Stationary distributions of fading evolutions
We study fading random walks on the line. We determine stationary distributions of the fading Markov evolution and investigate the special semi-Markov case where the sojourn times of the renewal process have Erlang distributions.
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| Datum: | 2009 |
|---|---|
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| Format: | Artikel |
| Sprache: | Ukrainisch Englisch |
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Institute of Mathematics, NAS of Ukraine
2009
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509052996419584 |
|---|---|
| author | Pogorui, A. О. Погоруй, А. О. |
| author_facet | Pogorui, A. О. Погоруй, А. О. |
| author_sort | Pogorui, A. О. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:43:35Z |
| description | We study fading random walks on the line. We determine stationary distributions of the fading Markov evolution and investigate the special semi-Markov case where the sojourn times of the renewal process have Erlang distributions. |
| first_indexed | 2026-03-24T02:34:59Z |
| format | Article |
| fulltext |
UDK 519.21
A. O. Pohoruj (Ûytomyr. un-t)
STACIONARNI ROZPODILY ZHASAGÇYX EVOLGCIJ
We study fading random walks on the line. We compute stationary distributions of the Markov fading
evolution and study the special semi-Markov case where sojourn times of the renewal process are
Erlang-distributed.
Rassmatryvagtsq zatuxagwye sluçajn¥e bluΩdanyq na prqmoj. V¥çyslen¥ stacyonarn¥e
raspredelenyq zatuxagwej markovskoj πvolgcyy, a takΩe yssledovan çastn¥j polumarkov-
skyj sluçaj, kohda vremena preb¥vanyq processa vosstanovlenyq ymegt πrlanhovskye raspre-
delenyq.
1. Vstup. Uperße telehrafnyj proces, qk model\ evolgci] çastynky na prq-
mij, vyvçavsq u robotax Hol\dßtejna [1] i Kaca [2]. Pislq c\oho telehrafnyj
proces doslidΩuvavsq bahat\ma matematykamy i fizykamy, oskil\ky takyj pro-
ces [ al\ternatyvog do vinerovo] modeli brounivs\koho procesu i ma[ vaΩlyve
znaçennq dlq praktyçnyx zastosuvan\ [3 – 9].
Rozhlqdalys\ takoΩ rizni uzahal\nennq telehrafnoho procesu na bahato-
vymirni prostory ta napivmarkovs\ki peremykagçi procesy (dyv. roboty [5 – 11]
ta navedenu v nyx bibliohrafig).
U roboti [12] doslidΩeno zhasagçu markovs\ku vypadkovu evolgcig, qka [
uzahal\nennqm modeli Hol\dßtejna � Kaca prqmolinijno] evolgci] çastynky na
vypadok, koly ]] ßvydkist\ z çasom prqmu[ do nulq. Zhasagça evolgciq mode-
lg[ rux çastynky na prqmij pid di[g zovnißn\o] syly, v rezul\tati qko] çastyn-
ka zupynq[t\sq u deqkij toçci, a otΩe, isnu[ hranyçnyj rozpodil koordynaty
procesu na prqmij.
U danij roboti doslidΩu[t\sq stacionarnyj rozpodil zhasagço] markovs\ko]
evolgci] iz zatrymkog u vidbyvagçomu ekrani, obçysleno hranyçnyj rozpodil
zhasagço] evolgci] z erlanhivs\kymy peremykannqmy.
2. Stacionarnyj rozpodil dlq markovs\koho vypadku z zatrymugçym
ekranom. Nexaj θk , k ≥ 0, � poslidovnist\ nezaleΩnyx vypadkovyx velyçyn,
qki magt\ pokaznykovi rozpodily P θk t≥{ } = e k t–λ I t ≥{ }0 , λk > 0. Vvedemo
vidpovidnyj cij poslidovnosti stoxastyçnyj potik τn =
k
n
k=∑ 0
θ , n ≥ 1. Nexaj
ξ( )t � proces vidnovlennq, qkyj zada[t\sq formulog ξ( )t = max n{ ≥ 0 : τn ≤
≤ t}, t > 0.
