Random process from the class V(φ,ψ): exceeding a curve
Random processes from the class V (φ, ψ) which is more general than the class of ψ-sub-Gaussian random process. The upper estimate of the probability that a random process from the class V (φ, ψ) exceeds some function is obtained. The results are applied to generalized process of fractional Brownian...
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irk-123456789-45262010-03-01T17:26:32Z Random process from the class V(φ,ψ): exceeding a curve Yamnenko, R. Vasylyk, O. Random processes from the class V (φ, ψ) which is more general than the class of ψ-sub-Gaussian random process. The upper estimate of the probability that a random process from the class V (φ, ψ) exceeds some function is obtained. The results are applied to generalized process of fractional Brownian motion. 2007 Article Random process from the class V(φ,ψ): exceeding a curve / R. Yamnenko, O. Vasylyk // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 4. — С.219–232. — Бібліогр.: 8 назв.— англ. 0321-3900 http://dspace.nbuv.gov.ua/handle/123456789/4526 en Інститут математики НАН України |
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Random processes from the class V (φ, ψ) which is more general than the class of ψ-sub-Gaussian random process. The upper estimate of the probability that a random process from the class V (φ, ψ) exceeds some function is obtained. The results are applied to generalized process of fractional Brownian motion. |
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Yamnenko, R. Vasylyk, O. |
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Yamnenko, R. Vasylyk, O. Random process from the class V(φ,ψ): exceeding a curve |
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Yamnenko, R. Vasylyk, O. |
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Yamnenko, R. |
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Random process from the class V(φ,ψ): exceeding a curve |
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Random process from the class V(φ,ψ): exceeding a curve |
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Random process from the class V(φ,ψ): exceeding a curve |
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Random process from the class V(φ,ψ): exceeding a curve |
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Random process from the class V(φ,ψ): exceeding a curve |
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random process from the class v(φ,ψ): exceeding a curve |
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Інститут математики НАН України |
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2007 |
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Random process from the class V(φ,ψ): exceeding a curve / R. Yamnenko, O. Vasylyk // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 4. — С.219–232. — Бібліогр.: 8 назв.— англ. |
| work_keys_str_mv |
AT yamnenkor randomprocessfromtheclassvphpsexceedingacurve AT vasylyko randomprocessfromtheclassvphpsexceedingacurve |
| first_indexed |
2025-07-02T07:45:00Z |
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2025-07-02T07:45:00Z |
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1836520362552066048 |
| fulltext |
Theory of Stochastic Processes
Vol.13 (29), no.4, 2007, pp.219–232
ROSTYSLAV YAMNENKO AND OLGA VASYLYK
RANDOM PROCESS FROM THE CLASS V (ϕ, ψ):
EXCEEDING A CURVE
Random processes from the class V (ϕ,ψ) which is more general than
the class of ψ-sub-Gaussian random process. The upper estimate of
the probability that a random process from the class V (ϕ,ψ) exceeds
some function is obtained. The results are applied to generalized
process of fractional Brownian motion.
1. Introduction
In this paper we consider random process from the class V (ϕ, ψ) defined
on compact set and the probability that this process exceeds some func-
tion. Recall that random process belongs to class V (ϕ, ψ) if its trajectories
belong to the space Subψ(Ω) and increments belong to the space Subϕ(Ω).
Properties of random variables and processes from the spaces Subϕ(Ω) and
SSubϕ(Ω) can be found in the book of Buldygin V.V. and Kozachenko Yu.V.
[1] and in the papers [2-7]. Here we generalize the results obtained earlier
in [6-8].
The paper is organized as follows. Basic definitions and some properties
of ϕ-sub-Gaussian and strictly ϕ-sub-Gaussian spaces of random variables
and processes are given in section 2. In section 3 we obtain general results
on estimates of probability that random process from the class V (ϕ, ψ)
overruns a level specified by a continuous function. The methods used in
the section are the same as in [6]. However for convenience of readers we
give here complete proofs. In section 4 we apply results from the previous
section to generalized process of fractional Brownian motion from the class
V (ϕ, ψ) and obtain the estimate of overcrossing by its trajectories the level
defined by function ct, where c > 0 is a given constant. Such estimate has
applications in the queuing theory as estimate of buffer overflow probability
or in the risk theory as estimate of ruin probability.
