Distribution of the lower boundary functional of the step process of semi-Markov random walk with delaying screen at zero
On the basis of a given sequence of independent identically distributed pairs of random variables, we construct the step process of semi-Markov random walk that is later delayed by a screen at zero. For this process, we obtain the Laplace transform of the distribution of the time of the first hit of...
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| Date: | 2007 |
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| Language: | Russian English |
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Institute of Mathematics, NAS of Ukraine
2007
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/3356 |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509432866144256 |
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| author | Nasirova, T. I. Omarova, K. K. Насирова, Т. И. Омарова, К. К. Насирова, Т. И. Омарова, К. К. |
| author_facet | Nasirova, T. I. Omarova, K. K. Насирова, Т. И. Омарова, К. К. Насирова, Т. И. Омарова, К. К. |
| author_sort | Nasirova, T. I. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:52:14Z |
| description | On the basis of a given sequence of independent identically distributed pairs of random variables, we construct the step process of semi-Markov random walk that is later delayed by a screen at zero. For this process, we obtain the Laplace transform of the distribution of the time of the first hit of the level zero. |
| first_indexed | 2026-03-24T02:41:01Z |
| format | Article |
| fulltext |
UDK 519.217
T. Y. Nasyrova (Bakyn. un-t, AzerbajdΩan),
K. K. Omarova (Yn-t kybernetyky NAN AzerbajdΩana, Baku)
RASPREDELENYE NYÛNEHO HRANYÇNOHO
FUNKCYONALA STUPENÇATOHO PROCESSA
POLUMARKOVSKOHO BLUÛDANYQ
S ZADERÛYVAGWYM ∏KRANOM V NULE
On the basis of a given sequence of independent identically distributed pairs of random variables, we
construct a step-type process of semi-Markov random walk, which is later delayed with a screen at the
zero. For this process, we obtain the Laplace transformation of time of the first attainment of the zero
level.
Za zadanog poslidovnistg nezaleΩnyx odnakovo rozpodilenyx par vypadkovyx velyçyn pobudo-
vano sxidçastyj proces napivmarkovs\koho blukannq, qkyj potim zatrymu[t\sq ekranom u nuli.
Dlq c\oho procesu znajdeno peretvorennq Laplasa rozpodilu perßoho momentu dosqhnennq riv-
nq nul\.
Vvedenye. NaxoΩdenyg raspredelenyq pervoho momenta dostyΩenyq zaderΩy-
vagweho πkrana v nule posvqweno nemalo rabot. V rabote [1, c. 91 – 93] dlq
processa polumarkovskoho bluΩdanyq s otrycatel\n¥m snosom poloΩytel\-
n¥my skaçkamy najdeno preobrazovanye Laplasa pervoho momenta dostyΩenyq
zaderΩyvagweho πkrana v nule. V rabotax [2 – 4; 5, c. 26 – 51; 6, c. 69 – 76] yzu-
çen¥ razlyçn¥e problem¥, svqzann¥e s hranyçn¥my funkcyonalamy sluçajn¥x
bluΩdanyj.
V dannoj rabote druhym, bolee prost¥m metodom, kohda bluΩdanye proysxo-
dyt po sloΩnomu laplasovomu raspredelenyg porqdka ( m, 1 ) , poluçen qvn¥j
vyd preobrazovanyq Laplasa raspredelenyq pervoho momenta dostyΩenyq za-
derΩyvagweho πkrana v nule.
1. Postanovka zadaçy. Pust\ zadana posledovatel\nost\ nezavysym¥x ody-
nakovo raspredelenn¥x par sluçajn¥x velyçyn { }( ), ( )ξ ω η ωk k k ≥1, v kotoroj
sluçajn¥e velyçyn¥ ξ ωk ( ) , k ≥ 1, nezavysym¥ meΩdu soboj, η ωk ( ) , k ≥ 1,
toΩe nezavysym¥ meΩdu soboj y ξ ωk ( ) > 0. Dalee, predpolahaem, çto
E ξ ω1( ) < ∞ , E η ω1( ) < ∞ y E η ω1( ) < 0. Postroym stupençat¥j process po-
lumarkovskoho bluΩdanyq [7, c. 11]
X1 ( t, ω ) = η ωi
i
k
( )
=
∑
0
, ξ ωi
i
k
( )
=
∑
0
≤ t < ξ ωi
i
k
( )
=
+
∑
1
1
,
hde
ξ ω0( ) = 0, k = 0, 1, 2, … .
