Distribution of the maximum of the Chentsov random field
Let D = [0, 1]^2 and X(s, t), (s, t) belongs D, be a two-parameter Chentsov random field. The aim of this paper is to find the probability distribution of the maximum of X(s, t) on a class of polygonal lines.
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Інститут математики НАН України
2008
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| Цитувати: | Distribution of the maximum of the Chentsov random field / N. Kruglova // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 1. — С. 76–81. — Бібліогр.: 8 назв.— англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859668784165421056 |
|---|---|
| author | Kruglova, N. |
| author_facet | Kruglova, N. |
| citation_txt | Distribution of the maximum of the Chentsov random field / N. Kruglova // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 1. — С. 76–81. — Бібліогр.: 8 назв.— англ. |
| collection | DSpace DC |
| description | Let D = [0, 1]^2 and X(s, t), (s, t) belongs D, be a two-parameter Chentsov random field. The aim of this paper is to find the probability distribution of the maximum of X(s, t) on a class of polygonal lines.
|
| first_indexed | 2025-11-30T12:30:39Z |
| format | Article |
| fulltext |
Theory of Stochastic Processes
Vol. 14 (30), no. 1, 2008, pp. 76–81
UDC 519.21
NATALIA KRUGLOVA
DISTRIBUTION OF THE MAXIMUM
OF THE CHENTSOV RANDOM FIELD
Let D = [0, 1]2 and X(s, t), (s, t) ∈ D, be a two-parameter Chentsov random field.
The aim of this paper is to find the probability distribution of the maximum of X(s, t)
on a class of polygonal lines.
1. Introduction
Let {X(s, t) : s, t ≥ 0} be a standard Chentsov field of two parameters that is a sepa-
rable real Gaussian stochastic process such that
1) X(0, t) = X(s, 0) = 0 for all s, t ∈ [0, 1];
2) E[X(s, t)] = 0 for all s, t ≥ 0;
3) E[X(s, t)X(s1, t1)] = min{s, s1}min{t, t1} for all (s, t) and (s1, t1) ∈ D.
This definition is given by Yeh [6] in 1960. Another (equivalent) definition is given by
Chentsov [7] in 1955 in terms of the probability density of X(s, t). Yeh showed that the
sample paths of this field are continuous with probability one andX(s, t) has independent
stationary increments in the plane.
The probability distributions of functionals of a Chentsov random field like M =
max(s,t)∈D X(s, t) are not yet known. Some trivial probability distribution theory for
X(s, t) can be obtained by using the known results about the standard Wiener process.
The distribution of the supremum of a Chentsov random field on the curve f(s),
where f(s) is a non-decreasing function of s, can be obtained, since a transformation of
X(s, f(s)) is equivalent to a one-dimensional standard Wiener process.
The probability distribution of the supremum of X(s, t) on the boundary of a unit
square is obtained by Paranjape and Park [1]. This probability is of its own interest, and
it gives a nice lower bound for the probability distribution of the supremum of X(s, t)
over the whole unit square D, which is unknown yet.
Park and Skoug [5] have found the probability that X(s, t) crosses a barrier of the
type ast+ bs+ ct+ d on the boundary ∂Λ, where Λ = [0, S]× [0, T ] is a rectangle. Later
on, I. Klesov [3] considered a probability of the form
(1) P (L, g) = P
{
sup
L
X(s, t)− g(s, t) < 0
}
,
whereX is a Chentsov random field on D = [0, 1]2, L ⊂ D, and g is an almost everywhere
Lebesgue continuous function onD. He presented results, where g(s, t) is a linear function
and L is a polygonal line with one point of break. Klesov and Kruglova [8] considered a
probability of the form (1), where L is a polygonal line with two points of break.
The main purpose of this paper is to evaluate the probability distribution of the form
(1), where g(s, t) = λ and L is a polygonal line with several points of break. we can
Key words and phrases. Two-parameter Chentsov random field, probability distribution, standard
Wiener process.
76
MAXIMUM OF THE CHENTSOV RANDOM FIELD 77
express this distribution in a very useful form: as an expression of the ”tail” of the
two-dimensional Gauss process.
2. Auxiliary results
Lemma 1. (Doob’s Transformation Theorem) [2]. Let X(t) be any Gaussian process
with covariance function R(s, t) = u(s)v(t), s ≤ t, if the ratio a(t) = u(t)/v(t) is con-
tinuous and strictly increasing with inverse a1(t), then w(t) and Y (a1(t))/v(a1(t)) are
stochastically equivalent processes.
