Almost critical branching processes and limit theorems
We study almost critical branching processes with infinitely increasing immigration and prove functional limit theorems for these processes.
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| Datum: | 2009 |
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| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | Russisch Englisch |
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Institute of Mathematics, NAS of Ukraine
2009
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509023600640000 |
|---|---|
| author | Khusanbaev, Ya. M. Хусанбаев, Я. М. Хусанбаев, Я. М. |
| author_facet | Khusanbaev, Ya. M. Хусанбаев, Я. М. Хусанбаев, Я. М. |
| author_sort | Khusanbaev, Ya. M. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:43:07Z |
| description | We study almost critical branching processes with infinitely increasing immigration and prove functional limit theorems for these processes. |
| first_indexed | 2026-03-24T02:34:31Z |
| format | Article |
| fulltext |
UDK 519.21
Q. M. Xusanbaev
(Yn-t matematyky y ynform. texnolohyj AN Respublyky Uzbekystan, Taßkent)
POÇTY KRYTYÇESKYE VETVQWYESQ PROCESSÁ
Y PREDEL|NÁE TEOREMÁ
Almost critical branching processes with infinitely increasing immigration are considered and functional
limit theorems for these processes are proved.
Rozhlqnuto majΩe krytyçni rozhaluΩeni procesy z imihraci[g, wo neskinçenno zrosta[, i dlq
takyx procesiv dovedeno funkcional\ni hranyçni teoremy.
Pust\ dlq kaΩdoho n ∈N ξk j
n
,
( ){ , k, j ∈ }N y εk
n( ){ , k ∈ }N � nezavysym¥e sovo-
kupnosty nezavysym¥x neotrycatel\n¥x celoçyslenn¥x y odynakovo rasprede-
lenn¥x sluçajn¥x velyçyn. Posledovatel\nost\ sluçajn¥x velyçyn Xk
n( ) ,
k ≥ 0 , opredelym sledugwymy rekurrentn¥my sootnoßenyqmy:
X n
0
( ) = 0, Xk
n( ) =
j
X
k j
n
k
n
=
∑
1
1–
( )
,
( )ξ + εk
n( ), k = 1, 2, … .
Tak opredelenn¥e process¥ çasto voznykagt v zadaçax teoryy vetvqwyxsq
processov (sm., naprymer, [1] y pryvedennug v nej byblyohrafyg) y naz¥vagtsq
vetvqwymysq processamy s ymmyhracyej. Esly ynterpretyrovat\ velyçynu
ξk j
n
,
( ) kak çyslo potomkov j-j çastyc¥ nekotoroj populqcyy çastyc v (k – 1)-m
pokolenyy, a velyçynu εk
n( ) kak çyslo çastyc, ymmyhryrugwyx v populqcyg v
k-m pokolenyy, to velyçyna Xk
n( ) budet predstavlqt\ soboj obwee çyslo ças-
tyc populqcyy v k-m pokolenyy.
Opredelym sluçajn¥j stupençat¥j process X tn( ) , t ≥ 0 , poloΩyv X tn( ) =
= X nt
n
[ ]
( ) , t ≥ 0 , hde znak a[ ] oznaçaet celug çast\ çysla a. Qsno, çto traekto-
ryq processa Xn prynadleΩyt prostranstvu Skoroxoda D 0, ∞[ ). Vsgdu v
dal\nejßem budem predpolahat\, çto E ξ1 1
2
,
( )n( ) < ∞ y E ε1
2( )n( ) < ∞ dlq vsex
n ∈N . Vvedem sledugwye oboznaçenyq:
mn = Eξ1 1,
( )n , σn
2 = varξ1 1,
( )n , λn = Eε1
( )n , bn
2 = varε1
( )n .
V rabote [2] rassmotren sluçaj, kohda mn = 1 + α/n → 1, α ∈R (poçty
krytyçeskyj sluçaj) y σn
2 → 0 pry n → ∞ , a srednee znaçenye y dyspersyq
ymmyhracyy εk
n( ) pry n → ∞ sxodqtsq k koneçn¥m velyçynam. Dokazano, çto v
πtom sluçae process n–1 X tn( ) slabo sxodytsq v prostranstve D 0, ∞[ ) k deter-
mynyrovannomu processu, y poluçena funkcyonal\naq predel\naq teorema dlq
n– /1 2 X tn( )( – EX tn( )), t ≥ 0 . V rabote [3] yssledovan¥ dostatoçn¥e uslovyq
sxodymosty processov X tn( ) (s sootvetstvugwej normyrovkoj) k determyny-
rovannomu processu pry razlyçn¥x uslovyqx na povedenye velyçyn mn , λn ,
σn
2 y bn
2 .
