The First Passage Time and Estimation of the Number of Level-Crossings for a Telegraph Process
It is a well-known result that almost all sample paths of a Brownian motion or Wiener process {W(t)} have infinitely many zero-crossings in the interval (0, δ) for δ > 0. Under the Kac condition, the telegraph process weakly converges to the Wiener process. We estimate the number of interse...
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| author | Kolomiets, T. Pogorui, A. О. Rodriguez-Dagnino, R. M. Коломиєць, Т. Погоруй, А. О. Родріжес-Дагніно, Р.М. |
| author_facet | Kolomiets, T. Pogorui, A. О. Rodriguez-Dagnino, R. M. Коломиєць, Т. Погоруй, А. О. Родріжес-Дагніно, Р.М. |
| author_sort | Kolomiets, T. |
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| datestamp_date | 2019-12-05T09:49:13Z |
| description | It is a well-known result that almost all sample paths of a Brownian motion or Wiener process {W(t)} have infinitely many zero-crossings in the interval (0, δ) for δ > 0. Under the Kac condition, the telegraph process weakly converges to the Wiener process. We estimate the number of intersections of a level or the number of level-crossings for the telegraph process. Passing to the limit under the Kac condition, we also obtain an estimate of the level-crossings for the Wiener process. |
| first_indexed | 2026-03-24T02:17:22Z |
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UDC 517.9
A. A. Pogorui (Zhytomyr State Univ., Ukraine),
R. M. Rodrı́guez-Dagnino (Electrical and Computer Engineering, Tecnológico de Monterrey, México),
T. Kolomiiets (Zhytomyr State Univ., Ukraine)
THE FIRST PASSAGE TIME AND AN ESTIMATE OF THE NUMBER
OF LEVEL-CROSSINGS FOR A TELEGRAPH PROCESS
ЧАС ПЕРШОГО ДОСЯГНЕННЯ ТА ОЦIНКА ЧИСЛА ПЕРЕТИНIВ РIВНЯ
ДЛЯ ТЕЛЕГРАФНИХ ПРОЦЕСIВ
It is a well-known result that almost all sample paths of Brownian motion or Wiener process {W (t)} have infinitely many
zero-crossings in the interval (0, δ) for δ > 0. Under the Kac condition, the telegraph process weakly converges to the
Wiener process. We estimate the number of intersections of a level or the number of level-crossings for the telegraph
process. Passing to the limit under the Kac condition, we also obtain an estimate for the level-crossings for the Wiener
process.
Вiдомо, що майже всi вибiрковi траєкторiї броунiвського руху чи вiнерiвського процесу {W (t)} мають нескiнченно
багато нульових перетинiв в iнтервалi (0, δ) при δ > 0. За умови Каца телеграфний процес слабко збiгається
до вiнерiвського процесу. В роботi оцiнюється число перетинiв рiвня для телеграфного процесу. Переходячи до
границi за умови Каца, ми також отримуємо оцiнку перетинiв рiвня для вiнерiвського процесу.
1. Introduction. Let us set the probability space (Ω,F ,P). On the phase space T = {0, 1} consider
an alternating Markov stochastic process {ξ(t), t ≥ t}, having the sojourn time τi corresponding to
the state x = i ∈ T, and generating matrix
Q = λ
[
−1 1
1 −1
]
.
Denote by {x(t), t ≥ 0} the associated telegraph process. Then
d
dt
x(t) = v(−1)ξ(t), v = constant > 0,
and x(0) = x0.
2. Distribution of the first passage time. In this section we will find the explicit mathematical
form for the distribution of the first passage time of a specific level L of a telegraph process on the
real line.
Let assume a fixed level L, then let us define ∆(t) = L − x(t). Furthermore, we assume that
z = L− x0 > 0.
Suppose ξ(0) = k and define
Tk(z) = inf {t ≥ 0 : ∆(t) = 0}, k ∈ {0, 1},
i.e., Tk(z) is the first passage time of the level L by the telegraph process {x(t)} after assuming
ξ(0) = k.
Now, let us denote as fk(t, z) dt = P (Tk(z) ∈ dt) the probability density function (pdf) of Tk(z).
c© A. A. POGORUI, R. M. RODRIGUEZ-DAGNINO, T. KOLOMIIETS, 2015
882 ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 7
THE FIRST PASSAGE TIME AND AN ESTIMATE OF THE NUMBER OF LEVEL-CROSSINGS . . . 883
Theorem 1. For t ≥ z
v
f0(t, z) = e−λt
δ(z − vt) +
λz
v2
I1
(
λ
v
√
v2t2 − z2
)
√
v2t2 − z2
,
f1(t, z) = e−λt
I1
(
λ
(
t− z
v
))
t− z
v
+ λz
t∫
z/v
I1(λ(t− u))I1
(
λ
v
√
v2u2 − z2
)
(t− u)
√
v2u2 − z2
du
,
where I1(·) is the modified Bessel function of the first kind.
