Weak convergence of integral functionals of random walks weakly convergent to fractional Brownian motion
We consider a random walk that converges weakly to a fractional Brownian motion with Hurst index H > 1/2. We construct an integral-type functional of this random walk and prove that it converges weakly to an integral constructed on the basis of the fractional Brownian motion.
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| Date: | 2007 |
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| Main Authors: | , , , |
| Format: | Article |
| Language: | Ukrainian 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/3368 |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509446861488128 |
|---|---|
| author | Mishura, Yu. S. Rode, S. H. Мішура, Ю. С. Роде, С. Г. |
| author_facet | Mishura, Yu. S. Rode, S. H. Мішура, Ю. С. Роде, С. Г. |
| author_sort | Mishura, Yu. S. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:52:34Z |
| description | We consider a random walk that converges weakly to a fractional Brownian motion with Hurst index H > 1/2. We construct an integral-type functional of this random walk and prove that it converges weakly to an integral constructed on the basis of the fractional Brownian motion. |
| first_indexed | 2026-03-24T02:41:14Z |
| format | Article |
| fulltext |
UDK 519.21
G. S. Mißura, S. H. Rode (Ky]v. nac. un-t im. T. Íevçenka)
SLABKA ZBIÛNIST| INTEHRAL|NYX FUNKCIONALIV
VID VYPADKOVYX BLUKAN|, WO SLABKO ZBIHAGT|SQ
DO DROBOVOHO BROUNIVS|KOHO RUXU
We consider a random walk that weakly converges to a fractional Brownian motion with the Hurst index
H > 1 / 2. We construct an integral-type functional of this random walk and prove that it weakly
converges to an integral constructed on the basis of the fractional Brownian motion.
Rassmotreno sluçajnoe bluΩdanye, slabo sxodqweesq k drobnomu brounovskomu dvyΩenyg s
yndeksom Xgrsta H > 1 / 2. Postroen funkcyonal yntehral\noho typa ot πtoho bluΩdanyq y
dokazana eho slabaq sxodymost\ k yntehralu, postroennomu po drobnomu brounovskomu dvy-
Ωenyg.
1. Vstup. Nexaj { ξ i } i ≥ 1 — poslidovnist\ vypadkovyx velyçyn, zadanyx na
jmovirnisnomu prostori ( Ω, F, P ), Sn = ξii
n
=∑ 1
— vidpovidne vypadkove blukan-
nq, { fn } n ≥ 1 — poslidovnist\ nevypadkovyx, hladkyx ta zbiΩnyx, u pevnomu ro-
zuminni, funkcij.
Dostatni umovy slabko] zbiΩnosti intehral\nyx funkcionaliv, tobto funk-
cionaliv vyhlqdu
f
S
n nn
i i
i
n
+
=
−
∑
ξ 1
1
1
,
u vypadku, koly ξ i — nezaleΩni odnakovo rozpodileni vypadkovi velyçyny
(n.o.r.v.v.), navedeno u knyzi [1].
U statti [2] dovedeno uzahal\nennq c\oho rezul\tatu, a same slabku zbiΩ-
nist\ funkcionaliv vyhlqdu
f
i
n
S
n n
f t W dWn
i i
i
n
t t, ,
→ ( )+
=
−
∑ ∫
ξ 1
1
1
0
1
,
de { Wt , t ∈ [ 0, 1 ]} — vineriv proces na ( Ω, F, P ), a velyçyny ξ i utvorggt\
martynhal-riznycg, tobto sumy Sn [ martynhalom.
U statti [3] rozhlqdagt\sq vypadkovi blukannq z intehral\nymy koefici[n-
tamy, wo magt\ vyhlqd
Z n z
nt
n
s dst
n
i
nt
i
n
i n
i n
( )
=
[ ]
( )
( − )
= [ ]
∑ ∫: ,
/
/
1 1
ξ , t ∈ [ 0, 1 ], (1)
de { }( )
≥ξi
n
i n, 1 — n.o.r.v.v., z ( t, s ) = C H s u u s duH
H H H
s
t
−
( − )− − −∫1
2
1 2 1 2 3 2/ / /
,
CH — stala, wo zaleΩyt\ lyße vid H. U cij statti dovedeno, wo taki vypadkovi
blukannq slabko zbihagt\sq do drobovoho brounivs\koho ruxu (DBR) { Zt , t ∈ [ 0,
1 ]} z indeksom Xgrsta H > 1 / 2. (Nahada[mo, wo DBR { Zt , t ∈ R+} z indeksom
Xgrsta H ∈ ( 0, 1 ) — ce haussivs\kyj proces zi stacionarnymy pryrostamy i ne-
perervnymy tra[ktoriqmy, Z0 = 0, E Zt = 0, E Zt Zs =
1
2
2 2 2t s t sH H H+ − −( ) .)
