉ MARÍA FERNÁNDEZ-PONCE
Depa amen o Es adís ica e In es igación Ope a i a
Facul ad de Ma emá icas
Uni e sidad de Se illa
41012 Se illa, Spain
E-mail: [email p o ec ed]
EVA MARÍA ORTEGA
Cen o de In es igación Ope a i a
Escuela Poli écnica Supe io de O ihuela
Uni e sidad Miguel He nández
03312 O ihuela (Alican e), Spain
E-mail: [email p o ec ed]
FRANCO PELLEREY
Dipa imen o di Ma ema ica
Poli ecnico diTo ino
c.so Duca Degli Ab uzzi 24
10129To ino, I aly
E-mail: [email p o ec ed]
Recen ly, Belzunce, O ega, Pelle ey, and Ruiz [3] ha e ob ained s ochas ic com-
pa isons in inc easing componen wise con ex o de sense o ec o s o andom
sums when he summands and numbe o summands depend on a common andom
en i onmen , which p o e how he dependence among he andom en i onmen al
pa ame e s in luences he a iabili y o ec o s o andom sums. The main esul s
p esen ed he e gene alize he esul s in Belzunce e al. [3] by conside ing ec o s o
pa ame e s ins ead o a couple o pa ame e s and he inc easing di ec ionally con ex
o de . Resul s on s ochas ic di ec ional con exi y o amilies o andom sums unde
app op ia e condi ions on he amilies o summands and numbe o summands a e
ob ained, which lead o he con ex compa isons be ween andom sums men ioned
ea lie . Di e en applica ions in ac ua ial science, eliabili y, and popula ion g ow h
a e also p o ided o illus a e he main esul s.
© 2008 Camb idge Uni e si y P ess 0269-9648/08 $25.00 389
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390 J. M. Fe nández-Ponce, E. M. O ega, and F. Pelle ey
1. INTRODUCTION
Much esea ch has been de o ed o s udy condi ions o he inc easing con ex o de
(also known as a iabili y o de , second s ochas ic dominance, o s op-loss o de ) o
andom sums (see Shaked and Shan hikuma [39], Pelle ey [28] and [29], Denui ,
Genes , and Ma ceau [7] o Kulik [17], among o he s). These esul s ha e ound a
wide ield o applica ions in ac ua ial science, eliabili y, epidemics, economics, o
queueing, whe e he andom sums ha e been used o desc ibe o al claim amoun s o e
a ixed ime, accumula ed wea o sys ems du ing ime in cumula i e damage shock
models, numbe o indi iduals in a popula ion ha g ows by means o a b anching
p ocess, numbe o in ec ed indi iduals in epidemic models, and so o h.
Dependencies be ween summands and numbe o summands a e common in
applica i e p oblems and se e al models o such dependence ha e been s udied in
he las ew yea s. In eal p oblems, he andom a iables in he sum usually depend
on some economical, physical, o geog aphical andom en i onmen . Recen ly, he
impac o dependencies among he andom en i onmen s on a iabili y compa isons
o mul i a ia e ec o s o andom sums has been s udied in Belzunce, O ega, Pelle ey
and Ruiz [3] and F os ig and Denui [12]. In addi ion, s ochas ic compa isons o
andom sums in ol ing Be noulli andom a iables ha e become o g owing in in e es
and ha e been applied in insu ance, enginee ing, and medicine (see Le è e and U e
[18], Hu and Wu [14], F os ig [11], o Hu and Ruan [13]).
In he li e a u e, he e a e di e en mul i a ia e ex ensions o he con ex o de
om se e al ex ensions o con exi y: in pa icula , he mul i a ia e con ex o de , he
componen wise con ex o de , and he di ec ionally con ex o de (see he monog aph
by Shaked and Shan hikuma [39]). The di ec ional con exi y akes in o accoun
he o de s uc u e on he space, which he usual no ion o con exi y does no . The
di ec ionally con ex o de was in oduced by Shaked and Shan hikuma [38] and
has been p o ed o be use ul in p oblems in ol ing dependence in se e al con ex s
o applied p obabili y (see, e.g., Mees e and Shan hikuma [23,24]), Bäue le and
Rolski [2], Li and Xu [19], o Rüschendo [35]). This o de is s ic ly weake han
he supe modula o de , which compa es only dependence s uc u e o ec o s wi h
ixed equal ma ginals. The di ec ionally con ex o de ells abou he dependence and
a iabili y o he ma ginals, which a e no necessa ily equal.
Belzunce e al. [3] ha e s udied a iabili y compa isons by means o he inc easing
componen wise con ex o de o wo ec o s o andom sums. In ha pape , he
summands and he numbe o summands a e dependen by means o a couple o
andom pa ame e s, which ep esen some en i onmen al condi ions. They ha e
conside ed andom sums de ined by
Zi(θ1,θ2)=
Ni(θ1)

k=1
Xk,i(θ2)(1.1)
o i=1, 2, ...,m, whe e (θ1,θ2)∈T⊆R2and Xi(θ2)={Xk,i(θ2),k∈N},
i=1, ...,m, is a sequence o nonnega i e andom a iables, (N1(θ1),...,Nm(θ1))
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CONVEX COMPARISONS FOR RANDOM SUMS 391
is a ec o o in ege - alued andom a iables, and X1(θ2),...,Xm(θ2)and
N1(θ1),...,Nm(θ1)a e mu ually independen .
In his a icle, we ex end he abo e se ing by conside ing dependence by means
o a mul i a ia e andom ec o o pa ame e s. A main mo i a ion o in oducing
mul i a ia e andom en i onmen s is clea om a p ac ical poin o iew. Fo example,
se e i y and numbe o claims in insu ance o na u e ca as ophes such as hu icanes
o ea hquakes depend on geog aphy as well as some o he physical ac o s; in mo o
hi d-pa y liabili y insu ance, he e a e se e al ac o s in luencing he d i ing abili ies
(see Denui , Dhaene, Goo ae s, and Kaas [6] o o he examples).
Fo mally, le T⊆Rn1and L⊆Rn2be wo subla ices in Rn1and Rn2, espec i ely,
and le θ=(θ1,...,θn1)∈Tand λ=(λ1,...,λn2)∈L. Conside he sums de ined by
Zi(θ,λ)=
Ni(θ)