ZaderΩym πtot process πkranom v nule [8, c. 41]
X ( t, ω ) = X1 ( t, ω ) – inf ( , ( , ))
0
10
≤ ≤s t
X s ω .
Process X ( t, ω ) budem naz¥vat\ stupençat¥m processom polumarkovskoho
bluΩdanyq s zaderΩyvagwym πkranom v nule (yly s zaderΩyvagwym πkranom
„0”).
Opredelym sluçajnug velyçynu τ ωη0
( ) = inf , ( , ){ }t X t ω = 0 .
PoloΩym τ ωη0
( ) = ∞, esly dlq vsex t X ( t, ω ) > 0.
Oçevydno, çto τ ωη0
( ) qvlqetsq perv¥m momentom dostyΩenyq zaderΩyva-
gweho πkrana „0” processom X ( t, ω ) .
© T. Y. NASYROVA, K. K. OMAROVA, 2007
912 ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 7
RASPREDELENYE NYÛNEHO HRANYÇNOHO FUNKCYONALA … 913
Cel\g dannoj stat\y qvlqetsq naxoΩdenye preobrazovanye Laplasa raspre-
delenyq sluçajnoj velyçyn¥ τ ωη0
( ) . Preobrazovanye Laplasa raspredelenyq
sluçajnoj velyçyn¥ τ ωη0
( ) oboznaçym çerez
L ( s ) = Ee
s− τ ωη0
( )
, s > 0,
preobrazovanye Laplasa ee uslovnoho raspredelenyq — çerez
L ( s / z ) = E e X z
s− =[ ]τ ωη ω0 0
( )
( , ) .
Po formule polnoj veroqtnosty ymeem
L ( s ) =
z
L s z d X z
=
∞
∫ <
0
0( ) ( , )/ { }P ω . (1)
Dalee, vvedem sledugwye oboznaçenyq:
N ( t / z ) = P{ / }( ) ( , )τ ω ωη0
0> =t X z
y
Ñ ( s / z ) =
0
∞
−∫ e N t z dtst ( )/ .
Oçevydno, çto
˜ ( )/N s z =
1 − L s z
s
( )/
yly
L s z( )/ = 1 − M s z( )/ ,
hde
M s z( )/ = s N s z˜ ( )/ .
Poskol\ku L s z( )/ v¥raΩaetsq çerez
˜ ( )/N s z , sostavym yntehral\noe uravne-
nye dlq posledneho.
Vvedem ewe odno oboznaçenye
ϕ ( s ) = Ee s− ξ ω1( ).
2. Sostavlenye yntehral\noho uravnenyq dlq Ñ (((( s //// z )))) . Yntehral\noe
uravnenye dlq
˜ ( )/N s z dano v sledugwej teoreme.
Teorema221. Funkcyq
˜ ( )/N s z udovletvorqet sledugwemu yntehral\nomu
uravnenyg:
˜ ( )/N s z =
1
0
1
− + + <
=
∞
∫ϕ ϕ η ω( )
( ) ˜ ( ) ( )/ { }s
s
s N s y d z y
y
y P . (2)
Dokazatel\stvo. Po formule polnoj veroqtnosty ymeem
P{ / }( ) ( , )τ ω ωη0
0> =t X z = P{ / }( ) ; ( ) ( , )τ ω ξ ω ωη0 1 0> > =t t X z +
+
u
t
y
du z z dy
= =
∞
∫ ∫ ∈ + > + ∈
0 0
1 1 10P{ }( ) , ( ) , ( )ξ ω η ω η ω ×
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 7
914 T. Y. NASYROVA, K. K. OMAROVA
× P{ / }( ) ( , )τ ω ωη0
0> − =t u X y . (3)
UmnoΩaq obe çasty (3) na e st− , yntehryruq po t ot 0 do ∞ y uçyt¥vaq obo-
znaçenye
˜ ( )/N s z , poluçaem uravnenye (2).
Teorema dokazana.
Uravnenye (2) dlq proyzvol\no raspredelenn¥x sluçajn¥x velyçyn ξ ωk ( ) ,
η ωk ( ) , k ≥ 1, moΩno reßyt\ metodom posledovatel\n¥x pryblyΩenyj. No ta-
koe reßenye nepryhodno dlq pryloΩenyj. ∏to uravnenye ymeet reßenye v qv-
nom vyde v klasse sloΩn¥x laplasov¥x raspredelenyj. Opredelenye sloΩnoho
laplasovoho raspredelenyq porqdka ( n, 1 ) dano v dokazatel\stve teorem¥JJ2.