Invited lecture.
2000 Mathematics Subject Classifications. 60G20, 60G18, 60K25.
Key words and phrases. Sub-Gaussian process, generalized fractional Brownian mo-
tion, metric entropy, buffer overflow probability, ruin probability.
219
220 ROSTYSLAV YAMNENKO AND OLGA VASYLYK
2. Class V (ϕ, ψ): essential definitions and properties
Let (Ω,B, P ) be a standard probability space and T be some paramet-
rical space.
Definition 2.1.[1] Function u = {u(x), x ∈ R} is called an Orlicz N-
function if u is a continuous even convex function such that u(0) = 0,
u(x) monotonically increases as x > 0, u(x)
x
→ 0 as x → 0 and u(x)
x
→ ∞ as
x → ∞.
Definition 2.2.[1] Let ϕ be such an Orlicz N-function that ϕ(x) = cx2 as
|x| ≤ x0 for some x0 > 0 and c > 0. Centered random variable ξ belongs to
the space Subϕ(Ω), the space of ϕ-sub-Gaussian random variables, if for all
λ ∈ R there exists a constant rξ ≥ 0 which satisfies the following inequality
E exp (λξ) ≤ exp {ϕ(λrξ))}.
Theorem 2.1.[1] The space Subϕ(Ω) is a Banach space with respect to the
norm
τϕ(ξ) = sup
λ>0
ϕ(−1) (log E exp{λξ})
λ
,
where ϕ(−1) is an inverse function to the function ϕ, and for all λ ∈ R the
following inequality holds
E exp(λξ) ≤ exp(ϕ(λτϕ(ξ))) . (1)
Moreover, there exist constants r > 0, cr > 0 such that
(E|ξ|r) 1
r ≤ crτϕ(ξ).
Lemma 2.1.[2] Let ξ ∈ Subϕ(Ω). Then for all ε > 0 the following inequality
holds true
P {|ξ| > ε} ≤ 2 exp
{
−ϕ
(
ε
τϕ(ξ)
)}
.
Definition 2.3. Random process X = (X(t), t ∈ T ) belongs to the space
Subϕ(Ω), if for all t ∈ T : X(t) ∈ Subϕ(Ω) and sup
t∈T
τϕ(X(t)) < ∞.
Let (T, ρ) be a pseudometrical (metrical) compact space with pseudo-
metric (metric) ρ.
Definition 2.4.[3] Metric entropy in relation to pseudometric (metric) ρ,
or just metric entropy is a function
H(T,ρ)(u) = H(u) =
{
log N(T,ρ)(u), if N(T,ρ)(u) < +∞
+∞, if N(T,ρ)(u) = +∞ ,
RANDOM PROCESS FROM THE CLASS V (ϕ,ψ) 221
where N(T,ρ)(u) = N(u) denotes the least the least number of closed ρ-balls
with radius u.
Definition 2.5. [3] A family of random variables Δ from the space Subϕ(Ω)
is called strictly Subϕ(Ω), if there exists a constant CΔ > 0 such that for
arbitrary finite set I : ξi ∈ Δ, i ∈ I, and for any λi ∈ R the following
inequality takes place
τϕ
(∑
i∈I
λiξi
)
≤ CΔ
⎛
⎝E
(∑
i∈I
λiξi
)2
⎞
⎠
1
2
. (2)
If Δ is a family of strictly Subϕ(Ω) random variables, then linear clo-
sure Δ of the family Δ in the space L2(Ω) also is strictly Subϕ(Ω) family
of random variables. Linearly closed families of strictly Subϕ(Ω) random
variables form a space of strictly ϕ-sub-Gaussian random variables. This
space is denoted by SSubϕ(Ω).
When ϕ(x) = x2
2
the space SSubϕ(Ω) is called the space of strictly sub-
Gaussian random variables and is denoted by SSub(Ω). The space of jointly
Gaussian random variables belongs to the space SSub(Ω) and τ 2(ξ) = Eξ2.