Lemma 2. (Malmquist’s Theorem 1) [4]. For a standard Wiener process w(t) and for
b > 0, a ≥ 0, s1 ≤ at′ + b,
P
{
w(t) ≤ at+ b, 0 < t < t
′ |w(t
′
) = s1
}
=
= P
{
w(t) ≤ bt+ (at
′
+ b− s1)/t′ , 0 < t <∞
}
=
= 1− exp
{
−2b(at
′
+ b− s1/t′
}
.
Lemma 3. (Malmquist’s Theorem 2) [4]. For a standard Wiener process w(t) and for
b > 0, a ≥ 0,
P {w(t) ≤ at+ b, x < t ≤ y|w(x) = s1, w(y) = s2} =
= 1− exp
{
− 2R
1−R2
· P1 − s1√
x
· P2 − s2√
y
}
,
where R =
√
x
y , s1 ≤ P1 = ax+ b, s2 ≤ P2 = ay + b.
Let L be a line as shown in Fig. 1 and given by the formula
t
s
A
B
0 1
1
Q
Figure 1
(2) L =
{
(s, t) : sa−1 + t = 1, s ≤ k; s+ tb−1 = 1, s > k, (s, t) ∈ D} ,
where tanα = a, tanβ = b, k = a(b−1)
ab−1 , α, β > π
4 .
78 NATALIA KRUGLOVA
Theorem 1. (Paranjape and Park)[1]. Let {X(s, t) : s, t ≥ 0} be a standard Chentsov
field. Then
(3)
P
{
sup
(s,t)∈L
X(s, t) ≤ λ
}
= Φ
(
λ(a+ c)
a
√
c
)
− exp
{−2λ2
a
}
Φ
(
λ(c− a)
a
√
c
)
−
− exp
{−2λ2
b
}
Φ
{
λ(1 − bc)
b
√
c
}
+ exp
{−2λ2(a−1 + b−1 − 2)
}
× Φ
{
λc−1/2(b−1 − c− 2)
}
.
3. Main results and proofs
Let L be a line as shown in Fig. 2 and given by the formula
t
s
A
B
0 1
1
Figure 2
Q1(x1, y1)
Q2(x2, y2)
(4) L =
⎧⎪⎪⎨⎪⎪⎩
t = 1− s(1−y1)
x1
, s ∈ [0, x1]
t = − s(y1−y2)
x2−x1
+ x2y1−x1y2
x2−x1
, s ∈ (x1, x2]
t = − sy2
1−x2
+ y2
1−x2
, s ∈ (x2, 1].
Theorem 2. Let X(s, t) be a standard Chentsov random field on a unit square. Let
the polygonal line L have two points of break Q1(x1, y1) and Q2(x2, y2) and be given by
formula (4). Let the coordinates of Q1 and Q2 satisfy the conditions
1) y2 < y1;
2) x2
y2
> x1
y1
.
Then
P2 = P
{
sup
(s,t)∈L
X(s, t) < λ
}
×
∫ λ
y1
−∞
∫ λ
y2
−∞
1
2π
√
x1
y1
(
x2
y2
− x1
y1
) exp
{
− u2
1
2x1
y1
}
exp
⎧⎨⎩− (u2 − u1)2
2
(
x2
y2
− x1
y1
)
⎫⎬⎭
×
(
1− exp
{
−2λy1
x1
(
λ
y1
− u1
)})(
1− exp
{
−2λ
(
λ
y2
− u2
)})
MAXIMUM OF THE CHENTSOV RANDOM FIELD 79
(5) ×
(
1− exp
{
−2(λ− u1y1)(λ − u2y2)
(x2y1 − x1y2)
})
du1du2
Corollary 1. Passing to the limit as Q1 −→ Q2 and using (5), we obtain a result which
agrees with Park’s result for a polygonal line with a single point of break (Theorem 1).
Let us denote that x0 = 0, xn+1 = 1, y0 = 1, yn+1 = 0. Let L be a line given by the
formula
(6) L = {(s, t) : t = v(s), s ∈ [0, 1]} .
For which (x1, y1), . . . , (xn, yn) are the points of break where
v(s) =
n+1∑
i=1
(
−s(yi−1 − yi)
xi − xi−1
+
xiyi−1 − xi−1yi
xi − xi−1
)
I(xi−1;xi](s).
Let us denote Δ0 = 0,Δi = xi
yi
, i = 1, n,Δn+1 =∞.
The following theorem is a generalization of Theorem 2.
Theorem 3. Let X(s, t) be a standard Chentsov random field on a unit square. Let
u0 = un+1 = 0. Let the polygonal line L have n points of break and be given by formula
(6). Let the coordinates of these points satisfy the conditions
1) y1 > · · · > yn;
2) x1
y1
< · · · < xn
yn
.