V dannoj rabote rassmatryvaetsq poçty krytyçeskyj sluçaj, v kotorom
ymmyhracyq v populqcyg v srednem beskoneçno uvelyçyvaetsq (λn → ∞ pry
n → ∞), y yssleduetsq skorost\ rosta sluçajnoho processa Xn pry n → ∞ , a
takΩe yzuçaetsq asymptotyçeskoe povedenye Xn � EXn (sootvetstvugwym
© Q. M. XUSANBAEV, 2009
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 1 127
128 Q. M. XUSANBAEV
obrazom normyrovann¥x) pry n → ∞ . Vsgdu v dal\nejßem T > 0 , γ > 1 �
fyksyrovann¥e çysla, znak I A( ) budet oboznaçat\ yndykator sob¥tyq A.
Teorema 1. Pust\ v¥polnen¥ sledugwye uslovyq:
A) mn = 1 + αdn
–1, α ∈R, hde dn � posledovatel\nost\ poloΩytel\n¥x
çysel takaq, çto βn = n dn
–1 → β < ∞ pry n → ∞;
B) σn
2 → σ2 ≥ 0 pry n → ∞;
C) n1– γ λn → λ ≥ 0 pry n → ∞;
D) n1 2– γ bn
2 → 0 pry n → ∞.
Tohda
sup
( )
– ( )
0≤ ≤t T
nX t
n
tγ µ P → 0 pry n → ∞,
hde process µ qvlqetsq reßenyem uravnenyq
d tµ( ) = λ βαµ+( )( )t dt
s naçal\n¥m uslovyem µ( )0 = 0. Zdes\ znak P → oznaçaet sxodymost\ po
veroqtnosty.
Teorema 2. Pust\ v¥polnen¥ uslovyq A , B y C teorem¥ 1, pryçem σ2 >
> 0, y sledugwye uslovyq:
E) n–γ bn
2 → 0 pry n → ∞;
F) dlq lgboho θ > 0
E ξ ξ θ
γ
1 1
2
1 1
1
2
,
( )
,
( )– –n
n
n
nm I m n( ) >
+
→ 0 pry n → ∞.
Tohda pry n → ∞ ymeet mesto slabaq sxodymost\
X t X t
n
n n( ) – ( )
( )/
E
1 2+ γ , t T∈[ ]0, →
0
t
t se s dW s∫ βα ρ( – ) ( ) ( ) , t T∈[ ]0, ,
v prostranstve Skoroxoda D T0,[ ], hde W � standartn¥j vynerovskyj pro-
cess, ρ( )t = λσ2
0
t se ds∫ βα .
Teorema 3. Pust\ v¥polnen¥ uslovyq A y C teorem¥ 1. Pust\, krome to-
ho, v¥polnen¥ sledugwye uslovyq:
K) n σn
2 → σ2 > 0 pry n → ∞; L : n1– γ bn
2 → 0 pry n → ∞;
M) n nE ξ1 1,
( )( – mn)2
I nξ1 1,
( )( – mn > θ γn /2) → 0 pry n → ∞ dlq lgboho θ > 0.
Tohda pry n → ∞ ymeet mesto slabaq sxodymost\
X t X t
n
n n( ) – ( )
/
E
γ 2 , t T∈[ ]0, →
0
t
t se s dW s∫ βα ρ( – ) ( ) ( ) , t T∈[ ]0, ,
v prostranstve Skoroxoda D T0,[ ], hde funkcyq ρ ymeet tot Ωe vyd, çto
y v teoreme 2.