Rozhlqnemo zhasagçyj telehrafnyj proces η( )t = – ( )a t( )ξ , de 0 < a < 1 �
konstanta, t ≥ 0, i vidpovidnu jomu markovs\ku vypadkovu evolgcig
x t( ) =
0
t
sa ds∫ ( )– ( )ξ .
Cej proces vidriznq[t\sq vid rozhlqnutoho u roboti [12] tym, wo rizni θk ma-
gt\ rizni parametry λk , k ≥ 0. Nexaj u toçci x = 0 proces x t( ) ma[ vidbyvag-
çyj ekran iz zatrymkog, tobto qkwo x t( ) dosqh nulq i ξ( )t ma[ neparne zna-
çennq, to x t( ) = 0 do tyx pir, poky ξ( )t ne zminyt\ znaçennq. Naßa meta po-
lqha[ u doslidΩenni umov isnuvannq stacionarnoho rozpodilu ρ procesu x t( ) z
vidbyvagçym ekranom ta oderΩanni formuly dlq joho obçyslennq.
Rozhlqnemo dvokomponentnyj markovs\kyj proces ς( )t = x t( )( , ξ( )t ) . Infi-
nitezymal\nyj operator A c\oho procesu [ vidomym [5, 6]:
A x sϕ( , ) = C s
d x s
dx
( )
( , )ϕ
+ λ ϕ ϕs P x s x s( , ) – ( , )[ ],
© A. O. POHORUJ, 2009
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 3 425
426 A. O. POHORUJ
de x R∈ , s = 0, 1, 2, … , funkciq ϕ( , )x s [ neperervno dyferencijovnog po x
z obmeΩenog perßog poxidnog, P x sϕ( , ) = P ϕ (x, s + 1), a funkciq C s( ) za-
da[t\sq formulog
C s( ) = (– )a s.
Nexaj ρ( , )x s � stacionarnyj rozpodil procesu ς( )t , todi
s
x s A x s dx
=
∞ ∞
∑ ∫
0 0
ρ ϕ( , ) ( , ) = 0. (1)
Zvidsy oderΩu[mo rivnqnnq dlq ρ( , )x s , a same
d x
dx
ρ( , )0
+ λ ρ0 0( , )x = 0,
–
( , )
a
d x
dx
ρ 1
+ λ ρ1 1( , )x – λ ρ0 0( , )x = 0,
a
d x
dx
2 2ρ( , )
+ λ ρ2 2( , )x – λ ρ1 1( , )x = 0, (2)
………………………………………………
(– )
( , )–1 1n na
d x n
dx
ρ
+ λ ρn x n( , ) – λ ρn x n– ( , – )1 1 = 0,
……………………………………………………………
z hranyçnymy umovamy λ2 1n – ρ[0, 2n – 1] = a n2 1– ρ(0 +, 2n – 1), n = 1, 2, … , qki
otrymugt\ iz (1) z uraxuvannqm isnuvannq v x = 0 atomiv ρ[0, 2n – 1] stacio-
narnoho rozpodilu ς( )t . Rozv�qzugçy poslidovno rivnqnnq systemy (2), oder-
Ωu[mo:
dlq neperervno] çastyny miry ρ
ρ( , )x 0 = c e x
0
0–λ , ρ( , )x 1 = c
a
e x
0
0
1 0
0
λ
λ λ
λ
+
– ,
ρ( , )x 2 = c
a a
e x
0
0
1 0
1
2
2
0
0
λ
λ λ
λ
λ λ
λ
+ –
– ,
………………………………………………………
ρ( , )x n = c
a0
0
1 0
λ
λ λ+
…
λ
λ λ
λn
n
n n
x
a
e– –
– (– )
1
01
0
i dlq atomiv
ρ[0, 2n – 1] = a n
n
n
2 1
2 1
0 2 1
–
–
( , – )
λ
ρ + =
= c a
a
n
n
0
2 1
2 1
0
1 0
–
–λ
λ
λ λ+
…
λ
λ λ
2 2
2 1
2 1
0
n
n
na
–
–
–+
, n = 1, 2, … .
