Convergence of a semi-Markov process and an accompanying
We propose an approach to the proof of the weak convergence of a semi-Markov process to a Markov process under certain conditions imposed on local characteristics of the semi-Markov process.
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| Дата: | 2010 |
|---|---|
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| Мова: | Українська Англійська |
| Опубліковано: |
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2010
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508889695387648 |
|---|---|
| author | Malik, V. F. Samoilenko, I. V. Малик, І. В. Самойленко, І. В. |
| author_facet | Malik, V. F. Samoilenko, I. V. Малик, І. В. Самойленко, І. В. |
| author_sort | Malik, V. F. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:39:51Z |
| description | We propose an approach to the proof of the weak convergence of a semi-Markov process to a Markov process under certain conditions imposed on local characteristics of the semi-Markov process. |
| first_indexed | 2026-03-24T02:32:23Z |
| format | Article |
| fulltext |
UDK 519.21
I. V. Samojlenko (In-t matematyky NAN Ukra]ny, Ky]v),
I. V. Malyk (Çernivec. nac. un-t)
ZBIÛNIST| NAPIVMARKOVS|KOHO
I SUPROVODÛUGÇOHO MARKOVS|KOHO PROCESIV
DO MARKOVS|KOHO PROCESU
We propose a new approach to the proof of weak convergence of a semi-Markov process to a Markov
process under conditions on local characteristics of semi-Markov process.
Predlahaetsq podxod k dokazatel\stvu slaboj sxodymosty polumarkovskoho processa k mar-
kovskomu v uslovyqx, nalahaem¥x na lokal\n¥e xarakterystyky polumarkovskoho processa.
U roboti [1] umovy slabko] zbiΩnosti sim’] napimarkovs\kyx procesiv (NMP) do
markovs\koho procesu (MP) sformul\ovano v terminax kompensugçoho operato-
ra (KO) [2]. Pry c\omu napivmarkovs\kij sim’] ξh t( ) , t ≥ 0 , h ↓ 0 , postavleno
u vidpovidnist\ rozpodily, a same, sim’g mir Ps x
h
, . Dlq dovedennq slabko] zbiΩ-
nosti cyx mir pry h ↓ 0 spoçatku dovodyt\sq vidnosna kompaktnist\ sim’] Ps x
h
, ,
h > 0 , a potim [dynist\ hranyçnoho rozpodilu pry h ↓ 0 . Pry c\omu zastosovu-
[t\sq xarakteryzaciq miry za dopomohog martynhal\nyx zadaç (dyv.5[3]).
U danij roboti zaproponovano pidxid do dovedennq slabko] zbiΩnosti NMP do
MP v umovax, wo nakladagt\sq na lokal\ni xarakterystyky NMP. ZauvaΩymo,
wo vidnosna kompaktnist\ sim’] NMP (lema54) dovodyt\sq analohiçno roboti [1],
ale isnuvannq hranyçnoho procesu vstanovleno zovsim inßym metodom.
Dlq porivnqnnq nahada[mo, wo v [1] z metog otrymannq hranyçnoho operato-
ra i dovedennq isnuvannq hranyçnoho procesu vymahagt\sq nastupni umovy: po-
perße, ças perebuvannq procesu v pevnomu stani povynen prqmuvaty do 0 pry
h ↓ 0 , a po-druhe, velyçyny strybkiv procesu takoΩ magt\ prqmuvaty do 0
pry h ↓ 0 .
Nareßti, ostannq umova polqha[ v tomu, wo KO NMP zbiha[t\sq pry h ↓ 0
do deqkoho operatora A
0
na klasi obmeΩenyx neperervnyx funkcij.
Teorema z [1] stverdΩu[, wo operator A
0
zbiha[t\sq z heneratorom hranyç-
noho MP.