Definition 2.6. A random process X = (X(t), t ∈ T ) is a strictly ϕ-sub-
Gaussian process if the corresponding family of random variables belongs
to the space SSubϕ(Ω).
Definition 2.7.[7] ϕ is subordinated to an Orlizc N -function ψ (ϕ ≺ ψ)
if there are exist such numbers x0 > 0 and k > 0 that ϕ(x) < ψ(kx) for
x > x0.
Definition 2.8.[7] Let ϕ ≺ ψ are two Orlicz N -functions. Random process
X = (X(t), t ∈ T ) belongs to class V (ϕ, ψ) if for all t ∈ T the process X(t)
is from Subψ(Ω) and for all s, t ∈ T increments (X(t)−X(s)) belong to the
space Subϕ(Ω).
3. Main Results
Let (T, ρ) be a pseudometrical (metrical) compact space with pseudo-
metric (metric) ρ and Y = {Y (t), t ∈ T} be a separable random process
from the class V (ϕ, ψ).
Suppose there exists such continuous monotonically increasing function
σ = {σ(h), h > 0}, that σ(h) → 0, as h → 0, and the following inequality
for increments of the process is true
sup
ρ(t,s)≤h
τϕ(Y (t) − Y (s)) ≤ σ(h). (3)
222 ROSTYSLAV YAMNENKO AND OLGA VASYLYK
Let β > 0 be some number such that
β ≤ σ
(
inf
s∈T
sup
t∈T
ρ(t, s)
)
(4)
and let εk = σ(−1)(βpk), p ∈ (0, 1), k = 0, 1, 2, . . ., γ(u) = τψ(Y (u)).
Lemma 3.1. Let f = {f(t), t ∈ T} be a continuous function such that
|f(u) − f(v)| ≤ δ(ρ(u, v)), where δ = {δ(s), s > 0} is some monotonically
increasing nonnegative function, and X(t) = Y (t) − f(t). Let {qk, k =
1, 2, . . .} be such a sequence that qk > 1 and
∞∑
k=1
q−1
k ≤ 1. Then for all
λ ∈ R, p ∈ (0, 1) we have
E exp{λ sup
t∈T
X(t)} ≤ exp
{
1
q1
sup
u∈T
(ψ(λq1γ(u)) − λq1f(u))
}
× (5)
×
( ∞∏
k=1
(N(εk))
1
qk
)( ∞∏
k=2
exp
{
1
qk
ϕ
(
λqkβpk−1
)
+ λδ
(
σ(−1)
(
βpk−1
))})
.
Proof. Denote by Vεk
the set of the centers of the closed balls with radius
εk, which form minimal covering of the space (T, ρ). Number of elements
in the set Vεk
is equal to NT (εk) = N(εk).
The process Y (t) and, therefore, the process X(t) are separable pro-
cesses.
It follows from lemma 2.1 and condition (3) that for any ε > 0
P {|Y (t) − Y (s)| > ε} ≤ 2 exp
{
−ϕ
(
ε
τϕ(Y (t) − Y (s))
)}
≤ 2 exp
{
−ϕ
(
ε
σ(ρ(t, s))
)}
.
Therefore the process Y is continuous on probability and the process X is
continuous on probability as well. If a separable random process on (T, ρ) is
continuous on probability, then any set, which is countable and everywhere
dense with respect to ρ, can be taken as a set of separability of this process.
Therefore the set V =
∞⋃
k=1
Vεk
is a set of separability of the process X and
we have that with probability one
sup
t∈T
X(t) = sup
t∈V
X(t). (6)
Consider a mapping αn = {αn(t), n = 0, 1 . . .} of the set V in Vεn, where
αn(t) is such a point from the set Vεn, that ρ(t, αn(t)) < εn. If t ∈ Vεn
then αn(t) = t. If there exist several points from the set Vεn, such that
RANDOM PROCESS FROM THE CLASS V (ϕ,ψ) 223
ρ(t, αn(t)) < εn, then we choose one of them and denote it by αn(t). Then
it follows from the theorem 2.1 and (3) that
P
{|Y (t) − Y (αn(t))| > p
n
2
}
≤ E(Y (t) − Y (αn(t)))2
pn
≤ c2
2τ
2
ϕ(Y (t) − Y (αn(t)))
pn
≤ c2
2σ
2(εn)
pn
= c2
2β
2pn.