Konstanta c0 [ normugçym mnoΩnykom, qkyj vyznaça[t\sq z rivnosti
n
x n dx
=
∞ ∞
∑ ∫
0 0
ρ( , ) +
m
m
=
∞
∑ [ ]
1
0 2 1ρ , – = 1. (3)
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 3
STACIONARNI ROZPODILY ZHASAGÇYX EVOLGCIJ 427
Zvidsy dlq isnuvannq nevyrodΩenoho rozpodilu ρ( )⋅ neobxidno, wob zbihalys\
rqdy
1
0λ
+ 1
1 0λ λ+ a
+ 1
1 0
1
2
2
0λ λ
λ
λ λ+ a a–
+ … + 1
1 0λ λ+ a
…
…
λ
λ λ
n
n
n na
–
– (– )
1
01
+ … (4)
ta
n
n
n
a
a a=
∞
∑ + +1
2 1
2 1
0
1 0
2
3
3
0
–
–λ
λ
λ λ
λ
λ λ
…
λ
λ λ
2 2
2 1
2 1
0
n
n
na
–
–
–+
. (5)
NevaΩko perekonatys\, wo koly 0 < a < 1 i λ = λ0 = λ1 = λ2 = … , to rqd (4) [
rozbiΩnym. Zaznaçymo, wo bez vidbyvagçoho ekranu takyj proces ma[ stacio-
narnyj rozpodil [12]. Poznaçymo n-j çlen rqdu (4) çerez dn . Vykorystovug-
çy kryterij Raabe dlq zbiΩnosti rqdiv, moΩna sformulgvaty dostatng umovu
zbiΩnosti rqdu (4):
lim –
n
n
n
n
d
d→∞ +
1
1 = lim
– (– )
n
n n
n n
n
n
a
→∞
+
++λ λ λ
λ
1
1
01
= p > 1.
Poznaçymo n-j çlen rqdu (5) çerez sn . Dlq zbiΩnosti rqdu (5) dostatn\o, wob
lim –
n
n
n
n
s
s→∞ +
1
1 = lim
– –
–n
n n n
n
n
n n
n
a a
a→∞
+
+
++λ λ λ λ λ
λ λ
2 1
2 2
2 2 1
2 1
0 2 1
2
2 2 1
= p > 1.
Zokrema, rqdy (4), (5) zbihagt\sq, qkwo isnu[ N ≥ 1 take, wo dlq vsix n ≥ N
λn = bn , de b > a, i v c\omu vypadku isnu[ stacionarnyj rozpodil procesu x t( )
z neperervnog çastynog ρ( )x =
n
x n=
∞∑ 0
ρ( , ) ta atomom ρ 0[ ] =
n=
∞∑ [
1
0ρ , 2n –
– 1]. Poznaçymo çerez σ1, σ2 sumy rqdiv (4), (5) vidpovidno. Todi stacionarnyj
rozpodil procesu x t( ) ma[ vyhlqd
ρ( )x =
λ σ
σ σ
λ0 1
1 2
0
+
e x– , ρ 0[ ] =
σ
σ σ
1
1 2+
.
3. Erlanhivs\kyj vypadok. Nexaj proces vidnovlennq ξ( )t zada[t\sq
formulog ξ( )t = max n{ ≥ 0 : τn ≤ t}, t > 0, de τn =
k
n
k=∑ 0
θ , θk � nezaleΩ-
ni vypadkovi velyçyny z erlanhivs\kym rozpodilom zi wil\nistg
f tk ( ) =
dF t
dt
k ( )
= λ λ2 0te I tt– ≥{ }, λ > 0.