V danij roboti my proponu[mo zaminyty umovy wodo çasu perebuvannq ta ve-
lyçyny strybka umovamy puassonivs\ko] aproksymaci] Π1 – Π4 (dyv. [4]). Pry
c\omu vynyka[ sutt[va vidminnist\ vid [1], a same, umovy puassonivs\ko] aproksy-
maci] nakladagt\sq na velyçynu strybkiv ta jmovirnosti perexodu vkladenoho
lancgha Markova, a umova zbiΩnosti KO do deqkoho operatora vzahali ne
potribna. ZbiΩnist\ KO do hranyçnoho heneratora markovs\koho procesu vy-
plyva[ z umov Π1 – Π4 ta dovodyt\sq v lemax51, 2.
OtΩe, doslidΩu[t\sq nastupna zadaça: napivmarkovs\kyj proces ηε ( )t v ev-
klidovomu prostori Rd , d ≥ 1, u sxemi serij z malym parametrom seri] ε → 0 ,
ε > 0 , porodΩu[t\sq procesom markovs\koho vidnovlennq (PMV) (dyv., napry-
klad, [4])
ηε
n , τε
n , n ≥ 0,
de τ ετε
n n: = , θ εθε
n n: = , τε
0 0= , η ηε
ν
ε
ε( )
( )
t
t
= , ν τε ε( ) sup :{ }t n tn= ≥ ≤0 .
© I. V. SAMOJLENKO, I. V. MALYK, 2010
674 ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 5
ZBIÛNIST| NAPIVMARKOVS|KOHO I SUPROVODÛUGÇOHO MARKOVS|KOHO … 675
PMV vyznaça[t\sq stoxastyçnym qdrom, qke zada[ umovni jmovirnosti vely-
çyny strybkiv, ta funkciqmy rozpodilu çasiv perebuvannq v stanax
Γε ( , )u dv : = P η ηε ε
n nd u+ ∈ ={ }1 v , u R d∈ , d dv ∈� ,
F tu ( ) : = P θ ηε
n nt u+ ≤ ={ }1 = P( )θu t≤ , t ≥ 0 ,
b u( ) : = 1/ ( )f u , f u( ) : = F t dtu ( )
0
∞
∫ , F tu ( ) : = 1 − F tu ( ) .
OtΩe, napivmarkovs\ke qdro
Q u d t( , )v, = P η θ ηε ε
n n nd t u+ +∈ ≤ ={ }1 1v, = Γε ( , ) ( )u d F tuv .
Nexaj vykonugt\sq umovy puassonivs\ko] aproksymaci] [4]:
Π1 ) v vΓε ( , )u d
Rd∫ = ε ε[ ]( ) ( )a u R u
a
+ + + , v vΓε ( , )u d
Rd∫ = ε ε[ ]( ) ( )a u R ua+ ;
Π2 ) vv v∗∫ Γε ( , )u d
Rd = ε ε[ ]( ) ( )C u R uc+ ;
Π3 ) ψ ε( ) ( , )v vΓ u d
Rd∫ = ε ψ
ε[ ]( ) ( )Γ0 u R u+ , ψ ∈C R d
2( ) ,
de C R d
2( ) — prostir usix neperervnyx funkcij, wo dorivnggt\ 0 v okoli 0 i
takyx, wo magt\ hranycg na neskinçennosti. Zhidno z [5] funkci] z c\oho pro-
storu [ takymy, wo vyznaçagt\ miru, tobto mira (Radona) vidnovlg[t\sq za for-
mulog
Γψ
0 ( )u = ψ( ) ( , )v v
Rd
u d∫ Γ0 , ψ ∈C R d
2( ) .
Zalyßkovi çleny v umovax Π1 – Π3 prqmugt\ do 0 pry ε → 0 :
R u R u R uc a
ε ε ε( ) ( ) ( )+ ++ → 0 pry ε → 0 .