This inequality means that
∞∑
n=1
P
{|Y (t) − Y (αn(t))| > p
n
2
}
< ∞.
From the Borell-Kantelli’s lemma follows that Y (t) − Y (αn(t)) → 0
as n → ∞ with probability one. Since the function f is continuous then
X(t)−X(αn(t)) → 0 as n → ∞ with probability one as well. Since the set
V is countable, then X(t)−X(αn(t)) → 0 as n → ∞ for all t simultaneously.
Let t be an arbitrary point from the set V . Denote by tm = αm(t), tm−1 =
αm−1(tm), . . . , t1 = α1(t2) for any m ≥ 1. Then for all m ≥ 2 we have the
following inequality
X(t) = X(t1) +
m∑
k=2
(X(tk) − X(tk−1)) + X(t) − X(αm(t)) ≤ max
u∈Vε1
X(u) +
+
m∑
k=2
max
u∈Vεk
(X(u) − X(αk−1(u)) + X(t) − X(αm(t)). (7)
It follows from (7) and (6) that with probability one
sup
t∈T
X(t) = sup
t∈V
X(t)
≤ lim
m→∞
inf
(
max
u∈Vε1
X(u) +
m∑
k=2
max
u∈Vεk
(X(u) − X(αk−1(u)))
)
. (8)
From the Helder’s inequality, Fatu’s lemma and (8) follows that for all
λ > 0
E exp
{
λ sup
t∈T
X(t)
}
≤ E lim
m→∞
inf exp
{
λ
(
max
u∈Vε1
X(u) +
m∑
k=2
max
u∈Vεk
(X(u) − X(αk−1(u)))
)}
≤ lim
m→∞
inf E exp
{
λ
(
max
u∈Vε1
X(u) +
m∑
k=2
max
u∈Vεk
(X(u) − X(αk−1(u)))
)}
≤ lim
m→∞
inf
((
E exp
{
q1λ max
u∈Vε1
X(u)
}) 1
q1 ×
224 ROSTYSLAV YAMNENKO AND OLGA VASYLYK
×
m∏
k=2
(
E exp
{
qkλ max
u∈Vεk
(X(u) − X(αk−1(u)))
}) 1
qk
)
≤
(
E exp
{
q1λ max
u∈Vε1
X(u)
}) 1
q1 ×
×
∞∏
k=2
(
E exp
{
qkλ max
u∈Vεk
(X(u) − X(αk−1(u)))
}) 1
qk
= I1 ·
∞∏
k=2
Ik. (9)
Let’s consider each term in (9). It follows from the theorem 2.1 that
E exp{q1λY (u)} ≤ exp {ψ(q1λγ(u))}. Therefore
I1 ≤
⎛
⎝∑
u∈Vε1
E exp
{
q1λY (u)
}
exp
{
− q1λf(u)
}⎞⎠
1
q1
≤
⎛
⎝∑
u∈Vε1
exp
{
ψ(q1λγ(u)) − q1λf(u)
}⎞⎠
1
q1
≤
(
N(ε1) exp
{
sup
u∈T
(
ψ(q1λγ(u)) − q1λf(u)
)}) 1
q1
≤
(
N(ε1)
) 1
q1 exp
{
1
q1
sup
u∈T
(
ψ(q1λγ(u)) − q1λf(u)
)}
. (10)
It also follows from the theorem 2.1 and assumption (3) that
E exp{qkλ(Y (u) − Y (αk−1(u)))} ≤ exp{ϕ(qkλσ(εk−1))}.