k=1
Xk,i(λ)(1.2)
o i=1, 2, ...,m, whe e X1,1(λ),X2,1(λ),...,X1,m(λ),X2,m(λ),... and N1(θ),...,
Nm(θ)a e mu ually independen .
Now, le (,)=(1,...,n1,1,...,n2)be a andom ec o aking on al-
ues in T×L. We a e in e es ed in s ochas ic compa isons o ec o s o andom sums
gi en by
Z(,)=(Z1(,),...,Zm(,)).(1.3)
He e, he andom sum
Zi(,)=
Ni()

k=1
Xk,i()(1.4)
can be conside ed as a mix u e o {Zi(θ,λ)|(θ,λ)∈T×L}, wi h espec o a ec o
(,)o andom pa ame e s desc ibing he en i onmen al condi ions.
Ano he gene aliza ion ha we will conside in he a icle gi es ise when some
o he pa ame e s o he andom sum appea bo h in he summands and he numbe
o summands. The p esence o duplica es o pa ame e s is use ul in some applica i e
con ex s (see, e.g., Sec ion 4.3). Fo mally, le D⊆Rnbe a subla ice in Rnand le
δ=(δ1,...,δn)∈D. Conside he sums de ined by
Zi(δ)=
Ni(δ)

j=1
Xj,i(δ)(1.5)
o i=1, 2, ...,m, whe e Xj,i(δ)≥0 a.s. and X1,1(δ),X2,1(δ),...,X1,m(δ),X2,m(δ),...
and N1(δ),...,Nm(δ)a e mu ually independen . No e ha (1.5) includes, as a pa icula
case, he case when he Xj,i(δ)o he Ni(δ)a e ac ually pa ame ized only by a subse
o he pa ame e s δ1,...,δn.
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392 J. M. Fe nández-Ponce, E. M. O ega, and F. Pelle ey
Assuming ha
=(1,...,n)
is a andom ec o aking on alues in D, i is in e es ing o s udy he s ochas ic
p ope ies o he ec o o andom sums
Z()=(Z1(),...,Zm()),(1.6)
whe e Zi()is a mix u e o {Zi(δ)|δ∈D}wi h espec o he ec o o andom
pa ame e s.
In his a icle we ob ain esul s on s ochas ic di ec ional con exi y (see Shaked
and Shan hikuma [38]) o amilies o andom sums, unde app op ia e condi ions on
he amilies o summands and numbe o summands. F om hese esul s, we s udy how
he dependence among mul i a ia e andom en i onmen s in luences he a iabili y
o andom sums and he dependence and a iabili y o ec o s o andom sums by
means o he inc easing di ec ionally con ex o de , which a e he main pu poses o
his a icle; ha is, we p o ide su icien condi ions o model, o compa e, and o
bound he a iabili y as well as he s eng h o dependence be ween wo ec o s o
andom sums pa ame e ized on mul i a ia e andom en i onmen s. In his way, his
a icle comple es he s udy s a ed in Belzunce e al. [3].
The a icle p oceeds as ollows. In Sec ion 2 we p o ide no a ion and ools on
s ochas ic compa isons and mul i a ia e s ochas ic con exi y ha will be used in he
a icle. In Sec ion 3 we s a e and p o e he main esul s men ioned ea lie conce ning
s ochas ic compa isons and s ochas ic di ec ional con exi y o amilies o andom
sums. Finally, applica ions o some models in insu ance, eliabili y, and popula ions
g ow h, de ined by means o andom sums, a e deal wi h in Sec ion 4.
2. UTILITY NOTIONS AND PRELIMINARIES
In his sec ion we ocus on p o iding no a ion and ma hema ical ools o he esul s in
he a icle. In pa icula , we will ecall he de ini ions o some s ochas ic o de s as well
as mul i a ia e no ions o s ochas ic con exi y o a amily o pa ame e ized andom
a iables. Fo ha , we will conside di e en no ions o con exi y in he mul i a ia e
se ing.
Some con en ions and no a ions ha a e used h oughou he a icle we e gi en
p e iously. Le ≤deno e he coo dina ewise o de ing (i.e., o any x,y∈Rn, hen
x≤yi xi≤yi o i=1, 2, ...,n) and [x,y]≤zas sho hand o x≤zand y≤z.
The ope a o s +,∨, and ∧deno e espec i ely he componen wise sum, maximum,
and minimum. The no a ion =s s ands o equali y in law and a.s. is sho hand o
almos su ely. Fo any amily o pa ame e ized andom a iables {Xθ|θ∈T}, wi h
T⊆R, such ha e e y θis a alue om a andom a iable , whose dis ibu-
ion is concen a ed on T, we deno e by X() he mix u e o he amily {Xθ|θ∈T}
wi h mixing dis ibu ion . Fo any andom a iable (o ec o ) Xand an e en A,
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CONVEX COMPARISONS FOR RANDOM SUMS 393
[X|A]deno es a andom a iable whose dis ibu ion is he condi ional dis ibu ion o
Xgi en A. Also, acco ding o mos o he eliabili y li e a u e, h oughou his a i-
cle we w i e “inc easing” ins ead o “non-dec easing” and “dec easing” ins ead o
“non-inc easing.”
2.1. Uni a ia e S ochas ic O de ings
Some o he main esul s in his a icle deal wi h he inc easing con ex o de o
andom sums. Le us ecall he de ini ion o his o de ing, also known as a iabili y
o de , second s ochas ic dominance o s op-loss o de , join ly wi h he s ochas ic o de .
Fo a comp ehensi e discussion on hese s ochas ic o de s, we e e o Shaked and
Shan hikuma [39] and Mülle and S oyan [26].
DEFINITION 2.1: Le X and Y be wo nonnega i e andom a iables, wi h su i al
unc ions FXand FY, espec i ely, hen X is said o be smalle han Y in he s ochas ic
(inc easing con ex) o de (deno ed by X ≤s (icx)Y) i
E[φ(X)]≤E[φ(Y)]
o all inc easing (inc easing con ex) unc ions φ o which he expec a ions exis .
Equi alen ly, X ≤s Y i o all ≥0i holds ha FX( )≤FY( ).
A cha ac e iza ion o he s ochas ic o de ing ha will play a c ucial ole in his
a icle is ecalled now (see Theo em 1.A.1 in Shaked and Shan hikuma [39]). Gi en
wo andom a iables Xand Y,X≤s Yi and only i he e exis wo andom a iables