Tomu pryrodno postavyty pytannq pro umovy zbiΩnosti intehral\nyx funkcio-
© G. S. MIÍURA, S. H. RODE, 2007
1040 ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 8
SLABKA ZBIÛNIST| INTEHRAL|NYX FUNKCIONALIV … 1041
naliv vyhlqdu f
i
n
Z Zn i n
n
i n
n
i
n
, / /
( ) ( )
=
−
∑ ∆
1
1
, de ∆Z Z Zi n
n
i n
n
i n
n
/ / /:( )
( + )
( ) ( )= −1 , do intehra-
la f t Z dZt t( )∫ ,
0
1
.
Zaznaçymo, wo f t Z dZt t( )∫ ,
0
1
isnu[ qk hranycq majΩe napevno intehral\nyx
sum Rimana – Stil\t\[sa, qkwo f [ hel\derovog funkci[g, f ∈ H
α
( R+ ) ×
× H
β
( R ), de β > 1 – H, α > H
–
1 – 1. Cej rezul\tat dovedeno v statti [4].
Danu robotu pobudovano takym çynom. Punkt 2 mistyt\ dopomiΩni tverd-
Ωennq pro zbiΩnist\ riznyx intehral\nyx funkcionaliv. Osnovnyj fakt, na
qkomu ©runtugt\sq dovedennq, — ce te, wo kvadratyçna variaciq procesiv Z (
n
)
asymptotyçno dorivng[ 0. U punkti 3 dovedeno osnovnyj rezul\tat dlq funkcij
vyhlqdu fn ( x ) ta sformul\ovano oçevydne uzahal\nennq dlq funkcij fn ( t, x ).
2. DopomiΩni rezul\taty. Dali budemo prypuskaty, wo n.o.r.v.v.
{ }( )
≥ξi
n
i n, 1 ta funkci] { fn } n ≥ 1 , fn : R → R, zadovol\nqgt\ vidpovidno umovy:
A) E ξi
n( ) = 0, D : = E( )( )ξi
n 2 = 1, i, n ≥ 1;
B1
) ∀n ≥ 1 : fn ∈ C
1
( R ), f ∈ C
1
( R ), pryçomu dlq bud\-qkoho R > 0 isnu[
MR > 0 taka, wo
sup sup
n x R
nf x f x
≥ ≤
′( ) + ′( ){ }
1
≤ MR ;
B2
) fn ( x ) ⇒ f ( x ) rivnomirno na koΩnomu vidrizku [ – R, R ] .
Nexaj πr : = { = < < … < = }( ) ( ) ( )0 10 1t t tr r
p
r
r
— deqka poslidovnist\ rozbyttiv
vidrizka [ 0, 1 ], | πr | → 0 pry r → ∞. Poznaçymo
∆Z Z Zj r
n
t
n
t
n
j
r
j
r, :( ) ( ) ( )= −
+
( ) ( )
1
i utvorymo poslidovnist\ intehral\nyx sum
S f Z Z
r j
r
r
n
n t
n
j
p
j r
n
π
( ) ( )
=
−
( )= ( )( )∑: ,
1
1
∆ .
Lema 1. Qkwo vykonu[t\sq umova A), to
P lim P lim lim/ ,− = −
→∞
( )
=
−
→∞ →∞
( )
=
−
( ) ( )∑ ∑
n
i n
n
i
n
r n
j r
n
j
p
Z Z
r
∆ ∆2
1
1
2
1
1
= 0.
Dovedennq. Naspravdi dovedemo troxy bil\ße, a same zbiΩnist\ do nulq v
L2 ( P ). Dlq c\oho vykorysta[mo rivnist\ (1) i zapyßemo dlq bud\-qkyx 0 ≤ t1 <
< t2 ≤ 1 riznycg
Z Z n z
nt
n
s z
nt
n
s dst
n
t
n
k
n
k n
k n
k
nt
2 1
1
2 1
11
( ) ( ) ( )
( − )=
[ ]
− = [ ]
− [ ]
∫∑ , ,
/
/
ξ +
+ n z
nt
n
s ds k
n
k n
k n
k nt
nt [ ]
( )
( − )=[ ]+
[ ]
∫∑ 2
111
2
,
/
/
ξ .