In that way since |f(u) − f(v)| ≤ δ(ρ(u, v)) then
Ik ≤
(
N(εk) max
u∈Vεk
E exp
{
qkλ[Y (u) − Y (αk−1(u))]
}
×
× exp
{
− qkλ[f(u) − f(αk−1(u))]
}) 1
qk
≤
(
N(εk)
) 1
qk
(
max
u∈Vεk
exp
{
ϕ(qkλσ(εk−1)) − qkλ[f(u) − f(αk−1(u))]
}) 1
qk
≤
(
N(εk)
) 1
qk
(
max
u∈Vεk
exp
{
ϕ(qkλσ(εk−1)) + qkλδ(ρ(u, αk−1(u)))
}) 1
qk
≤
(
N(εk)
) 1
qk exp
{
q−1
k ϕ(qkλβpk−1) + λδ(σ(−1)(βpk−1))
}
. (11)
From inequalities (9), (10) and (11) we have the assertion of the lemma.�
RANDOM PROCESS FROM THE CLASS V (ϕ,ψ) 225
Theorem 3.1. Let Y = {Y (t), t ∈ T} be a separable random process from
the class V (ϕ, ψ) and f = {f(t), t ∈ T} be such a continuous function that
|f(u) − f(v)| ≤ δ(ρ(u, v)), where δ = {δ(s), s > 0} is some monotonically
increasing nonnegative function, and X(t) = Y (t) − f(t). Let r1 = {r1(u) :
u ≥ 1} be such a continuous function that r1(u) > 0 as u > 1 and the
function s(t) = r1(exp{t}), t ≥ 0, is convex. If
β∫
0
r1(N(σ(−1)(u)))du < ∞, (12)
then for all p ∈ (0; 1) and x > 0 the following inequality holds true
P
{
sup
t∈T
X(t) > x
}
≤ inf
λ>0
Zr1(λ, p, β), (13)
where
Zr1(λ, p, β)
= exp
{
θψ(λ, p) + pϕ
(
λβ
1 − p
)
+ λ
( ∞∑
k=2
δ(σ(−1)(βpk−1)) − x
)}
×
×r
(−1)
1
⎛
⎝ 1
βp
βp∫
0
r1(N(σ(−1)(u)))du
⎞
⎠ , (14)
θψ(λ, p) = sup
u∈T
(
(1 − p)ψ
(
λγ(u)
1 − p
)
− λf(u)
)
. (15)
Proof. Let qk = ((1 − p)pk−1)−1 in the inequality (5) then
E exp
{
λ sup
t∈T
X(t)
}
≤ exp
{
θψ(λ, p) +
∞∑
k=2
(1 − p)pk−1ϕ
(
λβ
1 − p
)
+ λ
∞∑
k=2
δ
(
σ(−1)(βpk−1)
)}
× exp
{ ∞∑
k=1
(1 − p)pk−1 log N
(
σ(−1)(βpk)
)}
. (16)
Since
exp
{ ∞∑
k=1
(1 − p)pk−1 log N
(
σ(−1)(βpk)
)}
= r
(−1)
1
(
r1
(
exp
{ ∞∑
k=1
(1 − p)pk−1 log N
(
σ(−1)(βpk)
)}))
226 ROSTYSLAV YAMNENKO AND OLGA VASYLYK
≤ r
(−1)
1
( ∞∑
k=1
(1 − p)pk−1r1
(
N
(
σ(−1)(βpk)
)))
≤ r
(−1)
1
⎛
⎝ 1
βp
βp∫
0
r1
(
N
(
σ(−1)(u)
))
du
⎞
⎠ (17)
the assertion of the theorem follows from the lemma 3.1, (16) and Cheby-
shev’s inequality. �
Lemma 3.2. Suppose that all assumptions of lemma 3.1 are satisfied and
β∫
0
H(σ(−1)(u))
ϕ(−1)(H(σ(−1)(u)))
du < ∞, (18)
where H(ε) = log N(ε). Then for all p ∈ (0, 1) and λ > 0 we have that
E exp
{
λ sup
t∈T
X(t)
}
≤ Z(λ, p, β), (19)
where
Z(λ, p, β) = exp
{
W (λ, p, β) + pϕ
(
λβ
1 − p
)}
×
× exp
⎧⎪⎨
⎪⎩
2λ
p(1 − p)
βp2∫
0
H(σ(−1)(u))
ϕ(−1)(H(σ(−1)(u)))
du + λ
∞∑
k=2
δ
(
σ(−1)
(
βpk−1
))
⎫⎪⎬
⎪⎭ ,
W (λ, p, β) = inf
v≥(1−p)−1
(
1
v
H(σ(−1)(βp)) + sup
u∈T
(
ψ(λγ(u)v)
v
− λf(u)
))
.