ZauvaΩennq"1. Dlq porivnqnnq navedemo umovy, wo nakladagt\sq v roboti
[1] na ças perebuvannq ta velyçynu strybkiv NMP ( Ps x
h
, — vidpovidna procesu
sim’q mir):
1) dlq bud\-qkoho kompakta K ta bud\-qkoho ε > 0 h P ss x
h− − ≥1
1, { }( )τ ε →
→ 0, h ↓ 0 rivnomirno po s, x;
2) dlq bud\-qkoho kompakta K ta bud\-qkoho ε > 0 h P xs x
h− ≥1
1, { }( ( ), )ρ ξ τ ε →
→ 0, h ↓ 0 rivnomirno po s, x, de τ1 — markovs\kyj moment vidnovlennq.
Nexaj ma[ misce takoΩ umova
Π4 ) funkci] a u( ) , C u( ) , a u+ ( ) ta Λ Γ0 0( ) : ( , )u u R d= [ obmeΩenymy.
Oznaçennq"1 [2, 6]. KO NMP ηε ( )t , t ≥ 0 , vin Ωe henerator suprovod-
Ωugçoho markovs\koho procesu (SMP) ηε
0 ( )t , t ≥ 0 , wo di[ na test-funkciqx
ϕ ( , )u t , vyznaça[t\sq rivnistg
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 5
676 I. V. SAMOJLENKO, I. V. MALYK
Γε ϕ ( , )u t : = E Eϕ η τ ϕ η τ θε ε ε ε ε( , ) ( , ) ,n n n n nu t u t+ + +− = =
1 1 1 ηηε
n u=
.
(1)
Teorema (slabka zbiΩnist\ sim’] NMP do MP). Nexaj vykonugt\sq umo-
vy.Π1 – Π4 , a takoΩ nastupni umovy:
U1 ) rivnomirna intehrovnist\
sup ( )
u R
u
T
d
F t dt
∈
∞
∫ → 0 pry T → ∞ ;
U2 ) dlq bud\-qkoho u R d∈ ta ε > 0
E e u− εθ ≤ 1 − C ε ;
U3 ) ma[ misce zbiΩnist\ poçatkovyx umov
ηε ( )0 → η( )0 pry ε → 0
ta
sup ( )
ε
εη
> 0
0E ≤ C < + ∞ ,
a takoΩ umova, z qko] vyplyva[ kompaktnist\ procesu:
U4 ) Γε ϕ ( )u ≤ Cϕ
dlq test-funkcij ϕ ( ) ( )u C R d∈ 0
2
prostoru finitnyx obmeΩenyx neperervnyx
funkcij, wo magt\ poxidni do druhoho porqdku vklgçno.
Todi NMP ηε ( )t slabko zbiha[t\sq [7] do MP η0 ( )t pry ε → 0 :
ηε ( )t ⇒ η0 ( )t ,
do toho Ω η0 ( )t zada[t\sq heneratorom
Γ0 ϕ ( )u = b u u u u u u d
R d
0 0( ) ( ) ( ) [ ( ) ( )] ( , )′ + + −∫ϕ ϕ ϕΛ Γv v (2)
ta
Γε ϕ ( )u = Γ0 ϕ ϕε( ) ( )u R u+ ,
de
R uε ϕ ( ) → 0 pry ε → 0 ,
b u0 ( ) = b u a u( ) ( )0 = b u a u a u( ) ( ) ( )[ ]− 0 ,
Λ( )u = b u u R d( ) ( , )Γ0 , Γ0 ( , )u V = Γ Λ0 ( , ) ( )/u V u , V d∈� .
ZauvaΩennq."2. Umova U2 teoremy zbiha[t\sq z umovog51 z [1, c. 7].
3. Z umovy U4 zhidno z lemog56.4 z [4] vyplyva[ umova
′U4 ) lim sup sup ( )
l t T
t l
→ ∞ > ≤ ≤
>
ε
εη
0 0
P = 0.
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 5
ZBIÛNIST| NAPIVMARKOVS|KOHO I SUPROVODÛUGÇOHO MARKOVS|KOHO … 677
Dlq dovedennq teoremy nam znadoblqt\sq nastupni lemy.
Lema"1. KO (henerator SMP) ma[ vyhlqd
Γε ϕ ( , )u t = ε ϕ ϕ ε−
∞
∫ + + −1
0
b u F ds u t s u t u du( ) ( ) [ ( , ) ( , )] ( , )v vΓ
RR d
∫ .