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 8
1042 G. S. MIÍURA, S. H. RODE
Vvedemo poznaçennq
αn
m n
m n
m l z
l
n
s ds( ) =
( − )
∫, : ,
/
/
1
,
β
α α
αn
n n
n
m l l
m l m l m l l
m l l m l
( ) =
( ) − ( ) ≤ ≤
( ) < ≤
, , :
, , , ,
, , ,1 2
2 1 1 2
2 1 2
i perepyßemo navedenu vywe riznycg u vyhlqdi
Z Z n k nt ntt
n
t
n
n
k
nt
k
n
2 1
2
1 2
1
( ) ( )
=
[ ]
( )− = [ ] [ ]( )∑ β ξ, , .
Tomu, vraxovugçy umovu A), ma[mo
E , ,Z Z n k nt ntt
n
t
n
n
k
nt
2 1
22 2
1 2
1
( ) ( )
=
[ ]
− = [ ] [ ]( )∑ β =
= n z
nt
n
s z
nt
n
s ds
k n
k n
k
nt [ ]
− [ ]
( − )=
[ ]
∫∑ 2 1
11
2
1
, ,
/
/
+
+ n z
nt
n
s ds
k n
k n
k nt
nt [ ]
( − )=[ ]+
[ ]
∫∑ 2
1
2
11
2
,
/
/
≤
≤ z
nt
n
s z
nt
n
s ds z
nt
n
s ds
nt n
nt n
nt n
[ ]
− [ ]
+ [ ]
[ ]
[ ]
[ ]
∫ ∫2 1
2
0
2
21
1
2
, , ,
/
/
/
=
= C s u u s du dsH
H
nt n
H H
nt n
nt n
2 1 2
0
1 2 3 2
2
1
1
2
−
[ ]
− −
[ ]
[ ]
∫ ∫ ( − )
/
/ /
/
/
+
+ C s u u s du dsH
H
nt n
nt n
H H
s
nt n
2 1 2 1 2 3 2
2
1
2 2
−
[ ]
[ ]
− −
[ ]
∫ ∫ ( − )
/
/
/ /
/
. (2)
Teper vraxu[mo intehral\ne zobraΩennq DBR Zt çerez vineriv proces, qke mis-
tyt\sq u statti [5]: Zt = z t s dWs
t
( )∫ ,
0
, zhidno z qkym prava çastyna (2) dorivng[
E / /Z Z
nt
n
nt
n
t tnt n nt n
H
H
[ ] [ ]− = [ ] − [ ] ≤ ( − )
2 1
2 2 1
2
2 1
2
.
Tomu
E /∆Z ni n
n
i
n
H
i
n
( )
=
−
−
=
−
∑ ∑≤
2
1
1
2
1
1
→ 0, n → ∞,
E ,∆Z t tj r
n
j
p
j
r
j
r H
j
pr r
( )
=
−
+
( ) ( )
=
−
∑ ∑≤ ( − )
2
1
1
1
2
1
1
→ 0, r → ∞.
Lemu dovedeno.
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 8
SLABKA ZBIÛNIST| INTEHRAL|NYX FUNKCIONALIV … 1043
Lema 2. Qkwo vykonugt\sq umovy A) ta B1
), B2
), to
lim lim P / /
r n
n
n i n
n
i n
n
i
n
S f Z Z
r→∞ →∞ π
( ) ( ) ( )
=
−
− ( ) >
∑ ∆
1
1
δ = 0
dlq bud\-qkoho δ > 0.