Proof. It follows from lemma 3.1 (see inequality (5)) that for all qk > 1, k =
1, 2, . . . such that
∞∑
k=1
1
qk
≤ 1, and all λ > 0 the following inequality holds
true
E exp
{
λ sup
t∈T
X(t)
}
≤ exp
{
λ
∞∑
k=2
δ
(
σ(−1)(βpk−1)
)}
exp
{ ∞∑
k=2
H(εk) + ϕ(λqkβpk−1)
qk
}
×
× exp
{
1
q1
(
H(ε1) + sup
u∈T
(ψ(λq1γ(u)) − λq1f(u))
)}
. (20)
Let q1 = v, where v is such a number that v ≥ 1
1−p
and
qk =
1
λβpk−1
ϕ(−1)
(
ϕ
(
λβ
1 − p
)
+ H(εk)
)
, k = 2, 3 . . . (21)
RANDOM PROCESS FROM THE CLASS V (ϕ,ψ) 227
Since
1
qk
≤ λβpk−1
ϕ(−1)
(
ϕ
(
λβ
1−p
)) = pk−1(1 − p)
as k = 2, 3 . . ., then
∞∑
k=1
1
qk
≤
∞∑
k=1
pk−1(1 − p) = 1.
Consider
Z̃ =
∞∑
k=2
H(εk) + ϕ(λqkβpk−1)
qk
.
For the sequence qk defined in (21) we have
Z̃ =
∞∑
k=2
H(εk)
qk
+
∞∑
k=2
1
qk
ϕ
⎛
⎝λβpk−1
ϕ(−1)
(
ϕ
(
λβ
1−p
)
+ H(εk)
)
λβpk−1
⎞
⎠
=
∞∑
k=2
H(εk)
qk
+
∞∑
k=2
H(εk)
qk
+ ϕ
(
λβ
1 − p
) ∞∑
k=2
1
qk
≤ 2
∞∑
k=2
H(εk)
λβpk−1
ϕ(−1)(H(εk))
+ ϕ
(
λβ
1 − p
) ∞∑
k=2
pk−1(1 − p)
= ϕ
(
λβ
1 − p
)
p + 2λ
∞∑
k=2
H(σ(−1)(βpk))βpk−1
ϕ(−1)(H(σ(−1)(βpk)))
. (22)
The function ϕ(x)
x
increases as x > 0 (see, for example, [1]) therefore the
function x
ϕ(−1)(x)
increases as well. Then
βpk∫
βpk+1
H(σ(−1)(u))
ϕ(−1)(H(σ(−1)(u)))
du ≥ H(σ(−1)(βpk))
ϕ(−1)(H(σ(−1)(βpk)))
βpk(1 − p). (23)
And from (22) and (23) it follows that
Z̃ ≤ ϕ
(
λβ
1 − p
)
p +
2λ
p(1 − p)
βp2∫
0
H(σ(−1)(u))
ϕ(−1)(H(σ(−1)(u)))
du. (24)
Therefore the assertion of the lemma follows from (5) and (24). �
Theorem 3.2. Let Y = {Y (t), t ∈ T} be a separable random process
from the class V (ϕ, ψ) and f = {f(t), t ∈ T} be a continuous function
such that |f(u) − f(v)| ≤ δ(ρ(u, v)), where δ = {δ(s), s > 0} is some
228 ROSTYSLAV YAMNENKO AND OLGA VASYLYK
monotonically increasing nonnegative function, and X(t) = Y (t) − f(t).