Teorema222. Pust\
η ωk ( ) = η ω η ω η ω η ωk k kn k1 2
+ + + −+ + … + −( ) ( ) ( ) ( ), k = 1, ∞ ,
hde η ωki
+ ( ), i = 1, n , y η ωk
− ( ) ymegt πrlanhovskoe raspredelenye n -ho po-
rqdka y pervoho porqdka s parametramy λ y µ sootvetstvenno y E η ωk ( ) <
< 0.
Tohda
L ( s / z ) =
λ ϕ
λ
n
n
k s zs
k s
e
( )
[ ( )]
( )
− 1
1 ,
L ( s ) =
λ ϕ
λ
2
1
2
n
n
s
k s
( )
[ ( )]−
,
E τ ωη0
( ) =
λ µ
λ µ
ξ ω+
−
n
n
E 1( ),
Dτ ωη0
( ) =
λ µ
λ µ
ξ ω λµ λ µ
λ µ
ξ ω+
−
+ +
−
n
n
n
n
D E1 3 1
22
( )
( )
[ ]
[ ( )] .
Dokazatel\stvo. Lehko pokazat\, çto funkcyq raspredelenyq sluçajnoj
velyçyn¥ η ωk ( ) ymeet vyd
F x
nη η η η1 2 1
+ + + −+ +…+ − ( ) =
λ
λ µ
λ λ
λ µ
µ
λ
n
n
x
x
k n k
k
n
e x
e
x
k
x
( )
, ,
( )
!
, ,
+
<
− −
+
>
−
−
=
−
∑
0
1 1 0
0
1 . (4)
a plotnost\ raspredelenyq
ρη η η η1 2 1
+ + + −+ +…+ −n
x( ) =
=
λ µ
λ µ
λ λ λ λ
λ µ
λ
λ µ
µ
λ
n
n
x
x
k
k
n n k
n k
n
n
e x
e
x
k
x
k
x
( )
, ,
( )
( )! ( ) ( )
, .
+
<
−
−
−
+
+ −
+
>
−
−
=
− −
−∑
0
1
1 1 1 0
1
1
1 (5)
Funkcyg raspredelenyq, zadannug formuloj (4), budem naz¥vat\ sloΩnoj
laplasovoj funkcyej raspredelenyq porqdka ( n, 1 ) . Uçyt¥vaq v¥raΩenye
plotnosty raspredelenyq sluçajnoj velyçyn¥ η ω1( ) v (2), poluçaem ynteh-
ral\noe uravnenye otnosytel\no M s z( )/ :
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 7
RASPREDELENYE NYÛNEHO HRANYÇNOHO FUNKCYONALA … 915
M s z( )/ = 1
0
− +
+ ∫ −ϕ λ µ
λ µ
ϕ µ( )
( )
( ) ( )( )
/s s e M s y dy
n
n
z
y z +
+ λϕ λ λ λ
λ µ
( )
( ( ))
( )!
( )
( )
s
y z
k
y z
k
z
k
k
n n k
n k
∞ −
=
− −
−∫ ∑ −
−
− −
−
+
1
1
1
1
1 1 +
+ 1 −
+
− −λ
λ µ
λ
n
n
y ze M s y dy
( )
( )( )
/ .
Yz πtoho yntehral\noho uravnenyq poluçym dyfferencyal\noe uravnenye
k
n
n
k k k n k n kC M s z M s z
=
+ − −∑ +[ ] −
0
1 1µ λ( ) ( )( ) ( ) ( )/ / +
+ ( ) ( ) ( )/− +1 1n n s M s zλ µϕ = ( ) ( ( ))− −1 1n n sλ µ ϕ
s hranyçn¥my uslovyqmy
M s s
s
k
x e M s x dx
n
n
k
x
k x
k
n
( ) ( )
( )
( )
( )
!
( )/ /0 1
00
1
= − +
+
+
=
∞
−
=
−
∫∑ϕ λ µϕ
λ µ
λ µ λ
,
′ = − +
−
=
∞
− −∫M s M s
s
n
x e M s x dx
n
x
n x( ) ( )
( )
( )!