Then
Pn = P
{
sup
(s;t)∈L
X (s; t) < λ
}
=
∫ λ
y1
−∞
. . .
∫ λ
yn
−∞
n∏
i=1
ϕ0,Δi−Δi−1 (ui − ui−1)
×
n+1∏
i=1
⎛⎝1− exp
⎧⎨⎩−2
(
λ
yi−1
− ui−1
)(
λ
yi
− ui
)
(Δi −Δi−1)
⎫⎬⎭
⎞⎠du1 . . . dun
where ϕ0,Δ(u) is the density of the Gaussian random variable with variance Δ.
Proof. Let the restriction of X(s, t) over L be denoted by w1(s). Then
w1(s) = X (s, v(s))
Let us find the derivation of v(s).
v′(s) =
n+1∑
i=1
− (yi−1−yi)
xi−xi−1
I(xi−1;xi](s) < 0 because of conditions over coordinates of points.
This means that v(s) is monotone decreasing function.
Using the covariance property of X(s, t), we can write
cov(w1(s1), w1(s2)) = cov (X(s1, v(s1), X(s2, v(s2)) = s1v(s2), 0 < s1 � s2 � 1
a(s) = s
v(s) is continuous monotone increasing function. We can write a(s) in an explicit
form:
a(s) =
n+1∑
i=1
s
− s(yi−1−yi)
xi−xi−1
+ xiyi−1−xi−1yi
xi−xi−1
I(xi−1,xi](s)
It is enough to prove a continuity a(s) in the points xi, i = 1, n:
a(xi) =
xi
−xi(yi−1−yi)
xi−xi−1
+ xiyi−1−xi−1yi
xi−xi−1
=
xi
yi
.
80 NATALIA KRUGLOVA
a(xi+) =
xi
−xi(yi−yi+1)
xi+1−xi
+ xi+1yi−xiyi+1
xi+1−xi
=
xi
yi(xi+1−xi)
xi+1−xi
=
xi
yi
.
That is a(xi) = a(xi+) continuous in the point xi. That is a(s) continuous in (0; 1). s
is a monotone increasing function and v(s) is a monotone decreasing function. That is
why a(s) is a monotone increasing function. For a(s) the inverse will be the function:
a−1(s) =
n+1∑
i=1
s(xiyi−1 − xi−1yi)
s(yi−1 − yi) + xi − xi−1
I[Δi−1,Δi).
It is necessary to notice that v(0) = −x0(y0−y1)
x1−x0
+ x1y0−x0y1
x1−x0
= y0 = 1 and v(1) =
−xn+1(yn−yn+1)
xn+1−xn
+ xn+1yn−xnyn+1
xn+1−xn
= yn+1 = 0. That is why a(0) = 0 and lim
t→1
a(t) =∞.
1
v(a−1(s))
=
n+1∑
i=1
(
s(yi−1 − yi) + xi − xi−1
xiyi−1 − xi−1yi
)
I[Δi−1,Δi)(s)
The functions a(s) and v(·) satisfy the conditions of Doob’s transformation theorem.
Thus,
w∗(s) =
n+1∑
i=1
(
s(yi−1 − yi) + xi − xi−1
xiyi−1 − xi−1yi
)
×w1
(
s(xiyi−1 − xi−1yi)
s(yi−1 − yi) + xi − xi−1
)
I[Δi−1,Δi)(s)
and w(t) are stochastically equivalent processes.
Pn(λ) = P
{
sup
(s;t)∈L
X (s; t) < λ
}
= P
{
sup
s∈[0,1]
X (s; v(s)) < λ
}
= P
{
sup
s∈[0,1]
w1(s) < λ
}
= P
{
sup
s∈[0,∞)
w1
(
a−1(s)
)
< λ
}
= P
⎛⎝⋂
s�0
{
w1
(
a−1(s)
)
< λ
}⎞⎠
= P
(⋂
s>0
{
w1
(
a−1(s)
)
< λ
}⋂{
X
(
a−1(0), v
(
a−1(0)
))
< λ
})
= P
(⋂
s>0
{
w1
(
a−1(s)
)
< λ
}⋂
Ω
)
Because X
(
a−1(0), v
(
a−1(0)
))
= X(0, 1) = 0 and that is why{
X
(
a−1(0), v
(
a−1(0)
))
< λ
}
= Ω
Pn = P
(⋂
s>0
{
w1
(
a−1(s)
)
< λ
})
= P
{⋂
s>0
w1
(
a−1(s)
)
v (a−1(s))
− λ
v (a−1(s))
< 0
}
=
= P
{
sup
s∈(0,∞)
w1
(
a−1(s)
)
v (a−1(s))
− λ
v (a−1(s))
< 0
}
= P
{
sup
s∈(0,∞)
w(t) − λ
v (a−1(s))
< 0
}
=
= P
{
w(t) <
λ(xi − xi−1)
xiyi−1 − xi−1yi
+
λt(yi−1 − yi)
xiyi−1 − xi−1yi
; t ∈ (Δi−1; Δi] , i = 1, n+ 1
}
MAXIMUM OF THE CHENTSOV RANDOM FIELD 81
=
∫ λ
y1
−∞
. . .