Xand 
Y, de ined on he same p obabili y space, such ha X=s 
X,Y=s 
Y, and

X≤
Y, a.s.
The inc easing con ex o de has been applied in se e al con ex s, such as elia-
bili y and ac ua ial science. I allows one o compa e he s op-loss ans o ms o wo
insu ance policies o a kind o einsu ance con ac (see Mülle and S oyan [26] o
applica ions in isk heo y).
2.2. Mul i a ia e No ions o Con exi y
Nex , we ecall he concep s o con ex, di ec ionally con ex, and supe modula unc-
ions. Fo a comple e discussion on con ex unc ions, we e e o he monog aph by
Rocka ella [31]. Fo a de ini ion and p ope ies o di ec ionally con ex unc ions,
see Shaked and Shan hikuma [38] o Mees e and Shan hikuma [23]. Fo a discus-
sion and backg ound on supe modula unc ions ( ha a e also called supe addi i e
unc ions in he li e a u e) we e e o Ma shall and Olkin [22].

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394 J. M. Fe nández-Ponce, E. M. O ega, and F. Pelle ey
DEFINITION 2.2: A eal- alued unc ion φde ined on Rnis said o be he ollowing:
(i) Con ex (conca e) (deno ed by φ∈cx(c ))i
φ(αx+(1−α)y)≤(≥)αφ(x)+(1−α)φ(y)
o all x,y∈Rnand α∈[0, 1]. I in addi ion, φis inc easing (dec easing),
[i.e., o all x≤y, hen φ(x)≤(≥)φ(y)], hen we say ha φis inc easing
(dec easing) and con ex (deno ed by φ∈icx(ic )).
(ii) Inc easing componen wise con ex (deno ed by φ∈iccx) i i is inc easing
and i is con ex in each a gumen when he o he s a e held ixed.
(iii) Supe modula (deno ed by φ∈sm) i
φ(x∨y)+φ(x∧y)≥φ(x)+φ(y)
o all x,y∈Rn.
(i ) Di ec ionally con ex (conca e) (deno ed by φ∈dcx(dc )) i o any xi∈Rn,
i=1, 2, 3, 4, such ha x1≤[x2,x3]≤x4and x1+x4=x2+x3, hen
φ(x1)+φ(x4)≥(≤)φ(x2)+φ(x3).
I , in addi ion, φis inc easing (dec easing), hen we say ha φis inc easing
(dec easing) and di ec ionally con ex (deno ed by φ∈idcx(idc )).
A unc ion φ:Rn−→ Rmde ined by φ(x)=(φ1(x),...,φm(x)) is di ec ionally
con ex (conca e) i each o he coo dina e unc ions φi,i =1, 2, ...,m, is di ec ionally
con ex (conca e).
Di ec ional con exi y nei he implies no is implied by usual con exi y (see
Shaked and Shan hikuma [38]). The composi ion o unc ions p ese es inc easing
di ec ional con exi y (see Lemma 2.4 in Mees e and Shan hikuma [23]). In pa ic-
ula , he composi ion o an icx unc ion wi h an idcx unc ion is an idcx unc ion (see
Co olla y 2.5 in Mees e and Shan hikuma [23]). A use ul cha ac e iza ion o dcx
unc ions is gi en now (see P oposi ion 2.1 in Shaked and Shan hikuma [38]). Gi en
φ:Rn−→ R,φ∈dcx i and only i φis supe modula and coo dina ewise con ex.
Rema k 2.1: We no e ha φis a supe modula unc ion i and only i φis supe -
modula in any couple o a gumen s when he o he s a e held ixed (see Ma shall and
Olkin [22]). F om his p ope y and he p e ious cha ac e iza ion, obse e ha a unc-
ion φ:Rn−→ Ris inc easing and di ec ionally con ex in (θ1,...,θn)i and only
i φis inc easing, supe modula in any couple (θi,θl), whene e all o he a gumen s
a e held ixed, and con ex in any θi, whene e all o he a gumen s a e held ixed.
LEMMA 2.1: Le T⊆Rnand le g :T−→ Nbe an inc easing and di ec ionally
con ex unc ion. I {xj,j∈N}is any inc easing sequence o eal numbe s, hen he
unc ion ψ(θ):=g(θ)
j=1xjis inc easing and di ec ionally con ex.
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CONVEX COMPARISONS FOR RANDOM SUMS 395
PROOF: Fi s , le us w i e he unc ion ψas ψ(θ)=Sg(θ), whe e Sn=n
j=1xj. No e
ha Snis inc easing and con ex when {xj,j∈N}is an inc easing sequence o eal
numbe s.
Thus, he composi ion ψ=S◦gis inc easing and di ec ionally con ex by
Co olla y 2.5 in Mees e and Shan hikuma [23] and he asse ion ollows. 
2.3. Mul i a ia e No ions o he Inc easing Con ex O de
The inc easing con ex o de can be ex ended o he mul i a ia e case in se e al ways.
He e, we conside h ee o hem. Fo a su ey on hese s ochas ic o de ings, we e e
o Shaked and Shan hikuma [39].
DEFINITION 2.3: Le X=(X1,...,Xn)and Y=(Y1,...,Yn)be wo n-dimensional
andom ec o s; hen Xis said o be smalle han Yin he inc easing con ex
(inc easing componen wise con ex, inc easing di ec ionally con ex) o de (deno ed
by X≤icx(iccx,idcx)Y)i
E[φ(X)]≤E[φ(Y)]
o all inc easing con ex [inc easing componen wise con ex, inc easing di ec ionally
con ex] eal- alued unc ions φde ined on Rn o which he expec a ions exis .
Inc easing (componen wise, di ec ionally) conca e o de s a e de ined analo-
gously. Clea ly, he iccx o de is s onge han he icx o de ; ha is, i X≤iccx Y,
hen X≤icx Y. Also, i X≤iccx Y, hen Xi≤icx Yi.
S ochas ic o de s de ined abo e by means o unc ionals ake in o accoun a i-
abili y. The ollowing dependence o de is de ined in e ms o supe modula unc ions.
The supe modula o de s ic ly implies he inc easing di ec ionally con ex o de ,
al hough he supe modula o de compa es only dependence s uc u e o ec o s wi h