Dovedennq. Spravdi,
E [ ]θ ηε ε
n n u+ =1 = ε F t dtu ( )
0
∞
∫ = ε f u( ) ,
tobto
b uε ( ) = ε−1b u( ) .
Todi
Γε ϕ ( , )u t = b u F ds u t s u t u du
R d
ε εϕ ϕ( ) ( ) [ ( , ) ( , )] ( , )
0
∞
∫ + + −v vΓ∫∫ =
= ε ϕ ϕ ε−
∞
∫ + + −1
0
b u F ds u t s u t u du( ) ( ) [ ( , ) ( , )] ( , )v vΓ
RR d
∫ .
Lemu51 dovedeno.
Vvedemo poznaçennq
a u0 ( ) = v vΓ0 ( , )u d
R d
∫ .
Lema"2. Na test-funkciqx ϕ ( )u , wo magt\ obmeΩeni poxidni bud\-qkoho
porqdku, KO ma[ asymptotyçne zobraΩennq
Γε ϕ ( )u = Γ0 ϕ ε( ) ( )u R u+ ,
de Γ0
vyznaçeno v (2).
Dovedennq. Poznaçymo
ψu ( )v = ϕ ϕ ϕ( ) ( ) ( )u u u+ − − ′v v .
Vykorystovugçy umovy Π1 – Π3 , otrymu[mo
Γε ϕ ( )u = b u u u u d
R d
ε εϕ ϕ( ) [ ( ) ( )] ( , )+ −∫ v vΓ =
= ε ψ ϕε ε− ∫ ∫+ ′
1b u u d u u du
R Rd d
( ) ( ) ( , ) ( ) ( , )v v v vΓ Γ
=
= ε ψ ψε
δ
ε
δε ε
−
> ≤
∫ +1b u u d u du u( ) ( ) ( , ) ( ) ( , )v v v v
v v
Γ Γ∫∫ ∫+ ′
ϕ ε( ) ( , )u u d
R d
v vΓ ,
de δε → 0 pry ε → 0 .
Dlq druhoho intehrala vnaslidok obmeΩenosti poxidnyx funkci] ϕ ( )u çys-
lom K ta za umovog Π4 ma[mo
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 5
678 I. V. SAMOJLENKO, I. V. MALYK
ε ψ ε
δε
−
≤
∫1b u u du( ) ( ) ( , )v v
v
Γ = ε
ψ ε
δε
−
≤
∫1b u u du( )
( )
( , )v
v
v
v
v
Γ ≤
≤ ε δε
ε
δε
−
≤
∫1b u K u d( ) ( , )v v
v
Γ ≤ ε δε
ε− ∫1b u K u d
R d
( ) ( , )v vΓ =
= b u K a u R ua( ) ( ) ( )( )δε
ε+ + . (3)
Oçevydno, za umovy Π4 obmeΩenosti funkcij a u+ ( ) ta b u( ) ostannij vy-
raz prqmu[ do 0 pry ε → 0 .
ProdovΩyvßy v perßomu intehrali pidintehral\nu funkcig nulem v okoli
0, dlq novo] funkci] ψu
dC R0
2( ) ( )v ∈ za umovog Π3 budemo maty
ε ψ ε
δε
−
>
∫1b u u du( ) ( ) ( , )v v
v
Γ = ε ψ ε− ∫1 0b u u du
R d
( ) ( ) ( , )v vΓ =
= b u u du
R d
( ) ( ) ( , )ψ0 0v vΓ∫ + R uε( ) =
= b u u d b u u du
R
u
d
( ) ( ) ( , ) ( ) ( ) ( , )ψ ψ
δε
v v v v
v
Γ Γ0 0∫ ∫−
<
++ R uε( ) .