∫ λ
yn
−∞
1
(2π)n/2
P
{
w(s) < λ+
(1− y1)sλ
x1
, s ∈ (0; Δ1] |w(Δ1) = u1
}
×
×
n∏
i=2
P
{
w(t) <
λ(xi − xi−1)
xiyi−1 − xi−1yi
+
λt(yi−1 − yi)
xiyi−1 − xi−1yi
; t ∈ (Δi−1; Δi]∣∣∣ w(Δi−1) = ui−1, w(Δi) = ui
}
×P
{
w(t) <
λ(1− xn)
yn
+ λt, t > Δn|w(Δn) = un
}
×
n∏
i=1
ϕ0,Δi−Δi−1(ui − ui−1)√
Δi −Δi−1
dui.
Then, by using Lemma 2 and Lemma 3, we get
Pn = P
{
sup
(s;t)∈L
X (s; t) � λ
}
=
∫ λ
y1
−∞
. . .
∫ λ
yn
−∞
n+1∏
i=1
⎛⎝1− exp
⎧⎨⎩−2
(
λ
yi−1
− ui−1
)(
λ
yi
− ui
)
(Δi −Δi−1)
⎫⎬⎭
⎞⎠
×
n∏
i=1
ϕ0,Δi−Δi−1 (ui − ui−1)du1 . . . dun
Bibliography
1. S.R. Paranjape and C. Park, Distribution of the supremum of the two-parameter Yeh-Wiener
process on the boundary, Appl. Probab. 10 (1973), no. 4, 875–880.
2. J.L. Doob, Heuristic approach to Kolmogorov–Smirnov theorems, Ann. Math. Statist. 20
(1949), 393–403.
3. I.I. Klesov, On the probability of attainment of a curvilinear level by a Wiener field, Probab.
and Math. Statist. 51 (1995), 63–67.
4. S. Malmquist, On certain confidence contours for distribution functions, Ann. Math. Statist.
25 (1954), 523–533.
5. C. Park and D.L. Skoug, Distribution estimates of barrier-crossing probabilities of the Yeh-
Wiener process, Pacific Journal of Mathematics 78 (1978), 455–466.
6. J. Yeh, Wiener Measure in a Space of Functions of Two Variables, Transactions of the American
Mathematical Society 95 (1960), 433–450.
7. N.N. Chentsov, Wiener random fields depending on several parameters, Doklady AN SSSR 46
(1956), no. 4, 607–609. (Russian)
8. O.I. Klesov and N.V. Kruglova, Distribution of the maximum of the two-parameter Chentsov
random field, Naukovi visti NTUU ”KPI” 4 (2007), 136–141. (Ukrainian)
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|
| id | nasplib_isofts_kiev_ua-123456789-4538 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0321-3900 |
| language | English |
| last_indexed | 2025-11-30T12:30:39Z |
| publishDate | 2008 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Kruglova, N. 2009-11-25T11:04:54Z 2009-11-25T11:04:54Z 2008 Distribution of the maximum of the Chentsov random field / N. Kruglova // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 1. — С. 76–81. — Бібліогр.: 8 назв.— англ. 0321-3900 https://nasplib.isofts.kiev.ua/handle/123456789/4538 519.21 Let D = [0, 1]^2 and X(s, t), (s, t) belongs D, be a two-parameter Chentsov random field. The aim of this paper is to find the probability distribution of the maximum of X(s, t) on a class of polygonal lines. en Інститут математики НАН України Distribution of the maximum of the Chentsov random field Article published earlier |
| spellingShingle | Distribution of the maximum of the Chentsov random field Kruglova, N. |
| title | Distribution of the maximum of the Chentsov random field |
| title_full | Distribution of the maximum of the Chentsov random field |
| title_fullStr | Distribution of the maximum of the Chentsov random field |
| title_full_unstemmed | Distribution of the maximum of the Chentsov random field |
| title_short | Distribution of the maximum of the Chentsov random field |
| title_sort | distribution of the maximum of the chentsov random field |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/4538 |
| work_keys_str_mv | AT kruglovan distributionofthemaximumofthechentsovrandomfield |