A Differential Analog of the Main Lemma of the Theory of Markov Branching Processes and Its Applications
We obtain a differential analog of the main lemma in the theory of Markov branding processes $\mu(t),\quad t \geq 0$, of continuous time. We show that the results obtained can be applied in the proofs of limit theorems in the theory of branching processes by the well-known Stein - Tikhomirov method...
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| Date: | 2005 |
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| Main Authors: | , |
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| Language: | Russian English |
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Institute of Mathematics, NAS of Ukraine
2005
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509710046724096 |
|---|---|
| author | Imomov, A. A. Имомов, А. А. Имомов, А. А. |
| author_facet | Imomov, A. A. Имомов, А. А. Имомов, А. А. |
| author_sort | Imomov, A. A. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:59:22Z |
| description | We obtain a differential analog of the main lemma in the theory of Markov branding processes $\mu(t),\quad t \geq 0$, of continuous time.
We show that the results obtained can be applied in the proofs of limit theorems in the theory of branching processes by the well-known Stein - Tikhomirov method.
In contrast to the classical condition of nondegeneracy of the branching process $\{\mu(t) > 0\}$,
we consider the condition of nondegeneracy of the process in distant $\{\mu(\infty) > 0\}$ and justify in terms of generating functions.
Under this condition, we study the asymptotic behavior of trajectory of the considered process. |
| first_indexed | 2026-03-24T02:45:25Z |
| format | Article |
| fulltext |
UDK 519.21
A. A. Ymomov (Karßyn. un-t, Uzbekystan)
DYFFERENCYAL|NÁJ ANALOH OSNOVNOJ LEMMÁ
TEORYY MARKOVSKYX VETVQWYXSQ PROCESSOV
Y EHO PRYMENENYQ
We obtain a differential analog of the main lemma in the theory of Markov brancling processes µ ( t ),
t ≥ 0, of continuous time. We show that the results obtained can be applied in the proofs of limit
theorems in the theory of branching processes by the well-known Stein – Tikhomirov method. In
contrast to the classical condition of nondegeneracy of the branching process { µ ( t ) > 0 }, we consider
the condition of nondegeneracy of the process in distant { µ ( ∞ ) > 0 } and justify in terms of generating
functions. Under this condition, we study the asymptotic behavior of trajectory of the considered
process.
Otrymano dyferencial\nyj analoh osnovno] lemy teori] markovs\kyx hillqstyx procesiv µ ( t ),
t ≥ 0, neperervnoho çasu. Pokazano moΩlyvist\ zastosuvannq otrymanyx rezul\tativ pry dove-
denni hranyçnyx teorem teori] hillqstyx procesiv vidomym metodom Stejna – Tyxomyrova. Krim
c\oho, na vidminu vid klasyçno] umovy nevyrodΩennq hillqstoho procesu { µ ( t ) > 0 } rozhlqnu-
to i ob©runtovano movog tvirnyx funkcij umovu nevyrodΩennq procesu v dalekomu majbut-
n\omu { µ ( ∞ ) > 0 }. Za ci[] umovy vyvçeno asymptotyçnu povedinku tra[ktori] rozhlqduvanoho
procesu.
Rassmotrym markovskyj vetvqwyjsq sluçajn¥j process µ ( t ), t ≥ 0, neprer¥v-
noho vremeny. Pust\ P { µ ( 0 ) = 1 } = 1 y
Pi j ( t ) = P { µ ( t + τ ) = j | µ ( τ ) = i }, τ ≥ 0.
Yz uslovyq vetvlenyq sleduet, çto dlq yzuçenyq yzmenenyq sostoqnyj processa
µ ( t ) dostatoçno zadat\ veroqtnosty P1j ( t ), dlq kotor¥x predpolahaetsq v¥-
polnenye uslovyq
P1j ( t ) = δ1j + aj t + o ( t ), t → 0,
hde δ1j — znak Kronekera, plotnosty veroqtnostej perexoda aj ≥ 0 pry j ≠ 1 y
a1 < 0, a takΩe
aj
j≥
∑
0
= 0.
Vvedem v rassmotrenye proyzvodqwye funkcyy (p. f.)
Φ ( t; x ) = P1
0
j
j
j
t x( )
≥
∑ , f ( x ) = a xj
j
j≥
∑
0
, | x | ≤ 1.