Let r2 = {r2(u) : u ≥ 1} be such a continuous function that r2(u) > 0 as
u > 1, r2(1) = 0 and the function s(t) = r2(exp{t}), t ≥ 0, is convex. If
β∫
0
r2(N(σ(−1)(u)))
ϕ(−1)(log N(σ(−1)(u)))
du < ∞ (25)
then for all p ∈ (0; 1) and x > 0 the following inequality holds true
P
{
sup
t∈T
X(t) > x
}
≤ inf
λ>0
Zr2(λ, p, β), (26)
where
Zr2(λ, p, β)
= exp
{
W (λ, p, β) + pϕ
(
λβ
1 − p
)
+ λ
( ∞∑
k=2
δ
(
σ(−1)
(
βpk−1
))− x
)}
×
×
⎛
⎜⎝r
(−1)
2
⎛
⎜⎝ λ
p(1 − p)
βp2∫
0
r2(N(σ(−1)(u)))
ϕ(−1)(log N(σ(−1)(u)))
du
⎞
⎟⎠
⎞
⎟⎠
2
, (27)
W (λ, p, β) = inf
v≥(1−p)−1
(
1
v
H(σ(−1)(βp)) + sup
u∈T
(
ψ(λγ(u)v)
v
− λf(u)
))
.(28)
Proof. Let q1 and qk, k = 2, 3, . . . be defined as in the proof of the lemma 3.2.
It follows from (20) and (22) that for λ > 0, p ∈ (0; 1) and v ≥ 1
1−p
E exp
{
λ sup
t∈T
X(t)
}
≤ exp
{
1
v
H(σ(−1)(u)) + sup
u∈T
(
ψ(λvγ(u))
v
− λf(u)
)
+ λ
∞∑
k=2
δ(σ(−1)(βpk−1)) + pϕ
(
λβ
1 − p
)
+ 2
∞∑
k=2
H
(
σ(−1)(βpk)
)
qk
}
.(29)
From the convexity of the function s(t) = r2(exp{t}) it follows that for all
δi > 0, i ≥ 1, such that
∞∑
i=1
δi = 1 and all xi ≥ 0
s
( ∞∑
i=1
δixi
)
≤
∞∑
i=1
δis(xi).
If
∞∑
i=1
δi < 1 remembering s(0) = 0 we have
s
( ∞∑
i=1
δixi
)
= s
( ∞∑
i=1
δixi + 0(1 −
∞∑
i=1
δi)
)
RANDOM PROCESS FROM THE CLASS V (ϕ,ψ) 229
≤
∞∑
i=1
δis(xi) +
(
1 −
∞∑
i=1
δi
)
s(0) =
∞∑
i=1
δis(xi). (30)
It follows from (30) that
exp
{
2
∞∑
k=2
1
qk
H
(
σ(−1)(βpk)
)}
=
(
r
(−1)
2
(
r
(
exp
{ ∞∑
k=2
q−1
k log N
(
σ(−1)(βpk)
)})))2
≤
(
r
(−1)
2
( ∞∑
k=2
q−1
k s
(
log N
(
σ(−1)(βpk)
))))2
≤
(
r
(−1)
2
(
λ
∞∑
k=2
βpk−1 r2(N(σ(−1)(βpk)))
ϕ(−1)(log N(σ(−1)(βpk)))
))2
. (31)
The function a(t) = r2(exp{ϕ(t)}), t ≥ 0, is a convex function and a(0) = 0,
that is a(t) is an Orlicz function and the function a(t)
t
increases as t > 0 [?].
Therefore the function r2(exp{u})/ϕ(−1)(u) increases as well. Consequently
we have the following inequality
βpk∫
βpk+1
r2(N(σ(−1)(u)))
ϕ(−1)(log N(σ(−1)(u)))
du ≥ r2(N(σ(−1)(βpk)))
ϕ(−1)(log N(σ(−1)(βpk)))
βpk(1 − p)
and
∞∑
k=2
βpk−1 r2(N(σ(−1)(βpk)))
ϕ(−1)(log N(σ(−1)(βpk)))
≤ 1
p(1 − p)
∞∑
k=2
βpk∫
βpk+1
r2(N(σ(−1)(u)))
ϕ(−1)(log N(σ(−1)(u)))
du
≤ 1
p(1 − p)
βp2∫
0
r2(N(σ(−1)(u)))
ϕ(−1)(log N(σ(−1)(u)))
du. (32)
Using the Chebyshev’s inequality the assertion of the theorem follows
from the (29), (31), (32). �
230 ROSTYSLAV YAMNENKO AND OLGA VASYLYK
4. Examples
Definition 4.1.[4] Let ϕ ≺ ψ are two Orlicz N -functions. We call the pro-
cess ZH = (ZH(t), t ∈ T ) generalized fractional Brownian motion from the