U c\omu vypadku vypadkova evolgciq x t( ) =
0
t sa ds∫ (– ) ( )ξ [ napivmarkovs\kog.
Obçyslymo hranyçnyj rozpodil c\oho ne markovs\koho procesu.
Rozhlqnemo vypadkovu velyçynu σ =
0
∞
∫ (– ) ( )a dssξ . Oskil\ky θk odnakovo
rozpodileni i σ = θ1( + a2θ3 + … ) – a θ2( + a2θ4 + … ), to dlq znaxodΩennq
funkci] rozpodilu F xσ( ) = P σ ≤{ }x dosyt\ znajty rozpodil velyçyny η =
= θ1 + a2θ3 + … . Dlq sprowennq vykladok poklademo λ = 1. Poznaçymo
F x( ) = P η ≤{ }x , todi
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61 # 3
428 A. O. POHORUJ
F x( ) = P θ η1
2+ ′ ≤{ }a x =
0
2
∞
∫ + ′ ≤{ }ue u a x duu– P η =
=
0
2
∞
∫ ′ ≤{ }ue
x u
a
duu– P η –
,
de ′η � vypadkova velyçyna, odnakovo rozpodilena z η.
OtΩe,
F x( ) =
0
2
∞
∫ { }ue F
x u
a
duu– –
. (6)
NevaΩko perekonatys\, wo F( )0 = 0. Budemo ßukaty F x( ) u vyhlqdi rqdu
F x( ) = 1 + a x a e x
01 02+( ) – + a x a e x a
11 12
2
+( ) – / + …
… + a x a en n
x a n
1 2
2
+( ) – / + … . (7)
Zvidsy z uraxuvannqm (6) ma[mo
F x( ) =
0
01 2 02 11 2 121
2 4
∞
∫ + +
+ +
ue a
x u
a
a e a
x u
a
a eu
x u
a
x u
a– – ––
–
–
–
+
+ a
x u
a
a e du
x u
a
21 2 22
6– –
–
+
+ …
=
= 1 – xe x– – e x– + a a
a x x a e a x x a e
a
x x a
01
2
2 2 2 2
2 3
2 2
1
2
( – ) ( – – )
( – )
– – /+ +
+
+ a a
x a x a e a e
a
x x a
02
2
2 2 2
2 2
2
1
( – – )
( – )
– – /+
+
+ a a
a x x a e a x x a e
a
x x a
11
6
4 4 4 4
4 3
2 2
1
4
( – ) ( – )
( – )
– – /+ + +
+
+ a a
x a x a e a e
a
x x a
12
4
4 4 4
4 2
4
1
( – – )
( – )
– – /+
+
+ a a
a x x a e a x x a e
a
x x a
21
10
6 6 6 6
6 3
2 2
1
6
( – ) ( – )
( – )
– – /+ + +
+
+ a a
x a x a e a e
a
x x a
22
6
6 6 6
6 2
6
1
( – – )
( – )
– – /+
+ … , (8)
zvidky, v svog çerhu, z uraxuvannqm (7) oderΩu[mo:
pry xe x–
a01 = – 1 + a a
a
01
2
2 21( – )
+ a a
a
02
2
21 –
+ a a
a
11
6
4 21( – )
+ a a
a
12
4
41 –
+
+ a a
a
21
10
6 21( – )
+ a a
a
22
6
61 –
+ … + a a
a
n
n
n1
4 2
2 2 21
+
+( – )
+ a a
a
n
n
n2
2 2
2 21
+
+–
+ … ,
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 3
STACIONARNI ROZPODILY ZHASAGÇYX EVOLGCIJ 429
pry e x–
a02 = – 1 + a a
a
01
4
2 3
2
1( – )
– a a
a
02
4
2 21( – )
+ a a
a
11
10
4 3
2
1( – )
– a a
a
12
8
4 21( – )
+
+ a
a
a
21
16
6 3
2
1( – )
– a a
a
22
12
6 21( – )
+ … + a a
a
n
n
n1
6 4
2 2 3
1
+
+( )–
– a a
a
n
n
n2
4 4
2 2 2
1
+
+( )–
+ … ,
(9)
pry xe
x
a n– ( )2 1+
a n( )+1 1 = a a
a
n
n
n1
4 2
2 2 2
1
+
+( )–
,
pry e
x
a n– ( )2 1+
a n( )+1 2 = –
–
a
a
a
n
n
n1
6 4
2 2 3
2
1
+
+( )
+ a a
a
n
n
n2
4 4
2 2 2
1
+
+( )–
, n = 1, 2, … .