Dlq druhoho intehrala v ostann\omu vyrazi ocinka [ analohiçnog (3). Ostatoçno
ma[mo
Γε ϕ ( )u = b u u u u u d a u
R d
( ) [ ( ) ( ) ( )] ( , ) ( ) (ϕ ϕ ϕ ϕ+ − − ′ + ′∫ v v vΓ0 uu R u) ( )
+ ε
=
= b u u u u d a u u a u
R d
( ) [ ( ) ( )] ( , ) ( ) ( ) (ϕ ϕ ϕ+ − − ′ +∫ v vΓ0 0 )) ( ) ( )′
+ϕ εu R u =
= Γ0 ϕ ε( ) ( )u R u+ .
Lemu52 dovedeno.
Dovedennq teoremy. V lemi52 my dovely, wo pry ε → 0 KO prqmu[ do
Γ0
. Dlq toho wob dovesty slabku zbiΩnist\, zalyßa[t\sq pokazaty vidnosnu
kompaktnist\ sim’] ηε ( )t i vstanovyty, wo hranyçnyj operator zada[ martynhal
[5, 7, 8]
µt : = ϕ ϕ( ( )) ( ( ))u t u s ds
t
− ∫ Γ0
0
.
Rozhlqnemo matematyçne spodivannq procesu
µε
t = ϕ η ϕ ηε ε( ( )) ( ( ))t s ds
t
− ∫ Γ0
0
.
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 5
ZBIÛNIST| NAPIVMARKOVS|KOHO I SUPROVODÛUGÇOHO MARKOVS|KOHO … 679
Vvedemo vypadkovi procesy
ηε
+ ( )t : = η τε ε( ( ))+ t , ητ
ε ( )t : = η τε ε( ( ))t , t ≥ 0 ,
de τε
+ ( )t = τε ( )t + 1.
Ma[mo
E µε
t
= E ϕ η ϕ ηε ε( ( )) ( ( ))t s ds
t
−
∫ Γ0
0
= E ϕ η ϕ ηε ε( ( )) ( ( ))t t−
+ +
+ E ϕ η ϕ ηε ε
τ
ε
τε
( ( )) ( ( ))
( )
+ −
+
+
∫t s ds
t
Γ
0
EE Γ Γε
τ
ε
τ
ε
τ
εϕ η ϕ η
ε
( ( )) ( ( ))
( )
s ds s ds
t t
0 0
+
∫ ∫−
+
+ E EΓ Γε
τ
ε ε ε εϕ η ϕ η ϕ η( ( )) ( ( )) ( ( ))s s ds s
t
−
+ −∫
0
ΓΓ0
0
ϕ ηε( ( ))s ds
t
∫ .
Tretij dodanok zadovol\nq[ spivvidnoßennq
E Γε
τ
ε
τ
ϕ η
ε
( ( ))
( )
s ds
t
t+
∫ → 0 pry ε → 0
zavdqky vlastyvosti momentiv vidnovlennq
E τε
+ −
( )t t → 0 pry ε → 0
rivnomirno po t na koΩnomu skinçennomu intervali [ , ]0 T .
Dlq toho wob dovesty ostannij fakt, skorysta[mos\ lemog C. 1 z [4].
Lema"3. Nexaj sim’q momentiv vidnovlennq θu , u Rd∈ , wo magt\ funkci]
rozpodilu Fu , zadovol\nq[ umovy U1 , U2 . Todi ma[ misce spivvidnoßennq
P max ( )
0 ≤ ≤
≥
t T
tγ δε → 0, ε → 0,
dlq bud\-qkyx δ > 0 ta T > 0 , de γ τε ε( ) : ( )t t t= − .
Dovedennq. Vlastyvist\ rehulqrnosti dlq NMP da[ zbiΩnist\
P ( )/τ ε
ε
N T≤ → 0, N → ∞ ,
pry ε > 0 .
Dijsno, ma[mo
P ( )/τ ε
ε
N T≤ = P ( )/e eN T− −≥τ ε
ε
≤ Ee eN T− τ ε
ε
/ = Ee eN T− ετ ε/ .