Teorema 4. Pust\ v¥polnen¥ uslovyq A y C teorem¥ 1. Pust\ n σn
2 → 0
y n1– γ bn
2 → b2 > 0 pry n → ∞. Esly, krome toho, dlq lgboho θ > 0
n I nn
n
n
n
1
1
2
1
2– ( ) ( ) /– –γ γε λ ε λ θE( ) >( ) → 0
pry n → ∞, to ymeet mesto slabaq sxodymost\
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 1
POÇTY KRYTYÇESKYE VETVQWYESQ PROCESSÁ Y PREDEL|NÁE TEOREMÁ 129
X t X t
n
n n( ) – ( )
/
E
γ 2 , t T∈[ ]0, → b e dW s
t
t s
0
∫ βα( – ) ( ), t T∈[ ]0, ,
v prostranstve Skoroxoda D T0,[ ].
Dokazatel\stvo teorem¥ 1. Netrudno vydet\, çto
Xk
n( ) = Xk
n
–
( )
1 + ( – ) –
( )m Xn k
n1 1 + λn + Mk
n( ) , (1)
hde Mk
n( ) =
j
X
k j
nk
n
=∑ (1
1–
( )
,
( )ξ – mn) + εk
n( ) – λn . Razdelqq poçlenno sootnoßenye (1)
na nγ y oboznaçaq ηnk = Xk
n( )
/ nγ , v sylu uslovyq A poluçaem
ηnk = ηnk –1 + β αη λγ
n nk nn
n–
–
1
1 1+( ) + 1
n
Mk
n
γ
( ). (2)
Oboznaçym çerez Fk
n( ) σ-alhebru, poroΩdennug velyçynamy X n
1
( ) , … , Xk
n( ) .
Netrudno vydet\, çto Mk
n( ) , k ≥ 1, obrazuet kvadratyçno-yntehryruemug mar-
tynhal-raznost\ otnosytel\no potoka Fk
n( ) , k ≥ 1. Tohda v sylu neravenstva
Duba dlq martynhalov pry lgbom ε > 0 ymeem
P 1
0 1n
M
t T k
nt
k
n
γ εsup ( )
≤ ≤ =
[ ]
∑ >
≤ 1
2 2
1
2
ε γn
M
k
nT
k
n
=
[ ]
∑ ( )E ( ) . (3)
V dal\nejßem nam ponadobytsq sledugwaq lemma, kotoraq qvlqetsq çast\g
lemm¥ 2.1 yz [2].
Lemma. Esly mn ≠ 1, to dlq vsex k ≥ 1
EXk
n( ) =
m
m
n
k
n
n
–
–
1
1
λ ,
y esly mn = 1, to EXk
n( ) = kλn . Krome toho,
E Mk
n
k
n( )
–
( )/( )( )2
1F = σn k
nX2
1–
( ) + bn
2 .
Pust\ αβ ≠ 0. Tohda, prymenqq lemmu y sootnoßenye mn
nt[ ] ∼ e n tβ α pry
n → ∞, sohlasno uslovyqm teorem¥ poluçaem
1
2
1
2
n
M
k
nt
k
n
γ
=
[ ]
∑ ( )E ( ) =
λ σ
γ
n n
n
n
nt
nn m
m
m
nt
2
2 1
1
1( – )
–
–
–
[ ]
[ ]
+ nt
n
bn
[ ]
2
2
γ ∼
∼ 1 1
1
2
2n
e
t
t
γ
βαλσ
αβ α
β–
–
–
+ n b tn
1 2 2– γ → 0 pry n → ∞. (4)
Esly α = 0, to, snova prymenqq lemmu, naxodym
1
2
1
2
n
M
k
nt
k
n
γ
=
[ ]
∑ ( )E ( ) =
nt nt
n
n n
[ ]( )[ ]– 1
2 2
2
γ λ σ + nt
n
bn
[ ]
2
2
γ → 0 (5)
pry n → ∞. Dalee, esly β = 0, to, uçyt¥vaq, çto
mn
nt[ ] = 1 + β αn t + 1
2
2β αn t( ) + o nβ2( ) ,
kak v (4), ymeem
1
2
1
2
n
M
k
nt
k
n
γ
=
[ ]
∑ ( )E ( ) ∼ 1
21
2 2
n
t
γ
λσ
– + n b tn
1 2 2– γ → 0
pry n → ∞. Otsgda y yz (3) � (5) sleduet, çto
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 1
130 Q. M. XUSANBAEV
1
0 1n
M
t T k
nt
k
n
γ sup ( )
≤ ≤ =
[ ]
∑ P → 0 pry n → ∞.