( )/ / /0 0
1
0
1µ λ µϕ λ
,
……………………………………………………………
i
k
k
i i i k i k iC M s M s
=
+ − −∑ +[ ] −
0
10 0 1µ λ( ) ( )( ) ( ) ( )/ / =
=
( ) ( )
[ ( )]!
( )( )
/
−
− +
=
∞
− + −∫1
1
0
1
k n
x
n k xs
n k
x e M s x dx
λ µϕ λ
, (6)
……………………………………………………………
i
n
n
i i i n i n iC M s M s
=
−
−
+ − + − +∑ +[ ] −
0
2
2
1 2 20 0 1µ λ( ) ( ) ( ) ( )( ) ( ) ( )/ / =
= ( ) ( ) ( )/− −
=
∞
−∫1 2
0
n n
x
xs xe M s x dxλ µϕ λ
,
i
n
n
i i i n i n iC M s M s
=
−
−
+ − + − +∑ +[ ] −
0
1
1
1 1 10 0 1µ λ( ) ( ) ( ) ( )( ) ( ) ( )/ / =
= ( ) ( ) ( )/− −
=
∞
−∫1 1
0
n n
x
xs e M s x dxλ µϕ λ
,
xarakterystyçeskym uravnenyem
i
n
n
i i i n i n i n nC k s k s s
=
+ − − +∑ +[ ] − + −
0
1 11 1µ λ λ µϕ( ) ( ) ( ) ( ) ( ) = 0 (7)
y obwym reßenyem
M s z( )/ =
i
n
i
k s zc s e i
=
+
∑ +
1
1
1( ) ( ) , (8)
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 7
916 T. Y. NASYROVA, K. K. OMAROVA
hde c si ( ) — neyzvestn¥e postoqnn¥e y k si ( ), i = 1 1, n + , — korny xarakterys-
tyçeskoho uravnenyq (7). Yspol\zuq hranyçn¥e uslovyq (6) y obwyj vyd reße-
nyq (8), ymeem sledugwug systemu neodnorodn¥x lynejn¥x alhebrayçeskyx
uravnenyj otnosytel\no c sl ( ), l = 1 1, n + :
l
n n
n
i
n i
l
i l
s
k s
c s s
=
+
=
−
+∑ ∑−
+
+
−
= −
1
1
0
1
11 1
λ µϕ
λ µ
λ µ
λ
ϕ( )
( )
( )
[ ( )]
( ) ( ) ,
l
n
l
n
l
n lk s
s
k s
c s
=
+
∑ + −
−
= −
1
1
µ λ µϕ
λ
µ( )
( )
[ ( )]
( ) ,
…………………………………………………………
l
n
i
k
k
i
l
i
l
i k i k i
k n
l
n k lC k s k s
s
k s
c s
=
+
=
+ − −
−∑ ∑ +[ ] − − −
−
=
1
1
0
1 1
1
0µ λ λ µϕ
λ
( ) ( ) ( )
( ) ( )
[ ( )]
( ) ,
(9)
………………………………………………………………………
l
n
i
n
n
i
l
i
l
i n i n iC k s k s
=
+
=
−
−
+ − + − +∑ ∑ +[ ] −
1
1
0
2
2
1 2 21µ λ( ) ( ) ( )[ ( )] [ ( )] –
( ) ( )
[ ( )]
( )
−
−
=
−1
0
2
2
n n
l
l
s
k s
c s
λ µϕ
λ
,
l
n
i
n
n
i
l
i
l
i n i n iC k s k s
=
+
=
−
−
+ − + − +∑ ∑ +[ ] −
1
1
0
1
1
1 1 11µ λ( ) ( ) ( )[ ( )] [ ( )] –
( ) ( )
( )
( )
−
−
=
−1
0
1n n
l
l
s
k s
c s
λ µϕ
λ
.