∏ty p. f. svqzan¥ dyfferencyal\n¥m uravnenyem
∂ ( )
∂
Φ t x
t
;
= f ( Φ ( t; x ) ) (1)
s naçal\n¥m uslovyem Φ ( 0; x ) = x, kotoroe sootvetstvuet obratnomu uravne-
nyg Kolmohorova dlq perexodn¥x veroqtnostej markovskyx vetvqwyxsq pro-
cessov neprer¥vnoho vremeny (sm. [1]).
Vvedem sledugwye oboznaçenyq:
a = f ′ ( 1 ), b = f ′′ ( 1 ).
Budem rassmatryvat\ sluçay, kohda a < 0, a = 0; v πtyx sluçaqx process
µ ( t ) naz¥vaetsq dokrytyçeskym y krytyçeskym sootvetstvenno. Yzvestno, çto
© A. A. YMOMOV, 2005
258 ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 2
DYFFERENCYAL|NÁJ ANALOH OSNOVNOJ LEMMÁ … 259
v yssledovanyqx predel\n¥x svojstv processa µ ( t ) v rassmatryvaem¥x sluçaqx
osoboe znaçenye ymegt asymptotyçeskye svojstva funkcyy R ( t; x ) = 1 – Φ ( t; x ).
V çastnosty, dlq veroqtnosty prodolΩenyq processa Q ( t ) ≡ R ( t; 0 ) = P { µ ( t ) >
> 0 } spravedlyv¥ sledugwye utverΩdenyq (sm. [1]):
1. Pust\ a < 0. Dlq toho çtob¥ ymela mesto formula
Q ( t ) = K ea
t
( 1 + o ( 1 ) ), t → ∞, (2)
neobxodymo y dostatoçno, çtob¥ sxodylsq yntehral
au f u
uf u
du
+ ( − )
( − )∫ 1
1
0
1
. (3)
Yntehral (3) raven – ln K.
2. Pust\ a = 0, b < ∞. Tohda
Q ( t ) = 2
bt
( 1 + o ( 1 ) ), t → ∞. (4)
Dlq funkcyy R ( t; x ) spravedlyva sledugwaq lemma.
Lemma [1]. 1. Pust\ a < 0. Tohda pry koneçnosty yntehrala (3)
R ( t; x ) = Ke a du
f u
at
x
exp
( )
∫
0
( 1 + o ( 1 ) ), t → ∞. (5)
Zdes\ y dalee postoqnnaq K ta Ωe, çto y v razloΩenyy (2).
2. Pust\ a = 0, b < ∞. Tohda
R ( t; x ) =
1
1
2
1
−
+ ( − )
x
bt x
( 1 + o ( 1 ) ), t → ∞. (6)
Poslednqq lemma, po svoej znaçymosty, naz¥vaetsq osnovnoj lemmoj teoryy
markovskyx vetvqwyxsq processov neprer¥vnoho vremeny.
V nastoqwej rabote yzuçaetsq asymptotyçeskoe povedenye funkcyy
∂ ( )
∂
R t x
x
;
, t. e. dokaz¥vaetsq dyfferencyal\n¥j analoh osnovnoj lemm¥ y pred-
lahagtsq nekotor¥e eho pryloΩenyq.
Lemma A. 1. Pust\ a < 0. Tohda pry koneçnosty yntehrala (3)
∂ ( )
∂
=
( ) ( )
∫R t x
x
aKe
f x
a du
f u
at x
;
exp
0
( 1 + o ( 1 ) ), t → ∞. (7)
2. Pust\ a = 0, b < ∞. Tohda
∂ ( )
∂
= − ( − )
( ) + ( − )
R t x
x
b x
f x
bt
x
; 1
2 1
2
1
2
2 ( 1 + o ( 1 ) ), t → ∞. (8)
Dokazatel\stvo. Perepyßem dyfferencyal\noe uravnenye (1) v vyde
∂ ( )
∂
R t x
t
;
= – f ( 1 – R ( t; x ) ) (9)
s naçal\n¥m uslovyem R ( 0; x ) = 1 – x.