Dovedennq. Nexaj funkciq F
(
x
) : R → R [ pervisnog funkci] f, tobto
F ′
(
x
) = f (
x
) dlq vsix x ∈ R. Todi za formulog Tejlora ta na pidstavi umov B1
),
B2
) magt\ misce dva zobraΩennq odni[] i ti[] Ω riznyci F Z Fn( ) − ( )( )
1 0 :
F Z F F Z F Zn
i n
n
i n
n
i
n
( ) − ( ) = ( ) − ( )( )
( + )
( ) ( )
=
−
( )∑1 1
0
1
0 / / =
= f Z Z f Zi n
n
i n
n
i
n
i n i n
n
i
n
( ) + ′( )( )( ) ( )
=
−
( )
=
−
∑ ∑/ / , /∆ ∆
0
1
2
0
1
1
2
θ , (3)
F Z F F Z F Zn
t
n
t
n
j
p
j
r
j
r
r
( ) − ( ) = ( ) − ( )( ) ( ) ( )
=
−
( )
+
( ) ( )∑1
0
1
0
1
=
= f Z Z f Z
t
n
j r
n
j
p
j r n j r
n
j
p
j
r
r r
( ) + ′( )( )( )
( ) ( )
=
−
( )
=
−
∑ ∑∆ ∆, , , ,
0
1
2
0
1
1
2
θ , (4)
de toçky θi n, leΩat\ miΩ Zi n
n
/
( )
i Z i n
n
( + )
( )
1 / , a toçky θ j r n, , — miΩ Z
t
n
j
r( )
( )
i Z
t
n
j
r
+
( )
( )
1
.
Z (3) i (4) otrymu[mo
S f Z Z
r
n
i n
n
i n
n
i
n
π
( ) ( ) ( )
=
−
− ( )∑ / /∆
0
1
≤
≤
1
2
1
2
2
0
1
2
0
1
′( ) ( ) + ′( ) ( )( )
=
−
( )
=
−
∑ ∑f Z f Zi n i n
n
i
n
j r n j r
n
j
pr
θ θ, / , , ,∆ ∆ .
Tomu dlq bud\-qkoho δ > 0
P P sup/ /S f Z Z Z R
r
n
i n
n
i n
n
i
n
t
t
n
π
( ) ( ) ( )
=
−
≤ ≤
( )− ( ) >
≤ ≥
∑ ∆
0
1
0 1
δ +
+ P P/ ,( ) ≥
+ ( ) ≥
( )
=
−
( )
=
−
∑ ∑∆ ∆Z
M
Z
Mi n
n
i
n
R
j r
n
j
p
R
r
2
0
1
2
0
1
2 2δ δ
. (5)
Zhidno z rezul\tatom statti [3], Z(
n
)
zbiha[t\sq do Z v topolohi] Skoroxoda na
[ 0, 1 ]. Oskil\ky funkcionaly typu sup ta inf neperervni v topolohi] Skoroxo-
da, a rozpodil Zt — haussivs\kyj, to
P sup P sup
0 1 0 1≤ ≤
( )
≤ ≤
≥
→ ≥
t
t
n
t
tZ R Z R
dlq vsix R > 0. Ostannq jmovirnist\ prqmu[ do 0 pry R → ∞ [6], tomu dove-
dennq vyplyva[ z (5) ta lemy 1.
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 8
1044 G. S. MIÍURA, S. H. RODE
3. Slabka zbiΩnist\ intehral\nyx funkcionaliv. Teper sformulg[mo i
dovedemo osnovnyj rezul\tat.
Teorema 1. Qkwo vykonugt\sq umovy A) i B1
), B2
), to
f Z Z f Z dZn i n
n
i n
n
i
n
w
t t( ) → ( )( ) ( )
=
−
∑ ∫/ /∆
1
1
0
1
, n → ∞,
de
w→ oznaça[ slabku zbiΩnist\ za rozpodilom.
Dovedennq. Podamo riznycg
∆n : = f Z dZ f Z Zt t n i n
n
i n
n
i
n
( ) − ( )∫ ∑ ( ) ( )
=
−
0
1
1
1
/ /∆
u vyhlqdi ∆n = ∆n r
j
j ,
( )
=∑ 1
4
, de
∆ ∆n r t t t j r
j
p
f Z dZ f Z Z
j
r
r
, ,
( )
=
−
= ( ) − ( )∫ ∑ ( )
1
0
1
1
1
(ne zaleΩyt\ vid n ),
∆ ∆ ∆n r t j r
j
p
t
n
j r
n
j
p
f Z Z f Z Z
j
r
r
j
r
r
, , ,
( )
=
−
( ) ( )
=
−
= ( ) − ( )( ) ( )∑ ∑2
1
1
1
1
,
∆ ∆ ∆n r t
n
j r
n
j
p
n t
n
j r
n
j
p
f Z Z f Z Z
j
r
r
j
r
r
, , ,
( ) ( ) ( )
=
−
( ) ( )
=
−
= ( ) − ( )( ) ( )∑ ∑3
1
1
1
1
,
∆ ∆ ∆n r n t
n
j r
n
j
p
n i n
n
i n
n
i
n
f Z Z f Z Z
j
r
r
, , / /
( ) ( ) ( )
=
−
( ) ( )
=
−
= ( ) − ( )( )∑ ∑4
1
1
1
1
.