Uçyt¥vaq, çto (sm.J(8))
M s c s
l
n
l( ) ( )/ 0 1
1
1
= +
=
+
∑ ,
′ =
=
+
∑M s k s c s
l
n
l l( ) ( ) ( )/ 0
1
1
,
M s k s c sn
l
n
l
n
l
( ) ( ) ( ) ( )/ 0
1
1
=
=
+
∑ ,
[ ( )]λ −
=
+
∏ k si
i
n
1
1
= ( ) ( )− +1 1n n sλ µϕ
y
[ ( )]
,
λ −
= ≠
+
∏ k si
i i j
n
1
1
= ( ) [ ( )] [ ( )]− − +− −1 1 1n
j
n
jk s k sλ µ , j = 1 1, n + ,
ubeΩdaemsq, çto vse uravnenyq system¥ (9) svodqtsq k odnomu uravnenyg
[ ( )] ( ) [ ( )] ( ) [ ( )] ( )λ λ λ− + − + … + − + +k s c s k s c s k s c sn n
n
n
n1 1 2 2 1 1 = – λ ϕn s( ) . (10)
Tol\ko pry reßenyy
c s1( ) = –
λ ϕ
λ
n
n
s
k s
( )
[ ( )]− 1
, c si( ) = 0, i = 2, … , n + 1,
L s z( )/ qvlqetsq preobrazovanyem Laplasa raspredelenyq sluçajnoj velyçyn¥
τ ωη0
( ). Tohda ymeem
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 7
RASPREDELENYE NYÛNEHO HRANYÇNOHO FUNKCYONALA … 917
M s z( )/ = –
λ ϕ
λ
n
n
k s zs
k s
e
( )
[ ( )]
( )
−
+
1
1 1.
Ytak,
L s z( )/ = 1 − M s z( )/ =
λ ϕ
λ
n
n
k s zs
k s
e
( )
[ ( )]
( )
− 1
1 , (11)
hde k1 ( s ) — koren\ xarakterystyçeskoho uravnenyq (7), kotor¥j ymeet svojst-
vo k1 ( 0 ) = 0.
PokaΩem, çto funkcyq L s z( )/ (sm. formulu (11)) qvlqetsq preobrazova-
nyem Laplasa po s. Çtob¥ yzbeΩat\ texnyçeskyx trudnostej, nyΩe pokaΩem,
çto funkcyq L ( s ) (sm. formulu (12)) qvlqetsq preobrazovanyem Laplasa.
Yz (1) y (11) ymeem
L ( s ) =
0 1
∞
+
=
∫ ∑ <
L s z d zi
i
n
( ) ( )/ P η ω =
= λ λ
n
n z
n
z e L s z dz
( )!
( )/−
∞
− −∫1
0
1 =
λ ϕ
λ
2
1
2
n
n
s
k s
( )
[ ( )]−
,
hde
d ti
i
n
P η ω+
=
∑ <
( )
1
=
0 0
1
0 01
, ,
( )!
, , ,
t
n
t e t
n
n t
<
−
> >
− −λ λλ
— plotnost\ πrlanhovskoho raspredelenyq n-ho porqdka.
Takym obrazom, poluçaem
L ( s ) =
λ ϕ
λ
2
1
2
n
n
s
k s
( )
[ ( )]−
. (12)
Teper\ pokaΩem, çto L ( s ) est\ preobrazovanye Laplasa. Dlq πtoho po teo-
remeJJ1 (sm. [9, c. 62]) neobxodymo pokazat\, çto
L ( 0 ) = 1
y dlq lgboho natural\noho m
( )
( )−1 m
m
m
d L s
ds
> 0, s > 0.
Yz (12) ymeem
L ( s ) = λ ϕ λ2
1
2n ns k s( )[ ( )]− − .
Otsgda naxodym
L ( 0 ) = λ ϕ λ2
1
20 0n nk( )[ ( )]− − .
Poskol\ku ϕ ( s ) qvlqetsq preobrazovanyem Laplasa, ϕ ( 0 ) = 1. Yz xarakte-
rystyçeskoho uravnenyq (7) vydno, çto pry s = 0 odyn koren\ raven nulg. So-
hlasno (10) πtot koren\ k1 ( s ) . Znaçyt, L ( 0 ) = 1.
Zametym, çto λ − k si( ) > 0, i = 1 1, n + , v protyvnom sluçae yntehral¥
x
k x k s xx e e dx
=
∞
−∫
0
1λ ( ) =
x
k k s xx e dx
=
∞
− −∫
0
1[ ( )]λ
v (6) rasxodylys\ b¥, tak kak 0 < M s x( )/ < 1.
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 7
918 T. Y. NASYROVA, K. K. OMAROVA
Yz xarakterystyçeskoho uravnenyq pry λ > n µ sleduet, çto pry neçetnom i
k si
1 0( )( ) < , pry çetnom k si
1 0( )( ) > . V sylu toho, çto ϕ ( s ) — xarakterystyçe-
skaq funkcyq, ϕ( )( )i s < 0 pry neçetnom i y ϕ( )( )i s > 0 pry çetnom.