Tohda yz (2) y teorem¥ 3.1 [5] poluçaem
max –
1≤ ≤[ ]k nT
nk nkZη P → 0 (6)
pry n → ∞, hde velyçyn¥ Znk udovletvorqgt sootnoßenyg
Znk = Znk –1 + β α λγ
n nk nZ n
n–
–
1
1 1+( ) ⋅ .
Dalee, sohlasno uslovyqm A y C y v sylu teorem¥ 3.2 [5] ymeem
sup ( ) – ( )
0≤ ≤t T
nZ t tµ = max –
1≤ ≤
k nT
nkZ k
n
µ P → 0
pry n → ∞, hde Z tn( ) = Zn nt[ ] . Teper\ otsgda s uçetom (6) sleduet, çto
sup
( )
– ( )
0≤ ≤t T
nX t
n
tγ µ ≤ max –
1≤ ≤k nT
nk nkZη + max –
1≤ ≤
k nT
nkZ k
n
µ P → 0
pry n → ∞, çto y zaverßaet dokazatel\stvo teorem¥ 1.
Dokazatel\stvo teorem¥ 2. Sohlasno neravenstvu Kolmohorova dlq ne-
zavysym¥x sluçajn¥x velyçyn y v sylu uslovyq E dlq lgboho θ > 0 ymeem
P sup –( ) ( ) /
0 1
1 2
≤ ≤ =
[ ]
+∑ ( ) >
t T k
nt
k
n
n nε λ θ γ = nT
n
bn
[ ]
+θ γ2 1
2 ∼ θ γ– –2 2Tn bn → 0
pry n → ∞. Sledovatel\no,
sup –
– ( )
0
1
2
1≤ ≤
+
=
[ ]
∑ ( )
t T k
nt
k
n
nn
γ
ε λ P → 0 (7)
pry n → ∞. Teper\ dokaΩem, çto
n m
k
nt
j
X
k j
n
n
k
n
–
,
( )
–
( )
–
1
2
1 1
1+
=
[ ]
=
∑ ∑ ( )
γ
ξ → W T t( )( ) (8)
pry n → ∞ v prostranstve Skoroxoda D T0,[ ], hde T t( ) =
0
t
s ds∫ ρ( ) . Pust\
xn ∈ D T0,[ ] takye, çto sup ( )
0≤ ≤t T
nx t – µ( )t → 0 pry n → ∞. Snaçala pokaΩem,
çto
Φn nt x( , ) = n m
k
nt
j
n x
k
n
k j
n
n
n
–
–
,
( ) –
1
2
1 1
1
+
=
[ ]
=
∑ ∑ ( )
γ
γ
ξ → W T t( )( ) (9)
pry n → ∞ slabo v prostranstve Skoroxoda D T0,[ ]. Dejstvytel\no, pust\
ξ̃nk � nezavysym¥e sluçajn¥e velyçyn¥, ymegwye takoe raspredelenye, çto y
ξ1 1,
( )n – mn . Qsno, çto
n m
k
nt
j
n x
k
n
k j
n
n
n
–
–
,
( ) –
1
2
1 1
1
+
=
[ ]
=
∑ ∑ ( )
γ
γ
ξ =d.f
n
j
n x
k
n
n j
k
nt
n
–
–
,
˜
1
2
1
1
1+
=
=
[ ]∑
∑
γ
γ
ξ ,
hde znak =d.f
oznaçaet sovpadenye po raspredelenyg. Dalee netrudno vydet\,
çto
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 1
POÇTY KRYTYÇESKYE VETVQWYESQ PROCESSÁ Y PREDEL|NÁE TEOREMÁ 131
1 1 1
1
1n
n x
k
n
n x
k
nk
nt
n n+
=
[ ]
∑
γ
γ γ–
–
–
≤ nt
n
[ ]
+1 γ → 0 (10)
pry n → ∞. Poπtomu
k
nt
nn x
k
n=
[ ]
∑
1
1γ –
∼ n x s ds
nt n
n
1
0
+
[ ]
∫γ
/
( ) ∼ n s ds
t
1
0
+ ∫γ µ( ) (11)
pry n → ∞. Otsgda lehko poluçaem
varn
j
n x
k
n
n j