Zhidno z rezul\tatom Cel[ [4], zhadanym u vstupi,
P lim ,−
→∞
( )
r
n r∆1 = 0.
Na pidstavi lemy 2
P lim lim ,−
→∞ →∞
( )
r n
n r∆ 4 = 0.
Wo stosu[t\sq ∆n r,
( )2
, to vnaslidok slabko] zbiΩnosti Z(
n
)
do Z
f Z Z f Z Z
t
n
j r
n
j
p
w
t j r
j
p
j
r
r
j
r
r
( ) → ( )( ) ( )
( ) ( )
=
−
=
−
∑ ∑∆ ∆, ,
1
1
1
1
pry n → ∞ ta dlq bud\-qkoho fiksovanoho r ≥ 1.
Zalyßylos\ ocinyty ∆n r,
( )3
. Dlq c\oho vykorysta[mo pryjom, wo analohiçnyj
vykorystanomu pry dovedenni lemy 2.
Nexaj
F ( x ) = f t dt
x
( )∫
0
, Fn ( x ) = f t dtn
x
( )∫
0
.
Todi
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 8
SLABKA ZBIÛNIST| INTEHRAL|NYX FUNKCIONALIV … 1045
F Z F Z F Zn
t
n
t
n
j
p
j
r
j
r
r
( ) = ( ) − ( )( ) ( ) ( )
=
−
( )
+
( ) ( )∑1
1
1
1
=
= f Z Z f Z
t
n
j r
n
j
p
j r
n
j r
n
j
p
j
r
r r
( ) + ′( )( )( )
( ) ( )
=
−
( ) ( )
=
−
∑ ∑∆ ∆, , ,
1
1
2
1
1
1
2
θ (6)
i analohiçno
F Z f Z Z f Zn
n
n t
n
j r
n
j
p
n j r
n
j r
n
j
p
j
r
r r
( ) = ( ) + ′( )( )( ) ( ) ( )
=
−
( ) ( )
=
−
( )∑ ∑1
1
1
2
1
1
1
2
∆ ∆, , ,θ̃ , (7)
de θ j r
n
,
( )
i
˜
,θ j r
n( )
leΩat\ miΩ Z
t
n
j
r( )
( )
i Z
t
n
j
r
+
( )
( )
1
.
Teper
F Z F Z Z f t f tn
n
n n
t Z
n
n
( ) − ( ) ≤ ( ) − ( )( ) ( ) ( )
≤ ( )
1 1 1
1
sup ,
zvidky
P F Z F Zn
n
n( ) − ( ) ≥{ }( ) ( )
1 1 δ ≤ P P supZ R f t f t
R
n
t R
n1
( )
≤
≥{ } + ( ) − ( ) ≥
δ
(8)
(podiq pid znakom ostann\o] jmovirnosti ne [ vypadkovog). Oskil\ky fn rivno-
mirno prqmu[ do f na [ – R, R ] , to druhyj dodanok v (8) [ nul\ovym, poçynagçy z
deqkoho nomera n, a
lim P P
n
nZ R Z R
C
R→∞
( ) ≥{ } = ≥{ } ≤1 1 2 .
Tomu z (6) – (8) na pidstavi lem 1 i 2 ma[mo
P lim lim ,−
→∞ →∞r n
n r∆3 = 0,
wo i zaverßu[ dovedennq teoremy.
ZauvaΩennq. Vsi poperedni mirkuvannq lehko perenosqt\sq na vypadkovi
blukannq, vyznaçeni na dovil\nomu vidrizku [ 0, T ], T > 0. Nexaj teper f ( t, x ) i
fn ( t, x ) : R+ × R → R, n ≥ 1, — poslidovnist\ funkcij, wo zadovol\nq[ umovy:
C1
) f ∈ C
1
( R+ × R ), fn ∈ C
1
( R+ × R ) i dlq bud\-qkoho R > 0 isnu[ MR > 0
take, wo
sup , , , ,
,
, ,
0≤ ≤ ≤
′ ( ) + ′ ( ) + ′( ) + ′( ){ } ≤
t R x R
n t n x t x Rf t x f t x f t x f t x M ;
C2
) fn ⇒ f rivnomirno na bud\-qkomu prqmokutnyku [ 0, R ] × [ – R, R ] .