ixed equal ma ginals and he inc easing di ec ionally con ex o de addi ionally com-
pa es he a iabili y o he ma ginals, which migh be di e en . Fo a u he discussion
on supe modula o de o andom ec o s, see Ma shall and Olkin [22], Shaked and
Shan hikuma [40] and Mülle and S oyan [26].
DEFINITION 2.4: Le X=(X1,X2,...,Xn)and Y=(Y1,Y2,...,Yn)be wo
n-dimensional andom ec o s, wi h equal ma ginal dis ibu ions; hen Xis said
o be smalle han Yin he supe modula o de (deno ed by X≤sm Y)i
E[φ(X)]≤E[φ(Y)],
o e e y supe modula eal- alued unc ion φde ined on Rn o which he expec a-
ions exis .
Fo n=2, he supe modula o de is equi alen o he well-known posi i e quad-
an dependence o de ( o sho , PQD) (see Joe [15]). The supe modula o de has
been ecen ly used in se e al applied con ex s (see Shaked and Shan hikuma [40],
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396 J. M. Fe nández-Ponce, E. M. O ega, and F. Pelle ey
Mülle [25], Bäue le and Mülle [1], Denui e al. [7], Lillo, Pelle ey, Seme a o and
Shaked [20], F os ig [11], Rüschendo [35], Lillo and Seme a o [21], Belzunce e al.
[3] o Denui and Mülle [8], among o he s).
2.4. Mul i a ia e S ochas ic Con exi y
A his poin , we ecall some no ions o mul i a ia e s ochas ic con exi y o a amily
o pa ame e ized andom a iables. Shaked and Shan hikuma [36,37] in oduced
he no ion o s ochas ic con exi y. Mul i a ia e s ochas ic di ec ional con exi y was
in oduced in Shaked and Shan hikuma [38] and i was also s udied in Chang, Chao,
Pinedo, and Shan hikuma [4] and Mees e and Shan hikuma [23].
S ochas ic di ec ional con exi y was gene alized o a gene al space in Mees e
and Shan hikuma [24]. Below, we will conside a amily o mul i a ia e andom
a iables X(θ) o θ∈T, whe e Tis a subla ice o ei he Rno Nn.
DEFINITION 2.5: A amily {X(θ),θ∈T}o mul i a ia e andom a iables is said o
be he ollowing:
(i) S ochas ically inc easing (deno ed by {X(θ),θ∈T}∈SI) i o any θi∈T,
i=1, 2,θ1≤θ2, hen X(θ1)≤s X(θ2).
(ii) S ochas ically inc easing and di ec ionally con ex (deno ed by {X(θ),θ∈T}
∈SI −DCX) i {X(θ),θ∈T}∈SI and E[φ(X(θ))]is inc easing and di ec-
ionally con ex in θ o any φ∈idcx.
(iii) S ochas ically inc easing and di ec ionally con ex in he sample pa h
sense (deno ed by {X(θ),θ∈T}∈SI −DCX(sp)) i o any ou θi∈T,
i=1, ...,4, such ha θ1≤[θ2,θ3]≤θ4and θ1+θ4=θ2+θ3, he e exis
ou andom a iables Xi,i=1, ...,4, de ined on a common p obabili y
space, such ha Xi=s X(θi),i=1, ...,4and
[X2,X3]≤X4,a.s.(2.1)
and
X1+X4≥X2+X3,a.s.(2.2)
(i ) S ochas ically inc easing and di ec ionally linea in he sample pa h sense
(deno ed by {X(θ),θ∈T}∈SI −DL(sp)) i in (iii) he inequali y (2.2)is
eplaced by
X1+X4=X2+X3,a.s.(2.3)
In he case ha bo h he pa ame e and he andom a iables a e uni a ia e, hen
we will use he no a ion SI −CX(sp)ins ead o SI −DCX(sp).
No e ha s ochas ic di ec ional con exi y in he sample pa h sense s ic ly
implies s ochas ic di ec ional con exi y (see Coun e example 3.1 in Shaked and
Shan hikuma [38]).
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CONVEX COMPARISONS FOR RANDOM SUMS 397
S ochas ic inc easing di ec ional con exi y and s ochas ic inc easing di ec ional
con exi y in sample pa h sense a e closed by composi ion wi h idcx unc ions (see,
e.g., Lemma 2.15 in Mees e and Shan hikuma [23]).Also, bo h no ions o s ochas ic
con exi y a e closed by conjunc ion o independen andom a iables (see Lemma
2.16 in Mees e and Shan hikuma [23] o Theo em 3.3 and Theo em 4.4 in Mees e
and Shan hikuma [24]).
Some examples o s ochas ic di ec ional con exi y o pa ame e ized amilies
o andom a iables can be ound in he li e a u e: See Shaked and Shan hikuma
[38], Chang, Shan hikuma and Yao [5] o Mees e and Shan hikuma [24]. Fo
example, he Be noulli dis ibu ion and he Poisson dis ibu ion a e SI −DL(sp), he
mul inomial dis ibu ion and he gamma dis ibu ion a e SI −DCX(sp)and he mul-
i a ia e geome ic dis ibu ion is SD −DCX(sp). O he examples can be ob ained
by using he abo e p ese a ion p ope ies. Also, unde app op ia e condi ions, some
applied s ochas ic models ha e s ochas ic di ec ional con exi y p ope ies (see abo e
e e ences).
3. MAIN RESULTS
In his sec ion we p o ide esul s on s ochas ic di ec ional con exi y and s ochas ic
di ec ional con exi y in he sample pa h sense o a amily o pa ame e ized andom
sums, unde app op ia e condi ions on he pa ame e ized amilies o nonnega i e
summands and numbe o summands. F om hem, we p o ide esul s o compa ing
wo andom sums in he inc easing con ex o de and wo ec o s o andom sums in
he inc easing di ec ionally con ex o de sense when he summands and he numbe
o summands a e dependen by means o a mul i a ia e andom en i onmen .
THEOREM 3.1: Conside he amily o andom sums {Z(δ),δ∈D}de ined by
Z(δ)=
N(δ)