Netrudno ubedyt\sq v tom, çto uravnenye (9) s uçetom naçal\noho uslovyq
dopuskaet neqvnoe reßenye v vyde
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 2
260 A. A. YMOMOV
du
f u
R t x
x
( − )
( )
−
∫ 1
1
;
= t. (10)
Yspol\zuq formulu dyfferencyrovanyq yntehralov s peremenn¥my prede-
lamy, yz (10) poluçaem
∂ ( )
∂
= − − ( )
( )
( )R t x
x
f R t x
f x
; ;1
. (11)
Dalee, vospol\zuemsq razloΩenyem Tejlora dlq p. f. f ( 1 – R ( t; x ) ) v raven-
stve (11). Tohda v sylu toho, çto R ( t; x ) → 0, t → ∞, pry a < 0
∂ ( )
∂
=
( )
( )R t x
x
a
f x
R t x
;
; ( 1 + o ( 1 ) ), t → ∞, (12)
a pry a = 0, b < ∞
∂ ( )
∂
= − ( )
( )
R t x
x
bR t x
f x
; ;2
2
( 1 + o ( 1 ) ), t → ∞. (13)
Podstavlqq ravenstvo (5) v (12), poluçaem (7), a podstavlqq (6) v (13) — (8).
Lemma A dokazana.
Yzvestno, çto ′ ( )Φx t; 0 = P11 ( t ) predstavlqet soboj veroqtnost\ vozvrata
processa k naçal\nomu sostoqnyg { µ ( 0 ) = 1 } za vremq t. Polahaq v lemme A
x = 0, neposredstvenno poluçaem sledugwye dve lokal\n¥e predel\n¥e teo-
rem¥.
Teorema 1. Pust\ a < 0. Tohda pry koneçnosty yntehrala (3)
e t
a K
a
a t P11
0
( ) = ( 1 + o ( 1 ) ), t → ∞.
Teorema 2. Pust\ a = 0, b < ∞. Tohda
t2 P11 ( t ) =
2 1
0ba
O
t
+
, t → ∞.
Dyfferencyal\n¥j analoh osnovnoj lemm¥ budet polezen takΩe pry pry-
menenyy metoda Stejna – Tyxomyrova (S-T) pry dokazatel\stve predel\n¥x
teorem teoryy vetvqwyxsq sluçajn¥x processov. Yzvestno, çto metod S-T
osnov¥vaetsq na dyfferencyal\nom uravnenyy dlq sootvetstvugwyx xarakte-
rystyçeskyx funkcyj yly preobrazovanyy Laplasa (a takΩe dlq p. f.). V slu-
çae normal\noj approksymacyy πtot metod vperv¥e yspol\zovan v rabote [2]
(sm. takΩe [3]).
Dalee v soçetanyy s metodom S-T nam udobno yspol\zovat\ sledugwyj va-
ryant lemm¥ A.
Lemma B. 1. Pust\ a < 0. Tohda
∂ ( )
∂
=
( )
( )R t x
x
a
f x
R t x
;
; ( 1 + o ( 1 ) ), t → ∞. (14)
2. Pust\ a = 0, b < ∞. Tohda pry x → 1
∂ ( )
∂
= − − ( )
( )
R t x
x
R t x
Q t
; ;
1
2
( 1 + o ( 1 ) ), t → ∞. (15)
Prodemonstryruem prymenenye metoda S-T na prymere dvux klassyçeskyx
predel\n¥x teorem.
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 2
DYFFERENCYAL|NÁJ ANALOH OSNOVNOJ LEMMÁ … 261
Teorema 3 [1]. Pust\ a < 0. Tohda suwestvugt predel¥
lim
t→∞
P { µ ( t ) = k | µ ( t ) > 0 } = P*
k .
P. f. F ( x ) = P*
k
k
k
x
≥
∑
1
ymeet vyd
F ( x ) = 1
0
−
( )∫exp a
du
f u
x
.