Sohlasno teoreme Lejbnyca o proyzvodnoj proyzvedenyq dvux funkcyj dlq
lgboho natural\noho m
L sm( )( ) =
i
m
m
i n i m iC k s s
=
− −∑ −( )
0
1
2[ ( )] ( )
( ) ( )λ ϕ =
= [ ( )] ( ) [ ( )] ( )( ) ( ) ( )λ ϕ λ ϕ− + −( )−
=
− −∑k s s C k s sn m
i
m
m
i n i m i
1
2
1
1
2 .
Uçyt¥vaq znaky ϕ( )( )i s y k si
1
( )( ) pry çetnom y neçetnom i , poluçaem, çto pry
çetnom m L sm( )( ) > 0 , a pry neçetnom m L sm( )( ) < 0. Poπtomu pry lgbom m
( ) ( )( )−1 m mL s > 0.
Teper\ najdem Eτ ωη0
( ) y Dτ ωη0
( ). Lehko vydet\, çto
′L s( ) =
λ λ ϕ ϕ
λ
2
1 1
1
2 1
2n
n
k s s nk s s
k s
{ }[ ( )] ( ) ( ) ( )
[ ( )]
− ′ + ′
− +
y
′′L s( ) = λ λ ϕ ϕ ϕ λ
λ
2 1 1 1 1
1
2 2
2 1 2n
n
k s s n k s s nk s s k s
k s
{ }[ ( )] ( ) ( ) ( ) ( ) ( ) ( ) [ ( )]
[ ( )]
− ′′ + − ′ ′ + ′′ −
−
+ +
+
( ) ( ) ( ) ( )( ( )) ( ) ( )
[ ( )]
{ }2 1 21 1 1
1
2 2
n k s s s k s nk s s
k s n
+ ′ ′ − + ′
−
+
ϕ ϕ λ ϕ
λ
.
Pry s = 0 ymeem
′L ( )0 =
λϕ
λ
′ + ′( ) ( )0 2 01nk
(13)
y
′′L ( )0 =
λ ϕ λ ϕ λ
λ
2
1 1 1
2
2
0 4 0 0 2 0 2 2 1 0′′ + ′ ′ + ′′ + + ′( ) ( ) ( ) ( ) ( )[ ( )]n k n k n n k
. (14)
Yz xarakterystyçeskoho uravnenyq naxodym
′k1 0( ) =
λµ
λ µ
ϕ
−
′
n
( )0 , (15)
′′k1 0( ) = 1 0 2
1
2
02
1
2
λ λ µ
λ µϕ λ µ
( )
( ) [ ( )]
−
′′ + − −
′
n
n
n
k . (16)
Yz (13) – (16) poluçaem
′L ( )0 =
λ µ
λ µ
ξ ω+
−
n
n
E 1( )
y
′′L ( )0 =
λ µ
λ µ
ξ ω λ µ λµ µ
λ µ
ξ ω+
−
+ + −
−
n
n
n n n
n
E [E1
2
2 2 2 3
3 1
24 2 4
( )
( )
[ ]
( )] .
Sledovatel\no, pry λ > n µ
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 7
RASPREDELENYE NYÛNEHO HRANYÇNOHO FUNKCYONALA … 919
Eτ ωη0
( ) =
λ µ
λ µ
ξ ω+
−
n
n
E 1( ) ,
Dτ ωη0
( ) =
λ µ
λ µ
ξ ω λµ λ µ
λ µ
ξ ω+
−
+ +
−
n
n
n
n
D [E1 3 1
22
( )
( )
[ ]
( )] .
1. Ybaev ∏. A. Preobrazovanye Laplasa odnoho hranyçnoho funkcyonala processa polumar-
kovskoho bluΩdanyq s otrycatel\n¥m snosom, poloΩytel\n¥my skaçkamy y s zaderΩyva-
gwym πkranom v nule // Tr. II resp. konf. „Sovremenn¥e problem¥ ynformatyzacyy, kyber-
netyky y ynformacyonn¥x texnolohyj”. – 2004. – S. 91 – 93.
2. Husak D. V. O hranyçn¥x funkcyonalax dlq summ sluçajnoho çysla slahaem¥x // Sb. Yn-ta
matematyky AN USSR „Analytyçeskye metod¥ yssledovanyj v teoryy veroqtnostej”. –
1981. – S.J20 – 35.