Iz (9) otrymu[mo dva spivvidnoßennq:
1 = a a
a
a
a a
a
a a a
01
2
2 2
8
2 2 4 2
18
2 2 4 2 6 21
1 1 1 1 1 1
+
( )
+
( ) ( )
+
( ) ( ) ( )
+ …
– – – – – –
+
+ 2
1 1 1 1 1
8
2 3 4
18
2 3 4 2 6
a
a a
a
a a a– – – – –( ) ( )
+
( ) ( ) ( )
+
+ a
a a a
18
2 2 4 3 61 1 1– – –( ) ( ) ( )
+ …
+
+ a a
a
a
a a
a
a a a
02
2
2
8
2 2 4
18
2 2 4 2 61 1 1 1 1 1– – – – – –
+
( ) ( )
+
( ) ( ) ( )
+ …
(10)
ta
1 = a a
a
a
a a
01
2
2 3
12
2 2 4 3
2
1
2
1 1– – –( )
+
( ) ( )
+ 2
1 1
12
2 3 4 2
a
a a– –( ) ( )
+
+ 2
1 1 1
24
2 2 4 2 6 3
a
a a a– – –( ) ( ) ( )
+
2
1 1 1
24
2 2 4 3 6 2
a
a a a– – –( ) ( ) ( )
+
+ 2
1 1 1
24
2 3 4 2 6 2
a
a a a– – –( ) ( ) ( )
+ …
–
– a a
a
a
a a
02
4
2 2
12
2 2 4 21
1 1 1
+
( )
+
( ) ( )
– – –
+ a
a a a
24
2 2 4 2 6 2
1 1 1– – –( ) ( ) ( )
+ …
.(11)
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61 # 3
430 A. O. POHORUJ
NevaΩko perekonatys\, vykorystavßy, napryklad, oznaku Dalambera, wo vsi
rqdy pry koefici[ntax rivnostej (10), (11) [ zbiΩnymy.
Zaznaçymo, wo dlq rqdu pry koefici[nti a01 u formuli (10) ma[ misce for-
mula [13, s. 202]
1 + a
a
2
2 2
1–( )
+ a
a a
8
2 2 4 2
1 1– –( ) ( )
+ a
a a a
18
2 2 4 2 6 2
1 1 1– – –( ) ( ) ( )
+ …
… =
k
ka
=
∞
∏( )
1
2 1
1 –
–
.