Dokazatel\stvo. Uslovnug p. f. F ( t; x ) ≡ E { xµ
(
t
)
| µ ( t ) > 0 } zapyßem v
vyde
F ( t; x ) = 1 –
R t x
Q t
( )
( )
;
. (16)
Dyfferencyruq (16) s yspol\zovanyem sootnoßenyq (14), ymeem
∂ ( )
∂
= −
( )
( )
( )
F t x
x
a
f x
R t x
Q t
; ;
( 1 + o ( 1 ) ) =
=
a
f x( )
[ F ( t; x ) – 1 ] ( 1 + o ( 1 ) ), t → ∞. (17)
Poskol\ku p. f. F ( x ) udovletvorqet uravnenyg
∂ ( )
∂
=
( )
F x
x
a
f x
[ F ( x ) – 1 ]
s uslovyem F ( 0 ) = 0, uravnenye (17) svydetel\stvuet o sxodymosty p. f. F ( t; x )
k p. f. F ( x ).
Teorema 3 dokazana.
Teorema 4 [1]. Pust\ a = 0, b < ∞. Tohda dlq lgboho u > 0
lim P
Et
ut
t t
u t e
→∞
−( )
( ) ( ) >
≤ ( ) >
= −
( )
µ
µ µ
µ
0
0 1 .
Dokazatel\stvo. Poçty oçevydno, çto uslovnoe matematyçeskoe oΩyda-
nye E { µ ( t ) | µ ( t ) > 0 } = 1 / Q ( t ). Poskol\ku preobrazovanye Laplasa pokaza-
tel\noho zakona est\ reßenye dyfferencyal\noho uravnenyq ψ′ ( θ ) + ψ2
( θ ) = 0
s naçal\n¥m uslovyem ψ ( 0 ) = 1, nam neobxodymo dokazat\, çto preobrazovanye
Laplasa
ψt ( θ ) ≡ E( )− ( ) ( ) ( ) >e tQ t tθ µ µ 0 , θ ≥ 0,
asymptotyçesky udovletvorqet dyfferencyal\nomu uravnenyg
∂ ( )
∂
= − ( )ψ θ
θ
ψ θt
t
2
( 1 + o ( 1 ) ), t → ∞, (18)
s naçal\n¥m uslovyem ψt ( 0 ) = 1.
Polahaq θt = exp { – θ Q ( t ) }, preobrazovanyq Laplasa ψt ( θ ) zapys¥vaem v
vyde
ψt ( θ ) = 1 − ( )
( )
R t
Q t
t; θ
. (19)
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 2
262 A. A. YMOMOV
V sylu (4) θt → 1, t → ∞ . Tohda, uçyt¥vaq (15), neposredstvenn¥m dyfferen-
cyrovanyem yz (19) poluçaem
∂ ( )
∂
= ∂ ( )
∂
ψ θ
θ
θ
θ
θt t
t
R t;
=
= − − ( )
( )
+ ( ) = − ( ) + ( )( ) ( )1 1 1 1 1
2
2R t
Q t
o ot
t t
; θ θ ψ θ , t → ∞,
t. e. uravnenye (18), çto y dokaz¥vaet teoremu 4.
Obratymsq teper\ k uslovyg nev¥roΩdenyq processa v dalekom buduwem
{ µ ( ∞ ) > 0 }. Vvedem uslovnug p. f.
M ( t; x ) ≡ E lim E( ) ( )( )
→∞
( )(∞) > = ( + ) >x x tt tµ
τ
µµ µ τ0 0 .
Yz opredelenyq M ( t; x ) (dlq dyskretnoho sluçaq sm. [4]) lehko poluçaem v¥-
polnenye ravenstva
M ( t; x ) = − ∂ ( )
∂
−e x
R t x
x
at ;
. (20)
Snaçala v¥çyslym E { µ ( t ) | µ ( ∞) > 0 }. Dlq πtoho, loharyfmyruq uravnenye
(11) s uçetom (1), ymeem
ln
;
ln
; ln ;− ∂ ( )
∂
= ( )
( )
= ∂ ( )
∂
( ) ( )∫R t x
x
f t x
f x
f x
d
tΦ Φ τ
τ
τ
0
=
=
′ ( )
( )
∂ ( )
∂
= ′ ( )( )
( )
( )∫ ∫f x
f x
x
d f x d
t tΦ
Φ
Φ Φτ
τ
τ
τ
τ τ τ;
;
;
;
0 0
,
otkuda poluçaem
− ∂ ( )
∂
= ′ ( )( )∫R t x
x
f x d
t
;
exp ;Φ τ τ
0
. (21)
Podstavym ravenstvo (21) v (20):
M ( t; x ) = e x f x dat
t
− ′ ( )( )∫exp ;Φ τ τ
0
. (22)
Ytak, m¥ poluçyly qvn¥j vyd uslovnoj p. f. M ( t; x ). Oçevydno, çto pry ta-
kom neklassyçeskom uslovyy yzuçenyq processa µ ( t ), t ≥ 0, udaçno yspol\zu-
etsq predstavlenye funkcyy
∂ ( )
∂
R t x
x
;
.