j=1
Xj(δ),
whe e Dis a subla ice in Rn.I
(i) all o he amilies {Xj(δ),δ∈D},j∈N, and {N(δ),δ∈D}a e independen ,
(ii) {Xj(δ),δ∈D}∈SI −DCX(sp) o e e y ixed j ∈N,
(iii) {N(δ),δ∈D}∈SI −DCX(sp),
(i ) {Xj(δ),j∈N}∈SI o e e y ixed δ∈D,
hen {Z(δ),δ∈D}∈SI −DCX(sp).
PROOF: Le δi, wi h i=1, ..., 4, be such ha δ1≤[δ2,δ3]≤δ4and δ1+δ4=
δ2+δ3. By assump ions (i), (ii), and (iii), we can build on he same p obabili y
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404 J. M. Fe nández-Ponce, E. M. O ega, and F. Pelle ey
COROLLARY 3.3: Conside he amily o andom sums {Z(θ,λ,δ),(θ,λ,δ)∈
T×L×D}de ined by
Z(θ,λ,δ)=
N(θ,δ)

j=1
Xj(λ,δ).
I
(i) all o he amilies {Xj(λ,δ),(λ,δ)∈L×D},j∈N, and {N(θ,δ),(θ,δ)∈
T×D}a e independen ,
(ii) {Xj(λ,δ),(λ,δ)∈L×D}∈SI −DCX o e e y j ∈N,
(iii) {N(θ,δ),(θ,δ)∈T×D}∈SI −DCX,
(i ) {Xj(λ,δ),j∈N}∈SI o e e y ixed (λ,δ)∈L×D,
hen
(,,)≤idcx (,,)
implies
Z(,,)≤icx Z(,,)
PROOF: Fi s , we will p o e ha o any wo andom ec o s (1,1,1)and
(2,2,2),
(1,1,1)≤idcx (2,2,2)⇒((1,1),(1,1)) ≤idcx ((2,2),(2,2)).
(3.5)
Fo i , no e ha i g((θ,δ1),(λ,δ2)) is idcx, hen also he unc ion φ(θ,λ,δ)=
g((θ,δ),(λ,δ)) is idcx. The e o e, i (1,1,1)≤idcx (2,2,2), hen o any
idcx unc ion gwe ha e ha
E[g((1,1),(1,1))]=E[φ(1,1,1)]
≤E[φ(2,2,2)]
=E[g((2,2),(2,2))],
and his p o es (3.5).
Wewilldeno e ˜
Z(θ,λ,δ1,δ2)=N(θ,δ1)
j=1Xj(λ,δ2)andobse e ha Z(,,)=s
˜
Z(,,,).