Za umovy U2 otrymu[mo
Ee N− ετ ε/ = E Ee eN N− − −εθ ετε ε/ / 1 ≤ ( ) /1 1− − −C e Nε ετ εE ≤ ( ) /1 − C Nε ε ≤
≤ e C N− → 0 pry N → ∞ .
Za umovog U1
P max
/
/
1≤ ≤
≥
k N
kε
θ δ ε ≤ P ( / )
/
θ δ ε
ε
k
k
N
≥
=
∑
1
≤
N
F t dt
u R
u
dε δ ε
sup ( )
/∈
∞
∫ ≤
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 5
680 I. V. SAMOJLENKO, I. V. MALYK
≤
N
F
u R
u
dε
ε
δ
δ εsup ( / )
∈
=
N
F
u R
u
dδ
δ εsup ( / )
∈
→ 0 pry ε → 0 .
Ostatoçno ma[mo
P max ( )
0 ≤ ≤
≥
t T
tγ δε ≤ P max
( ) /
0 ≤ ≤ +
≥
t T t
θ δ ενε ≤
≤ P max ,
/
//
1
1≤ ≤ + ≥ >
k N
k N T
ε ε
εθ δ ε τ +
+ P ( )/τ ε
ε
N T≤ ≤ P Pmax
/
// ( )
1
1≤ ≤ + ≥
+ ≤
k N
k N T
ε ε
εθ δ ε τ → 0
pry ε → 0 , N → ∞ .
Lemu53 dovedeno.
Analohiçno dovodyt\sq zbiΩnist\ do 0 perßoho ta çetvertoho dodankiv zav-
dqky neperervnosti ϕ ( )u .
Ostannij dodanok prqmu[ do 0 za lemog52, oskil\ky
lim ( )
ε
ε ϕ
→ 0
Γ u = Γ0 ϕ ( )u
na test-funkciqx ϕ ( )u , wo magt\ rivnomirno obmeΩeni poxidni vsix porqdkiv.
Druhyj dodanok dorivng[
ζε
t = ϕ η ϕ ηε
τ
ε ε
ε
( ) ( )( ) ( )
( )
+ +−
+
∫
|
| id | umjimathkievua-article-2897 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:32:23Z |
| publishDate | 2010 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/0d/8fa767f3fa18f13463112d45b713b80d.pdf |
| spelling | umjimathkievua-article-28972020-03-18T19:39:51Z Convergence of a semi-Markov process and an accompanying Збіжність папівмарковського і супроводжуючого марковського процесії до марковського процесу Malik, V. F. Samoilenko, I. V. Малик, І. В. Самойленко, І. В. We propose an approach to the proof of the weak convergence of a semi-Markov process to a Markov process under certain conditions imposed on local characteristics of the semi-Markov process. Предлагается подход к доказательству слабой сходимости полумарковского процесса к марковскому в условиях, налагаемых на локальные характеристики полумарковского процесса. Institute of Mathematics, NAS of Ukraine 2010-05-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2897 Ukrains’kyi Matematychnyi Zhurnal; Vol. 62 No. 5 (2010); 674–681 Український математичний журнал; Том 62 № 5 (2010); 674–681 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/2897/2545 https://umj.imath.kiev.ua/index.php/umj/article/view/2897/2546 Copyright (c) 2010 Malik V. F.; Samoilenko I. V. |
| spellingShingle | Malik, V. F. Samoilenko, I. V. Малик, І. В. Самойленко, І. В. Convergence of a semi-Markov process and an accompanying |
| title | Convergence of a semi-Markov process and an accompanying |
| title_alt | Збіжність папівмарковського і супроводжуючого марковського процесії до марковського процесу |
| title_full | Convergence of a semi-Markov process and an accompanying |
| title_fullStr | Convergence of a semi-Markov process and an accompanying |
| title_full_unstemmed | Convergence of a semi-Markov process and an accompanying |
| title_short | Convergence of a semi-Markov process and an accompanying |
| title_sort | convergence of a semi-markov process and an accompanying |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2897 |
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