k
nt
n
–
–
,
˜
1
2
1
1
1+
=
=
[ ]∑
∑
γ
γ
ξ =
=
σ
γ
γn
k
nt
n
n
n x
k
n
2
1
1
1
+
=
[ ]
∑
–
→ σ µ2
0
t
s ds∫ ( ) =
0
t
s ds∫ρ( ) . (12)
Poskol\ku velyçyn¥ ˜
,ξn k odynakovo raspredelen¥ po k, uçyt¥vaq (11), ymeem
n I n
j
n x
k
n
n j n j
k
nt
n
– –
–
, ,
˜ ˜1
1
1
2
1
2
1
γ
γ
γ
ξ ξ θ
=
+=
[ ]∑
( ) >
∑ E ∼
∼
0
1 1
2
1 1
1
2
t
n
n
n
ns ds m I m n∫ ⋅ ( ) >
+
µ ξ ξ θ
γ
( ) – –,
( )
,
( )E → 0
pry n → ∞, sohlasno uslovyg F. Znaçyt, v¥polneno uslovye Lyndeberha dlq
˜
,ξn k . Tohda otsgda y yz (12), sohlasno funkcyonal\noj central\noj predel\-
noj teoreme (sm., naprymer, [5]), poluçaem (9). Dalee, prymenqq neravenstvo
Kolmohorova y uçyt¥vaq (10), dlq lgboho ε > 0 ymeem
P sup ( , ) – ( , )
0≤ ≤
>
t T
n n nt x tΦ Φ µ ε ≤
≤
σ
ε
µ
γ
γ
n
t T
n
nT n
n
x t t
2
2 1
0
[ ]
+
≤ ≤
sup ( ) – ( ) ∼
σ
ε
µ
2
2
0
T
x t t
t T
nsup ( ) – ( )
≤ ≤
→ 0
pry n → ∞. Otsgda dlq lgb¥x ε > 0 y δ > 0 netrudno poluçyt\, çto
P sup , – ( , )
0≤ ≤
>
t T
n
n
nt
X
n
tΦ Φγ µ ε ≤
σ
ε
δn T2
2 + P sup
( )
– ( )
0≤ ≤
>
t T
nX t
n
tγ µ δ .
V¥borom δ perv¥j çlen v poslednej summe moΩno sdelat\ skol\ uhodno
mal¥m, a vtoroj çlen, sohlasno teoreme 1, stremytsq k nulg. Teper\ otsgda y
yz (9), lemm¥ 8 yz [6, s. 19] poluçaem (8). V svog oçered\, yz (7) y (8) s uçetom
lemm¥ 8 yz [6, s. 19] poluçaem slabug sxodymost\
M tn( ) = n M
k
nT
k
n
– ( )
1
2
1
+
=
[ ]
∑
γ
→ M t( ) = W T t( )( ) (13)
pry n → ∞ v prostranstve Skoroxoda D T0,[ ]. Dalee, netrudno vydet\, çto
Xk
n( ) – EXk
n( ) = m X Xn k
n
k
n
–
( )
–
( )–1 1E( ) + Mk
n( ) .
Reßenye πtoho uravnenyq ymeet vyd
Xk
n( ) – EXk
n( ) =
j
k
n
k j
j
nm M
=
∑
1
– ( ).
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 1
132 Q. M. XUSANBAEV
Tohda
n X t X tn n
–
( ) – ( )
1
2
+
( )
γ
E = n m M
j
nt
n
nt j
j
n
– – ( )
1
2
1
+
=
[ ]
[ ]∑
γ
=
=
j
nt
n
nt j
n nm M
j
n
M
j
n=
[ ]
[ ]∑
1
1– –
–
=
0
nt n nt
n
s
ne dM s
n
[ ] [ ]
∫
/ –
( )
α
,
hde αn = n log mn → βα pry n → ∞. Poslednee sootnoßenye zapyßem v vyde
n X t X tn n
–
( ) – ( )
1
2
+
( )
γ
E = M nt
nn
[ ]
+ α
α
n
nt n nt
n
s
ne M s ds
n
0
[ ] [ ]
∫
/ –
( ) .