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CONVEX COMPARISONS FOR RANDOM SUMS 405
Now, le ube any inc easing and con ex unc ion and le hbe de ined as in
Theo em 3.2. Then, by Theo em 3.2 and inequali y (3.5) we ge
E[u(Z(,,))]=E[u(˜
Z(,,,))]
=E[E[u(˜
Z(,,,))|(,,,)]]
=E[h((,),(,))]
≤E[h((,),(,))]
=E[E[u(˜
Z(,,,))|(,,,)]]
=E[u(˜
Z(,,,))]
=E[u(Z(,,))]
(i.e., he asse ion). 
No e ha he abo e esul can be gene alized o a ec o o andom sum like o
Co olla y 3.2. In ac , he p oo o he ollowing co olla y is simila o he p oo o
Co olla y 3.2, bu he e we use Theo em 3.3 in Mees e and Shan hikuma [24] ins ead
o Theo em 4.4 in Mees e and Shan hikuma [24].
COROLLARY 3.4: Conside m ∈N andom sums de ined by
Zi(θ,λ,δ)=
Ni(θ,δ)

j=1
Xj,i(λ,δ),i=1, ...,m,
ha a e independen o any ixed alue o (θ,λ,δ)∈T×L×Dand le
Z(θ,λ,δ)=(Z1(θ,λ,δ),...,Zm(θ,λ,δ)).
I
(i) all o he amilies {Xj,i(λ,δ),(λ,δ)∈L×D},j∈N, and {Ni(θ,δ),
(θ,δ)∈T×D},i=1, ...,m, a e independen ,
(ii) {Xj,i(λ,δ),(λ,δ)∈L×D}∈SI −DCX o e e y j ∈Nand i =1, ...,m,
(iii) {Ni(θ,δ),(θ,δ)∈T×D}∈SI −DCX o any i =1, ...,m,
(i ) he sequence {Xj,i(λ,δ),j∈N}∈SI o e e y ixed (λ,δ)∈L×Dand
i=1, ...,m,
hen
(,,)≤idcx (,,)
implies
Z(,,)≤idcx Z(,,)
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406 J. M. Fe nández-Ponce, E. M. O ega, and F. Pelle ey
4. APPLICATIONS
In his sec ion we p o ide some examples o illus a e how he main esul s can be
applied.
4.1. Collec i e Risk Models in Ac ua ial Sciences
Conside an homogeneous po olio o n isks o e a single pe iod o ime and assume
ha du ing ha pe iod, each policyholde ican ha e a nonnega i e claim Xiwi h
p obabili y θi∈[0, 1]⊆R. Then he o al claim amoun S(θ1,...,θn)du ing ha
ime can be ep esen ed as
S(θ1,...,θn)=
n

i=1
Ii(θi)Xi,
whe e Ii(θi)deno es a Be noulli andom a iable wi h pa ame e θi.
As i is poin ed ou , o example, in F os ig [10], assump ion o independence
among he Be noulli andom a iables Ii(θi),i=1, ...,n, is no sui able o desc ibe
eal con ex s, since hei dis ibu ions migh ac ually depend on some common andom
en i onmen . Thus, one can eplace he ec o o eal pa ame e s (θ1,...,θn)by a
andom ec o =(1,...,n), wi h alues in [0, 1]n⊆Rnand desc ibing bo h
he andom en i onmen o occu ences o claims and he dependence among hem.
Some known esul s in he li e a u e deal wi h s ochas ic compa isons o andom sums
in ol ing Be noulli andom a iables (see Le è e and U e [18], Hu and Wu [14],
F os ig [10], o Hu and Ruan [13]).
He e, we s a e condi ions o he s ochas ic compa ison, in he inc easing con ex
sense, o wo o al claim amoun s de ined as abo e.
PROPOSITION 4.1: Le I(θ)=(I1(θ1),...,In(θn)), whe e he Ii(θi)a e indepen-
den Be noulli andom a iables wi h pa ame e s θi,i=1, ...,n. Conside
N(θ1,...,θn)=n
i=1Ii(θi). Then {N(θ1,...,θn),(θ1,...,θn)∈[0, 1]n⊆Rn}∈
SI −DCX(sp).
PROOF: Fi s , no e ha {N(θ1,...,θn),(θ1,...,θn)∈[0, 1]n⊆Rn}is clea ly s ochas-
ically inc easing.
Now, conside a amily o Be noulli andom a iables {Iθ:θ∈[0, 1]}. I is easy
o see ha his amily is SI −DL(sp) (see, e.g., Example 5.3.8 in Chang e al. [5]).
The e o e, o any ixed θi,k(k=1, ...,4,i=1, ...,n)such ha θi,1 ≤[θi,2,θi,3]≤
θi,4 and θi,1 +θi,4 =θi,2 +θi,3, we can build, on he same p obabili y space, andom
a iables
Ii(θk)=s Ii(θk) o k=1, ..., 4 and i=1, ...,n, such ha

Ii(θi,2),
Ii(θi,3)≤
Ii(θi,4), a.s.
and

Ii(θi,1)+
Ii(θi,4)=
Ii(θi,2)+
Ii(θi,3), a.s.
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CONVEX COMPARISONS FOR RANDOM SUMS 407
No e ha , by independence, we can build all o he a iables
Ii(θi,k), o all i=1, ...,n,
on he same p obabili y space.
Now, conside he andom a iables 
Nk=n
i=1
Ii(θi,k). We obse e ha
[
N2,
N3]≤
N4, a.s.
and