Prqm¥m dyfferencyrovanyem v toçke x = 1 yz (22) naxodym
E ( t ) ≡ E { µ ( t ) | µ ( ∞) > 0 } = 1
1
0
1 0
+ ( − ) <
+ =
b e
a
a
bt a
at
, ,
, .
(23)
Yz (23) neposredstvenno poluçaem sledugwug teoremu.
Teorema 5. Pust\ a < 0, b < ∞. Tohda uslovnoe matematyçeskoe oΩydanye
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 2
DYFFERENCYAL|NÁJ ANALOH OSNOVNOJ LEMMÁ … 263
E ( t ) → 1 + b
a
, t → ∞.
DokaΩem teper\ predel\n¥e teorem¥, analohyçn¥e teoremam 3, 4, pry us-
lovyy nev¥roΩdenyq processa v dalekom buduwem { µ ( ∞) > 0 }.
Teorema 6. Pust\ a < 0. Tohda suwestvugt predel¥
lim
t→∞
P { µ ( t ) = n| µ ( ∞) > 0 } = Pn .
P. f. M ( x ) = Pn
n
n
x
≥
∑
1
ymeet vyd
M ( x ) = a K
x
f x
a
du
f u
x
( ) ( )∫exp
0
.
Dokazatel\stvo. Yz (14), (20) ymeem
M ( t; x ) = −
( )
( )−ae
x
f x
R t xat ; . (24)
Vvydu (2), (16) ravenstvo (24) preobrazuem k vydu
M ( t; x ) = aK x
f x
F t x o
( )
( ) − +[ ]( ); ( )1 1 1 , t → ∞. (25)
V svog oçered\, sohlasno teoreme 3 F ( t; x ) → F ( x ), t → ∞. Tohda yz (25) ymeem
M ( t; x ) → aK
x
f x
F x
( )
( ) −[ ]1 , t → ∞,
çto ravnosyl\no utverΩdenyg teorem¥ 6.
Lehko ubedyt\sq v tom, çto M ( 1 ) = 1, a takΩe
M ′ ( 1 ) = 1 + b
a
.
Teorema 7. Pust\ a = 0, b < ∞. Tohda dlq lgboho u > 0
lim P
Et
u ut
t
u e ue
→∞
− −( )
( ) (∞) >
≤ (∞) >
= − −
( )
µ
µ µ
µ
0
0 1 22 2
.
Dokazatel\stvo. Vvedem v rassmotrenye preobrazovanye Laplasa
ϕt ( θ ) ≡ M t e E t( )− ( ); /θ , θ ≥ 0.
V sylu (23) e E t− ( )θ / → 1, t → ∞ , y pry x = e E t− ( )θ /
ymeet mesto razloΩenye
(15). Tak, pry a = 0 yz (15), (20) ymeem
ϕt ( θ ) = e
R t e
Q t
oE t
E t
− ( )
− ( )
− ( )
( )
+ ( )( )θ
θ
/
/;
1 1 1
2
, t → ∞. (26)
Poskol\ku Q ( t ) ∼ 2 / E ( t ), t → ∞ , s uçetom (19) ravenstvo (26) preobrazuetsq k
vydu
ϕt ( θ ) = ψ θ
t o2
2
1 1
+ ( )( ), t → ∞. (27)
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 2
264 A. A. YMOMOV
Yz teorem¥ 4 yzvestno, çto preobrazovanye Laplasa ψ t ( θ ) asymptotyçesky
udovletvorqet uravnenyg (18), reßenyem kotoroho est\ funkcyq 1 / ( 1 + θ ).
Takym obrazom, ravenstvo (27) oznaçaet, çto
ϕt ( θ ) → 1
1
2
2
+
θ
, t → ∞,
y πto sootvetstvuet zakonu ∏rlanha 1-ho porqdka, poluçaemomu yz kompozycyj
dvux pokazatel\n¥x zakonov s odynakov¥m parametrom λ = 2:
p ( x ) = 4 2xe x− , x > 0.