Otsgda y yz (13), prymenqq teoremu o neprer¥vnom otobraΩenyy [7], poluçaem
slabug sxodymost\ v D T0,[ ] pry n → ∞
n X t X tn n
–
( ) – ( )
1
2
+
( )
γ
E → Y t( ), (14)
hde
Y t( ) = M t( ) + βα βα
0
t
t se M s ds∫ ( – ) ( ) .
Prymenyv formulu Yto [8], netrudno ubedyt\sq v tom, çto process Y udov-
letvorqet lynejnomu stoxastyçeskomu dyfferencyal\nomu uravnenyg
dY t( ) = βαY t dt( ) + dM t( ) .
Po formule zamen¥ peremenn¥x
dM t( ) = ρ( ) ( )t dW t .
Znaçyt, process Y udovletvorqet uravnenyg
dY t( ) = βαY t dt( ) + ρ( ) ( )t dW t .
Yzvestno [8], çto reßenye posledneho uravnenyq ymeet vyd
Y t( ) =
0
t
t se s dW s∫ βα ρ( – ) ( ) ( ) .
Teper\ otsgda s uçetom (14) poluçaem utverΩdenye teorem¥ 2.
Dokazatel\stvo teorem¥ 3 analohyçno dokazatel\stvu teorem¥ 2, poπtomu
m¥ eho opuskaem.
Dokazatel\stvo teorem¥ 4. Netrudno vydet\, çto v¥polnen¥ vse uslo-
vyq teorem¥ 1.
RassuΩdaq, kak pry poluçenyy (4) y (5), netrudno ubedyt\sq v tom, çto
varn m
k
nt
j
X
k j
n
n
k
n
– /
,
( )
–
( )
–γ ξ2
1 1
1
=
[ ]
=
∑ ∑ ( ) → 0 pry n → ∞.
Tohda, prymenqq neravenstvo Duba dlq martynhalov, poluçaem
sup –– /
,
( )
–
( )
0
2
1 1
1
≤ ≤ =
[ ]
=
∑ ∑ ( )
t T k
nt
j
X
k j
n
nn m
k
n
γ ξ P → 0 pry n → ∞. (15)
Dalee, uçyt¥vaq nezavysymost\ y odynakovug raspredelennost\ velyçyn εk
n( ),
naxodym
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 1
POÇTY KRYTYÇESKYE VETVQWYESQ PROCESSÁ Y PREDEL|NÁE TEOREMÁ 133
varn
k
nt
k
n
n
– / ( ) –γ ε λ2
1=
[ ]
∑ ( ) = nt
n
bn
[ ]
γ
2 → tb2 pry n → ∞.
Teper\ proverym uslovye Lyndeberha dlq εk
n( ). Ymeem
n I n
k
nt
k
n
n k
n
n
– / ( ) ( ) /– –γ γε λ ε λ θ2
1
2 2
=
[ ]
∑ ( ) >( )E =
= tn I nn
n
n
n
1
1
2
1
2– γ γε λ ε λ θE ( ) ( ) /– –( ) >( ) → 0
pry n → ∞ sohlasno uslovyg teorem¥. Znaçyt, yz funkcyonal\noj central\-
noj predel\noj teorem¥ dlq nezavysym¥x sluçajn¥x velyçyn [5] sleduet, çto
n
k
nt
k
n
n
– / ( ) –γ ε λ2
1=
[ ]
∑ ( ) → W b t2( ), n → ∞,
slabo v prostranstve D T0,[ ]. Otsgda y yz (15) sohlasno lemme 8 [6, s. 19] po-
luçaem
n M
k
nt
k
n– / ( )γ 2
1=
[ ]
∑ → W b t2( ), n → ∞,
slabo v prostranstve D T0,[ ].
Netrudno vydet\, çto
Znk = Znk –1 + β αn nkZ
n–1
1⋅ + n Mk
n−γ / ( )2 ,
hde Znk = n−γ /2 Xk
n( )( – EXk
n( )) . Prymenqq teoremu 3.5 [5], poluçaem, çto pro-
cess Znt = Zn nt[ ] slabo sxodytsq v D T0,[ ] k reßenyg lynejnoho stoxastyçes-
koho dyfferencyal\noho uravnenyq
dZ t( ) = βαZ t dt( ) + bdW t( ) .