N1+
N4=
N2+
N3, a.s.
Then {N(θ1,...,θn),(θ1,...,θn)∈[0, 1]n⊆Rn}∈SI −DL(sp), since
(θ1,1,...,θn,1)≤[(θ1,2,...,θn,2),(θ1,3,...,θn,3)]≤(θ1,4,...,θn,4), a.s.
(θ1,1,...,θn,1)+(θ1,4,...,θn,4)=(θ1,2,...,θn,2)+(θ1,3,...,θn,3), a.s.
and 
Nk=s N(θ1,k,...,θn,k), o k=1, ..., 4. The asse ion ollows obse ing ha
SI −DL(sp)implies SI −DCX(sp).
As immedia e consequence, we ge he ollowing esul .
COROLLARY 4.1: Le X1,...,Xnbe independen and iden ically dis ibu ed nonneg-
a i e andom a iables and le I1(θ1),...,In(θn)be independen Be noulli andom
a iables wi h pa ame e s θ1,...,θn, espec i ely, and independen o Xi,i =1, ...,n.
Conside he o al claim amoun s S(θ1,...,θn)=n
i=1Ii(θi)Xi. Then
(1,...,n)≤idcx (
1,...,
n)
implies
S(1,...,n)≤icx S(
1,...,
n).
PROOF: Obse e ha since he claims Xia e assumed o be independen , hen
S(θ1,...,θn)=s
N(θ1,...,θn)

i=1
Xi.
The asse ion now ollows om P oposi ion 4.1 and Co olla y 3.1. 
4.2. Popula ion G ow h Models
B anching p ocesses ha e been conside ed an app op ia e ma hema ical model o he
desc ip ion o popula ions’ g ow h, whe e indi iduals p oduce o sp ings acco ding
o some s ochas ic laws. Se e al applica ions in ol e medicine, molecula and cel-
lula biology, human e olu ion, physics o ac ua ial science (see Rolski, Schmidli,
Schmid , and Teugels [32], Ross [33], o Kimmel and Axel od [16]). In his subsec-
ion, we p o ide a esul dealing wi h s ochas ic compa isons be ween wo b anching
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408 J. M. Fe nández-Ponce, E. M. O ega, and F. Pelle ey
p ocesses de ined on andom en i onmen s, which is closely ela ed o Theo em 2.2
in Pelle ey [30].
The b anching p ocesses on andom en i onmen s ha we conside he e a e
de ined as ollows. Le θ={θ0,θ1,...,}be a sequence o alues in Tdesc ibing
he e olu ions o he en i onmen , and de ine, ecu si ely, he s ochas ic p ocess
Z(θ)={Zn(θ0,...,θn),n∈N}by
Z0(θ0)=X1,0(θ0)
and
Zn(θ0,...,θn)=
Zn−1(θ0,...,θn−1)

j=1
Xj,n(θn),n≥1. (4.1)
In o de o deal wi h andom e olu ions o he en i onmen , we conside a
sequence =(0,1,...)o andom a iables aking on alues in Tand we conside
he s ochas ic p ocess Z()={Zn(0,...,n),n∈N}de ined by
Z0(0)=X1,0(0)
and
Zn(0,...,n)=
Zn−1(0,...,n−1)

j=1
Xj,n(n),n≥1, (4.2)
whe e, o e e y j,k∈N,Xj,k(k)is a nonnega i e andom a iable such ha
[Xj,k(k)|k=θ]=
s Xj,k(θ).
Fi s , we p o e he SI −DCX(sp) p ope y o such pa ame e ized amilies o
b anching p ocesses.
PROPOSITION 4.2: Le θ=(θ0,θ1,...)be a sequence o alues in T⊆Rand conside
he s ochas ic p ocess de ined by (4.1). I
(i) he a iables {Xj,k(θk)},j∈Nand k ∈Na e all mu ually independen ,
(ii) {Xj,k(θk),θk∈T}∈SI −CX(sp) o e e y ixed j ∈Nand k ∈N,
(iii) {Xj,k(θk),j∈N}∈SI o e e y ixed θk∈Tand k ∈N,
hen {Zn(θ0,...,θn),(θ0,...,θn)∈Tn+1}∈SI −DCX(sp) o e e y n ∈N.
PROOF: We will p oceed by induc ion. Fi s , obse e ha , i ially we ha e ha
{Z1(θ0),(θ0)∈T}is SI −CX(sp)and, hus, SI −DCX(sp). Now, assume ha asse -
ion is ue o n−1; ha is, assume ha {Zn−1(θ0,...,θn−1),(θ0,...,θn−1)∈Tn}is
SI −DCX(sp).
Then, by Theo em 3.1 and he induc i e hypo hesis, i ollows ha
Zn(θ0,...,θn)=
Zn−1(θ0,...,θn−1)

j=1
Xj,n(θn)(4.3)
is SI −DCX(sp)in (θ0,...,θn)∈Tn+1and, hus, he asse ion ollows. 
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CONVEX COMPARISONS FOR RANDOM SUMS 409
F om he p e ious esul , we can easily ge he ollowing compa ison esul o
wo b anching p ocesses de ined on wo di e en andom en i onmen s (see Pelle ey
[30] o u he de ails)).
COROLLARY 4.2: Conside he s ochas ic p ocesses Z(θ)={Zn(θ0,...,θn),n∈N}
and Z()={Zn(0,...,n),n∈N}de ined by (4.1) and (4.2), espec i ely. I he
assump ions o P oposi ion 4.2 hold, hen
(1,...,n)≤idcx (
1,...,
n)
implies
Zn(1,...,n)≤icx Zn(
1,...,
n).
4.3. Cumula i e Damage Shock Models
Shock models a e o g ea in e es in he con ex o eliabili y heo y since hey a e
commonly used o desc ibe he li e ime o he eliabili y o sys ems o i ems subjec ed
o shocks. In his con ex , compound Poisson p ocesses a e used o desc ibe he wea
accumula ed by sys ems du ing ime. Assume ha a sys em is subjec ed o shocks
a i ing acco ding o a Poisson p ocess Nθha ing a e θ>0 and ha he i h shock
causes a nonnega i e damage Xi, whe e he damages accumula e addi i ely. Then he
o al wea accumula ed up o ime ≥0 by he sys em is gi en by (see Esa y, Ma shall,
and P oschan [9])
Wθ( )=
Nθ( )