V cilomu analohiçno do teoremy 1 moΩna dovesty nastupnyj rezul\tat.
Teorema 2. Qkwo vykonugt\sq umovy A) i C1
), C2
), to
f
i
n
Z Z f t Z dZn i n
n
i n
n
i
n
w
t t, ,/ /
( ) ( )
=
−
→ ( )∑ ∫∆
1
1
0
1
, n → ∞.
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 8
1046 G. S. MIÍURA, S. H. RODE
1. Skoroxod A. V., Slobodengk N. P. Predel\n¥e teorem¥ dlq sluçajn¥x bluΩdanyj. – Kyev:
Nauk. dumka, 1970. – 304 s.
2. Yoshihara K.-I. A weak convergence theorem for functionals of sums of martingale differences //
Yokohama Math. J. – 1978. – 26. – P. 101 – 107.
3. Sottinen T. Fractional Brownian motion, random walks and binary market models // Finance and
Stochastics. – 2001. – 5, # 3. – P. 343 – 355.
4. Zaehle M. Integration with respect to fractal functionals and stochastic calculus // Probab. Theory
and Relat. Fields. – 1997. – 111. – P. 333 – 374.
5. Norros I., Valkeila E., Virtamo J. An elementary approach to a Girsanov formula and other
analytical results on fractional Brownian motions // Bernoulli. – 1999. – 5, # 4. – P. 571 – 587.
6. Synaj Q. H. O raspredelenyy maksymuma drobnoho brounovskoho dvyΩenyq // Uspexy mat.
nauk. – 1997. – 52, v¥p. 2(314). – S. 119 – 138.
OderΩano 06.10.2005,
pislq doopracgvannq — 27.02.2006
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 8
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| id | umjimathkievua-article-3368 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:41:14Z |
| publishDate | 2007 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/aa/6d9478ad4496f93d3be92d2dc76a8faa.pdf |
| spelling | umjimathkievua-article-33682020-03-18T19:52:34Z Weak convergence of integral functionals of random walks weakly convergent to fractional Brownian motion Слабка збіжність інтегральних функціоналів від випадкових блукань, що слабко збігаються до дробового броунівського руху Mishura, Yu. S. Rode, S. H. Мішура, Ю. С. Роде, С. Г. We consider a random walk that converges weakly to a fractional Brownian motion with Hurst index H > 1/2. We construct an integral-type functional of this random walk and prove that it converges weakly to an integral constructed on the basis of the fractional Brownian motion. Рассмотрено случайное блуждание, слабо сходящееся к дробному броуновскому движению с индексом Хюрста H > 1 / 2. Построен функционал интегрального типа от этого блуждания и доказана его слабая сходимость к интегралу, построенному по дробному броуновскому движению. Institute of Mathematics, NAS of Ukraine 2007-08-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3368 Ukrains’kyi Matematychnyi Zhurnal; Vol. 59 No. 8 (2007); 1040–1046 Український математичний журнал; Том 59 № 8 (2007); 1040–1046 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3368/3479 https://umj.imath.kiev.ua/index.php/umj/article/view/3368/3480 Copyright (c) 2007 Mishura Yu. S.; Rode S. H. |
| spellingShingle | Mishura, Yu. S. Rode, S. H. Мішура, Ю. С. Роде, С. Г. Weak convergence of integral functionals of random walks weakly convergent to fractional Brownian motion |
| title | Weak convergence of integral functionals of random walks weakly convergent to fractional Brownian motion |
| title_alt | Слабка збіжність інтегральних функціоналів від випадкових блукань, що слабко збігаються до дробового броунівського руху |
| title_full | Weak convergence of integral functionals of random walks weakly convergent to fractional Brownian motion |
| title_fullStr | Weak convergence of integral functionals of random walks weakly convergent to fractional Brownian motion |
| title_full_unstemmed | Weak convergence of integral functionals of random walks weakly convergent to fractional Brownian motion |
| title_short | Weak convergence of integral functionals of random walks weakly convergent to fractional Brownian motion |
| title_sort | weak convergence of integral functionals of random walks weakly convergent to fractional brownian motion |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3368 |
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