Vvedemo poznaçennq
S1 = 1
1 1 1 1 1 1
2
2 2
8
2 2 4 2
18
2 2 4 2 6 2+
( )
+
( ) ( )
+
( ) ( ) ( )
+ …
a
a
a
a a
a
a a a– – – – – –
+
+ 2
1 1 1 1 1
8
2 3 4
18
2 3 4 2 6
a
a a
a
a a a– – – – –( ) ( )
+
( ) ( ) ( )
+
+ a
a a a
18
2 2 4 3 61 1 1– – –( ) ( ) ( )
+ …
,
S2 = a
a
2
21 –
+ a
a a
8
2 2 41 1– –( ) ( )
+ a
a a a
18
2 2 4 2 61 1 1– – –( ) ( ) ( )
+ … ,
S3 = 2
1
2
2 3
a
a –( )
+ 2
1 1
12
2 2 4 3
a
a a– –( ) ( )
+ 2
1 1
12
2 3 4 2
a
a a– –( ) ( )
+
+ 2
1 1 1
24
2 2 4 2 6 3
a
a a a– – –( ) ( ) ( )
+ … ,
S4 = 1 + a
a
4
2 2
1–( )
+ a
a a
12
2 2 4 2
1 1– –( ) ( )
+ a
a a a
24
2 2 4 2 6 2
1 1 1– – –( ) ( ) ( )
+ … .
Rozv�qzugçy (10), (11), znaxodymo
a01 =
S S
S S S S
2 3
1 3 2 4
+
+
, a02 =
S S
S S S S
4 1
1 3 2 4
–
+
.
Zvidsy z uraxuvannqm (9) obçyslg[mo znaçennq inßyx koefici[ntiv rozkla-
du (7).
Iz (9) lehko baçyty, wo F( )0 = 1 + a02 + a12 + a22 + … = 0. Dali, iz (6) vy-
plyva[, wo F x( ) [ monotonnog, a iz (7) i (9) � lim ( )
x
F x
→ +∞
= 1. OtΩe, F x( ) �
funkciq rozpodilu. Lehko baçyty, wo ne isnu[ inßyx funkcij rozpodilu, qki b
zadovol\nqly rivnqnnq (6). Dijsno, qkwo F1 ≠ F � funkciq rozpodilu, wo [
rozv�qzkom (6), to i funkciq Φ = F1 – F [ rozv�qzkom (6), do toho Ω Φ [ nemo-
notonnog, wo nemoΩlyvo.
Funkciq rozpodilu dlq
σ = θ θ1
2
3+ + …( )a – a aθ θ2
2
4+ + …( )
ma[ vyhlqd
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 3
STACIONARNI ROZPODILY ZHASAGÇYX EVOLGCIJ 431
F xσ( ) = P σ ≤{ }x = P θ θ θ θ1
2
3 2
2
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.
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| id | umjimathkievua-article-3031 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:34:59Z |
| publishDate | 2009 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/a8/b2ca1948912a318abd49afb6e28f74a8.pdf |
| spelling | umjimathkievua-article-30312020-03-18T19:43:35Z Stationary distributions of fading evolutions Стаціонарні розподіли згасаючих еволюцій Pogorui, A. О. Погоруй, А. О. We study fading random walks on the line. We determine stationary distributions of the fading Markov evolution and investigate the special semi-Markov case where the sojourn times of the renewal process have Erlang distributions. Рассматриваются затухающие случайные блуждания на прямой. Вычислены стационарные распределения затухающей марковской эволюции, а также исследован частный полумарковский случай, когда времена пребывания процесса восстановления имеют эрланговские распределения. Institute of Mathematics, NAS of Ukraine 2009-03-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3031 Ukrains’kyi Matematychnyi Zhurnal; Vol. 61 No. 3 (2009); 425-431 Український математичний журнал; Том 61 № 3 (2009); 425-431 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3031/2811 https://umj.imath.kiev.ua/index.php/umj/article/view/3031/2812 Copyright (c) 2009 Pogorui A. О. |
| spellingShingle | Pogorui, A. О. Погоруй, А. О. Stationary distributions of fading evolutions |
| title | Stationary distributions of fading evolutions |
| title_alt | Стаціонарні розподіли згасаючих еволюцій |
| title_full | Stationary distributions of fading evolutions |
| title_fullStr | Stationary distributions of fading evolutions |
| title_full_unstemmed | Stationary distributions of fading evolutions |
| title_short | Stationary distributions of fading evolutions |
| title_sort | stationary distributions of fading evolutions |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3031 |
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