class V (ϕ, ψ) with Hurst index H ∈ (0, 1) (V (ϕ, ψ)-GFBM) if ZH is strictly
ψ-sub-Gaussian process with stationary strictly ϕ-sub-Gaussian increments
and covariance function
RH(t, s) = EZH(s)ZH(t) =
1
2
(
t2H + s2H − |s − t|2H
)
. (33)
Theorem 4.1. Let ZH = (ZH(t), t ∈ [a, b]), 0 ≤ a < b < ∞ be a generalized
fractional Brownian motion from the class V (ϕ, ψ) with Hurst index H ∈
(0, 1) and let c > 0 be a constant. Then for all p ∈ (0, 1), β ∈
(
0,
(
b−a
2
)H]
and λ > 0 the following inequlity holds true
P
{
sup
a≤t≤b
(
ZH(t) − ct
)
> x
}
≤ (b − a)
(
e
βp
) 1
H
×
× exp
{
λc(βp)
1
H
CΔ(1 − p
1
H )
+ pϕ
(
λβ
1 − p
)
+ (1 − p)θψ(λ, p) − λx
CΔ
}
, (34)
where θψ(λ, p) = sup
a≤u≤b
(
ψ
(
λuH
1−p
)
− λcu
CΔ(1−p)
)
, CΔ is the constant from defi-
nition 2.5 of the space SSubϕ(Ω).
Proof. Let’s apply theorem 3.1.
P
{
sup
a≤t≤b
(
ZH(t) − ct
)
> x
}
= P
{
sup
a≤t≤b
(Y (t) − Ct) > ε
}
, (35)
where ε = x
CΔ
and C = c
CΔ
. Since
τϕ(Yi(t) − Yi(s)) =
1
CΔ
τϕ(ZH
i (t) − ZH
i (s))
≤ (E(ZH
i (t) − ZH
i (s))2) 1
2 = |t − s|H,
put γ(u) = uH and σ(h) = hH , then 0 ≤ β ≤ ( b−a
2
)H
. Also we have that
|f(u) − f(v)| = |Cu − Cv| = C|u − v|, i.e. δ(h) = Ch. As function r1(u)
let’s choose r1(u) = uα, u ≥ 1, 0 < α < H. Then
θψ(λ, p) = sup
a≤u≤b
(
ψ
(
λuH
1 − p
)
− λCu
1 − p
)
,
∞∑
k=2
δ
(
σ(−1)(βpk−1)
)
=
∞∑
k=2
C(βpk−1)
1
H =
C(βp)
1
H
1 − p
1
H
.
RANDOM PROCESS FROM THE CLASS V (ϕ,ψ) 231
Since
log
(
max
{
b − a
2u
, 1
})
≤ H(u) ≤ ln
(
b − a
2u
+ 1
)
,
then for u ≤
(
b−a
2
)H
the following estimate is fulfilled
r1
(
NB
(
σ(−1)(u)
)) ≤ r1
(
b − a
2σ(−1)(u)
+ 1
)
=
(
b − a
2u
1
H
+ 1
)α
≤ (b − a)α
u
α
H
.
Since βp < β ≤ ( b−a
2
)H
then
r(−1)
(
1
βp
βp∫
0
r
(
NB
(
σ(−1)(u)
))
du
)
≤
(
1
βp
βp∫
0
(b − a)α
u
α
H
du
) 1
α
= (b − a)β− 1
H p−
1
H
(
1 − α
H
)− 1
α
. (36)
Infinum of the right of estimate (36) equals to
lim
α→0
(b − a)β− 1
H p−
1
H
(
1 − α
H
)− 1
α
= (b − a)
(
e
βp
) 1
H
. (37)
Therefore from (35)-(37) we obtain the assertion of the theorem. �
Acknowledgments. The authors thank professor Yuriy Kozachenko for
his supporting and valuable advices.
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232 ROSTYSLAV YAMNENKO AND OLGA VASYLYK
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Department of Probability Theory and Mathematical Statistics,
Kyiv National Taras Shevchenko University, Kyiv, Ukraine
E-mail address: yamnenko@univ.kiev.ua; vasylyk@univ.kiev.ua
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