i=1
Xi,(4.4)
wi h Wθ( )=0 in he case Nθ( )=0.
I he sys em ails when he accumula ed wea exceeds a ixed h eshold, hen
some p ope ies o he dis ibu ion o he sys em li e ime can be ob ained om
s ochas ic p ope ies o he p ocess Wθ={Wθ( ), ∈R}.
In li e a u e he e a e many a icles dealing wi h s ochas ic compa isons among
accumula ed wea p ocesses de ined as in (4.4). Howe e , almos all o hem assume
independence among all damages Xiand also independence be ween he damages
and he coun ing p ocess Nθ(see, e.g., Esa y e al. [9], Ross and Schechne [34], o
Pelle ey [27]). He e, we p o ide a gene aliza ion o hese esul s unde condi ional
independence among damages and he shock a i al p ocess.
Fo i , assume ha he sys em is subjec ed o shocks a i ing acco ding o a Pois-
son p ocess Nθ. Le Xj(θ,λ) deno e he damage caused by he j h shock, pa ame e ized
by he same pa ame e θo he p ocess Nθand a gene ic en i onmen al pa ame e λ
ha is common o all damages. Then he o al wea accumula ed up o ime ≥0by

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410 J. M. Fe nández-Ponce, E. M. O ega, and F. Pelle ey
he sys em is gi en by
Wθ,λ( )=
Nθ( )

j=1
Xj(θ,λ) (4.5)
(whe e Nθ( )
j=1Xj(θ,λ) =0 in he case Nθ( )=0).
Now, assume ha he pa ame e s a e gi en by andom en i onmen al ac o s (i.e.,
by a andom ec o (,)), and conside he wea p ocess
W,( )=
N( )

j=1
Xj(,),(4.6)
de ined as a mix u e o he amilies Wθ,λwi h espec o he ec o (,). Then by
Co olla y 3.1 and since Poisson andom a iables a e SI −DL(sp), we ob ain he
ollowing compa ison c i e ion.
COROLLARY 4.3: Conside he s ochas ic p ocesses Wθ,λand W,de ined by (4.5)
and (4.6), espec i ely. I
(i) Xj(θ,λ) a e independen o all j ∈N o any ixed alues o (θ,λ),
(ii) {Xj(θ,λ),(θ,λ) ∈R+×R+}∈SI −DCX(sp) o any j ∈N,
(iii) he amilies {Xj(θ,λ),(θ,λ) ∈R+×R+}and {Nθ,θ∈R+}a e independen ,
(i ) {Xj(θ,λ),j∈N}∈SI o any (θ,λ) ∈R+×R+,
hen
(,) ≤idcx (,)
implies
W,( )≤icx W,( )∀ ≥0.
Simila esul s can be s a ed in case he damages do no accumula e addi i ely.
Fo example, assume ha he damage caused by he i h shock is gi en by a unc ion
o he p e iously accumula ed damage and he in ensi y Xio he i h shock. Fo
ha , conside a cumula i e damage disc e e- ime p ocess W(λ)={Wn(λ1,...,λn),
n∈N,λi∈R+,i=1, ...,n}de ined ecu si ely as
W1(λ1)=X1(λ1)
and
Wn(λ1,...,λn)=Wn−1(λ1,...,λn−1)+g(Wn−1(λ1,...,λn−1),Xn(λn)),n>1.
Now, conside wo p ocesses de ined as abo e bu wi h pa ame e s gi en by
ealiza ions o wo ec o s (1,...,n)and (
1,...,
n)desc ibing di e en en i-
onmen al condi ions. P oceeding by induc ion and using a gumen s simila o hose
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CONVEX COMPARISONS FOR RANDOM SUMS 411
in he p e ious p oo and Lemma 2.4 in Mees e and Shan hikuma [23], one can
easily p o e he ollowing esul .
COROLLARY 4.4: Conside Wn(λ1,...,λn),n∈N,λi∈R+,i=1, ...,n de ined as
abo e. I
(i) he amilies {Xi(λi),λi∈R+}, wi h i =1, ...,n, a e independen ,
(ii) {Xi(λi),λi∈R+}∈SI −CX(sp), o e e y ixed alue i =1, ...,n,
(iii) {Xi(λi),i=1, ...,n}∈SI, o e e y ixed alue λi∈R+,
hen
(1,...,n)≤idcx (
1,...,
n)∀n∈N
implies
Wn(1,...,n)≤icx Wn(
1,...,
n)∀n∈N
whene e he unc ion g(w,x)is inc easing and di ec ionally con ex.
Acknowledgmen
We since ely hank P o esso Moshe Shaked o use ul commen s and sugges ions ega ding he p oo
o Theo em 3.1. F. Pelle ey is suppo ed by he I alian PRIN-Co in 2006 “Me odi S ocas ici in Finanza”
J. M. Fe nández-Ponce is suppo ed by Conseje ia de Inno acion, Ciencia y Emp esa o Jun a deAndalucia
unde g an Ayuda a la In es igación ( esol. 19 de sep iemb e de 2005). E. M. O ega is suppo ed by he
Ope a ions Resea ch Cen e and he Depa men o S a is ics, Ma hema ics and Compu e Sciences, bo h
a he Uni e si y Miguel He nández and Minis e io de Ciencia y Tecnologia unde g an BFM2003-02947.
She is since ely g a e ul o D . JoséAlouso and P o . Lau eauo F. Escude o by hei ime and encou agemen
on ela ion by his pape .
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