Poslednee ravnosyl\no utverΩdenyg teorem¥ 7.
Analoh teorem¥ 7 dlq processa Hal\tona – Vatsona dyskretnoho vremeny
dokazan v rabote [4].
1. Sevast\qnov B. A. Vetvqwyesq process¥. – M.: Nauka, 1971. – 436 s.
2. Tyxomyrov A. N. O skorosty sxodymosty v central\noj predel\noj teoreme dlq slabo
zavysym¥x velyçyn // Teoryq veroqtnostej y ee prymenenyq. – 1980. – 25, # 4. – S. 800 – 818.
3. Formanov Sh. K. On non-classical variant of central limit theorem // Abstrs Commun. 7th Vilnius
Int. Conf. Probab. Theory and Math. Statist. (1998, August 12 – 18, Vilnius, Lithuania). – P. 208.
4. Ymomov A. A. Ob odnom vyde uslovyq nev¥roΩdenyq vetvqwyxsq processov // Uzbek. mat.
Ωurn. – 2001. – # 2. – S. 46 – 51.
Poluçeno 20.11.2003
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 2
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| id | umjimathkievua-article-3593 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:45:25Z |
| publishDate | 2005 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/7e/35508e67f5d5a5deb61583d3cfdb797e.pdf |
| spelling | umjimathkievua-article-35932020-03-18T19:59:22Z A Differential Analog of the Main Lemma of the Theory of Markov Branching Processes and Its Applications Дифференциальный аналог основной леммы теории марковских ветвящихся процессов и его применения Imomov, A. A. Имомов, А. А. Имомов, А. А. We obtain a differential analog of the main lemma in the theory of Markov branding processes $\mu(t),\quad t \geq 0$, of continuous time. We show that the results obtained can be applied in the proofs of limit theorems in the theory of branching processes by the well-known Stein - Tikhomirov method. In contrast to the classical condition of nondegeneracy of the branching process $\{\mu(t) > 0\}$, we consider the condition of nondegeneracy of the process in distant $\{\mu(\infty) > 0\}$ and justify in terms of generating functions. Under this condition, we study the asymptotic behavior of trajectory of the considered process. Отримано диференціальний аналог основної леми теорії марковських гіллястих процесів $\mu(t),\quad t \geq 0$, неперервного часу. Показано можливість застосування отриманих результатів при доведенні граничних теорем теорії гіллястих процесів відомим методом Стейна - Тихомирова. Крім цього, на відміну від класичної умови невиродження гіллястого процесу $\{\mu(t) > 0\}$ розглянуто і обґрунтовано мовою твірних функцій умову невиродження процесу в далекому майбутньому $\{\mu(\infty) > 0\}$. За цієї умови вивчено асимптотичну поведінку траєкторії розглядуваного процесу. Institute of Mathematics, NAS of Ukraine 2005-02-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3593 Ukrains’kyi Matematychnyi Zhurnal; Vol. 57 No. 2 (2005); 258–264 Український математичний журнал; Том 57 № 2 (2005); 258–264 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3593/3918 https://umj.imath.kiev.ua/index.php/umj/article/view/3593/3919 Copyright (c) 2005 Imomov A. A. |
| spellingShingle | Imomov, A. A. Имомов, А. А. Имомов, А. А. A Differential Analog of the Main Lemma of the Theory of Markov Branching Processes and Its Applications |
| title | A Differential Analog of the Main Lemma of the Theory of Markov Branching Processes and Its Applications |
| title_alt | Дифференциальный аналог основной леммы теории марковских ветвящихся процессов и его применения |
| title_full | A Differential Analog of the Main Lemma of the Theory of Markov Branching Processes and Its Applications |
| title_fullStr | A Differential Analog of the Main Lemma of the Theory of Markov Branching Processes and Its Applications |
| title_full_unstemmed | A Differential Analog of the Main Lemma of the Theory of Markov Branching Processes and Its Applications |
| title_short | A Differential Analog of the Main Lemma of the Theory of Markov Branching Processes and Its Applications |
| title_sort | differential analog of the main lemma of the theory of markov branching processes and its applications |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3593 |
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