Yz [8, s. 222] yzvestno, çto poslednee uravnenye ymeet reßenye vyda
Z t( ) = b e dW st s
t
βα( – ) ( )
0
∫ .
Teorema 4 dokazana.
1. Haccou P., Jagers P., Vatutin V. A. Branching processes variation, growth and extinction of
populations. – Cambridge Univ. Press, 2005. – 317 p.
2. Ispany M., Pap G., Van Zuijlen M. C. A. Fluctuation limit of branching processes with immigration
and estimation of the means // Adv. Appl. Probab. – 2005. – 37. – P. 523 – 538.
3. Xusanbaev Q. M. Ob asymptotyçeskyx svojstvax processa Hal\tona � Vatsona s ymmyhracyej
// Uzb. mat. Ωurn. � 2007. � # 2. � S. 119 � 127.
4. Xusanbaev Q. M. O fluktuacyy vetvqwehosq processa Hal\tona � Vatsona s ymmyhracyej //
Tam Ωe. � 2008. � # 1.
5. Anysymov V. V., Lebedev E. A. Stoxastyçeskye sety obsluΩyvanyq. Markovskye modely. �
Kyev: Lybid\, 1992. � 208 s.
6. Syl\vestrov D. S. Predel\n¥e teorem¥ dlq sloΩn¥x sluçajn¥x funkcyj. � Kyev: Vywa
ßk., 1974. � 318 s.
7. Byllynhsly P. Sxodymost\ veroqtnostn¥x mer. � M.: Nauka, 1977. � 352 s.
8. Vatanabπ S., Ykπda N. Stoxastyçeskye dyfferencyal\n¥e uravnenyq y dyffuzyonn¥e
process¥. � M.: Nauka, 1986. � 448 s.
Poluçeno 29.01.08
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 1
|
| id | umjimathkievua-article-3008 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:34:31Z |
| publishDate | 2009 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/55/c9b0b208a39a63b35c4a45a8ddb6b655.pdf |
| spelling | umjimathkievua-article-30082020-03-18T19:43:07Z Almost critical branching processes and limit theorems Почти критические ветвящиеся процессы и предельные теоремы Khusanbaev, Ya. M. Хусанбаев, Я. М. Хусанбаев, Я. М. We study almost critical branching processes with infinitely increasing immigration and prove functional limit theorems for these processes. Розглянуто майже критичні розгалужені процеси з іміграцією, що нескінченно зростає, i для таких процесів доведено функціональні граничні теореми. Institute of Mathematics, NAS of Ukraine 2009-01-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3008 Ukrains’kyi Matematychnyi Zhurnal; Vol. 61 No. 1 (2009); 127-133 Український математичний журнал; Том 61 № 1 (2009); 127-133 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3008/2765 https://umj.imath.kiev.ua/index.php/umj/article/view/3008/2766 Copyright (c) 2009 Khusanbaev Ya. M. |
| spellingShingle | Khusanbaev, Ya. M. Хусанбаев, Я. М. Хусанбаев, Я. М. Almost critical branching processes and limit theorems |
| title | Almost critical branching processes and limit theorems |
| title_alt | Почти критические ветвящиеся процессы и предельные теоремы |
| title_full | Almost critical branching processes and limit theorems |
| title_fullStr | Almost critical branching processes and limit theorems |
| title_full_unstemmed | Almost critical branching processes and limit theorems |
| title_short | Almost critical branching processes and limit theorems |
| title_sort | almost critical branching processes and limit theorems |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3008 |
| work_keys_str_mv | AT khusanbaevyam almostcriticalbranchingprocessesandlimittheorems AT husanbaevâm almostcriticalbranchingprocessesandlimittheorems AT husanbaevâm almostcriticalbranchingprocessesandlimittheorems AT khusanbaevyam počtikritičeskievetvâŝiesâprocessyipredelʹnyeteoremy AT husanbaevâm počtikritičeskievetvâŝiesâprocessyipredelʹnyeteoremy AT husanbaevâm počtikritičeskievetvâŝiesâprocessyipredelʹnyeteoremy |