3. Husak D. V. Faktoryzacyonn¥e toΩdestva dlq summ sluçajnoho çysla slahaem¥x // Sb. Yn-
ta matematyky AN USSR „Prykladn¥e zadaçy teoryy veroqtnostej”. – 1982. – S.J25 – 44.
4. Husak D. V. Hranyçni zadaçi dlq procesiv z nezaleΩnymy pryrostamy na lancghax Markova
ta dlq napivmarkovs\kyx procesiv // Pr. In-tu matematyky NAN Ukra]ny. – 1998. – 320Js.
5. Lotov V. I. On the asymptotics of distributions in two-sided boundary problems for random walks
defined on a Markov chain // Sib. Adv. Math. – 1991. – 1, # 2. – P. 26 – 51.
6. Qpar DΩ., Nasyrova T. Y., Xapyev T. A. O veroqtnostn¥x xarakterystykax urovnq zapasa v
modely ( , )s S // Kybernetyka y system. analyz. – 1998. – # 5. – S.J69 – 76.
7. Nasyrova T. Y. Process¥ polumarkovskoho bluΩdanyq. – Baku: ∏lm, 1984. – 163 s.
8. Borovkov A. A. Veroqtnostn¥e metod¥ v teoryy massovoho obsluΩyvanyq. – M.: Nauka,
1972. – 362 s.
9. Korolgk V. S., Kovalenko Y. N., Skoroxod A. V., Turbyn A. F. Spravoçnyk po teoryy veroqt-
nostej y matematyçeskoj statystyke. – M.: Nauka, 1980. – 360 s.
Poluçeno 17.06.2005,
posle dorabotky — 25.04.2006
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 7
|
| id | umjimathkievua-article-3356 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:41:01Z |
| publishDate | 2007 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/e6/b0f7f20f09c6faae1fe018b69c35f5e6.pdf |
| spelling | umjimathkievua-article-33562020-03-18T19:52:14Z Distribution of the lower boundary functional of the step process of semi-Markov random walk with delaying screen at zero Распределение нижнего граничного функционала ступенчатого процесса полумарковского блуждания с задерживающим экраном в нуле Nasirova, T. I. Omarova, K. K. Насирова, Т. И. Омарова, К. К. Насирова, Т. И. Омарова, К. К. On the basis of a given sequence of independent identically distributed pairs of random variables, we construct the step process of semi-Markov random walk that is later delayed by a screen at zero. For this process, we obtain the Laplace transform of the distribution of the time of the first hit of the level zero. За заданою послідовністю незалежних однаково розподілених пар випадкових величин побудовано східчастий процес напівмарковського блукання, який потім затримується екраном у нулі. Для цього процесу знайдено перетворення Лапласа розподілу першого моменту досягнення рівня нуль. Institute of Mathematics, NAS of Ukraine 2007-07-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3356 Ukrains’kyi Matematychnyi Zhurnal; Vol. 59 No. 7 (2007); 912–919 Український математичний журнал; Том 59 № 7 (2007); 912–919 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3356/3456 https://umj.imath.kiev.ua/index.php/umj/article/view/3356/3457 Copyright (c) 2007 Nasirova T. I.; Omarova K. K. |
| spellingShingle | Nasirova, T. I. Omarova, K. K. Насирова, Т. И. Омарова, К. К. Насирова, Т. И. Омарова, К. К. Distribution of the lower boundary functional of the step process of semi-Markov random walk with delaying screen at zero |
| title | Distribution of the lower boundary functional of the step process of semi-Markov random walk with delaying screen at zero |
| title_alt | Распределение нижнего граничного функционала ступенчатого процесса полумарковского блуждания с задерживающим экраном в нуле |
| title_full | Distribution of the lower boundary functional of the step process of semi-Markov random walk with delaying screen at zero |
| title_fullStr | Distribution of the lower boundary functional of the step process of semi-Markov random walk with delaying screen at zero |
| title_full_unstemmed | Distribution of the lower boundary functional of the step process of semi-Markov random walk with delaying screen at zero |
| title_short | Distribution of the lower boundary functional of the step process of semi-Markov random walk with delaying screen at zero |
| title_sort | distribution of the lower boundary functional of the step process of semi-markov random walk with delaying screen at zero |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3356 |
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