Proof. Consider the Laplace transform of Tk(z), k ∈ {0, 1},
ϕk(s, z) = E
[
e−sTk(z)
]
, s ≥ 0,
and by using renewal theory concepts we can obtain the following system of integral equations for
these Laplace transforms, i.e.,
ϕ0(s, z) = e−
s+λ
v z +
λ
v
z∫
0
e−
s+λ
v uϕ1(s, z − u) du =
= e−
s+λ
v z
1 +
λ
v
z∫
0
e
s+λ
v uϕ1(s, u) du
,
ϕ1(s, z) =
λ
v
e
s+λ
v z
∞∫
z
e−
s+λ
v uϕ0(s, u) du.
We then differentiate these two equations to obtain the following system:
∂
∂z
ϕ0(s, z) = −s+ λ
v
ϕ0(s, z) +
λ
v
ϕ1(s, u),
∂
∂z
ϕ1(s, z) =
s+ λ
v
ϕ1(s, z)−
λ
v
ϕ0(s, u).
It is well-known, Pogorui and Rodrı́guez-Dagnino [1], that this set of equations relating ϕ0(s, u)
and ϕ1(s, u) can be represented as
Mf = 0,
where
M =
∂
∂z
+
s+ λ
v
−λ
v
λ
v
∂
∂z
− s+ λ
v
.
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 7
884 A. A. POGORUI, R. M. RODRIGUEZ-DAGNINO, T. KOLOMIIETS
Then f(z) satisfies the following equation:
(Det(M))f = 0,
where Det(M) is the determinant of the matrix M.
By calculating the determinant, we have
∂2
∂z2
f(z)− s2 + 2λs
v2
f(z) = 0.
The solution of this equation has the form
f(z) = C1 e
√
s2+2λs
z
v + C2 e
−
√
s2+2λs
z
v .
The constantsC1 andC2 are obtained from the conditions imposed on the system of integral equations,
and we can obtain the solutions
ϕ0(s, z) = e−
z
v
√
s(s+2λ) (1)
and
ϕ1(s, z) =
s+ λ−
√
s(s+ 2λ)
λ
e−
z
v
√
s(s+2λ). (2)
The inverse Laplace transform of ϕ0(s, z) with respect to s yields the following (generalized) pdf
(see [2, p. 239], formula 88):
f0(t, z) = L−1
(
e−
z
v
√
s(s+2λ), t
)
=
= e−λtδ(z − vt) + zλe−λt
I1
(
λ
v
√
v2t2 − z2
)
√
v2t2 − z2
, t ≥ z
v
. (3)
Hence,
P (T0(z) ∈ dt) = e−λtδ(z − vt) dt+ zλe−λt
I1
(
λ
v
√
v2t2 − z2
)
√
v2t2 − z2
dt, t ≥ z
v
.
Similarly, the inverse Laplace transform of the first term of ϕ1(s, z) can be obtained from Bateman
and Erdélyi [3, p. 237] and [4, p. 284] (formula (11))
L−1
(
s+ λ−
√
s(s+ 2λ)
λ
, t
)
=
1
λ
L−1
(
(s+ λ−
√
s(s+ 2λ)), t
)
=
e−λt
t
I1(λt). (4)
The inverse Laplace transform of ϕ1(s, z) is just the convolution of Eqs. (3) and (4). Thus, the
pdf f1(t, z) is given by
f1(t, z) =
t∫
z/v
e−λ(t−u)
(t− u)
I1(λ(t− u))
e−λuδ(z − vu) + zλe−λu
I1
(
λ
v
√
v2u2 − z2
)
√
v2u2 − z2
du =
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 7
THE FIRST PASSAGE TIME AND AN ESTIMATE OF THE NUMBER OF LEVEL-CROSSINGS . . . 885
=
e−λt
t− z
v
I1
(
λ
(
t− z
v
))
+ zλe−λt
t∫
z/v
I1(λ(t− u)) I1
(
λ
v
√
v2u2 − z2
)
(t− u)
√
v2u2 − z2
du
for t ≥ z
v
.
Theorem 1 is proved.
3. Estimation of the number of level-crossings for a telegraph process. We denote as Ck(t, z)
the number of intersections of level z made by the particle x(t) during the time interval (0, t), t > 0,
assuming that ξ(0) = k ∈ {0, 1}. We consider the renewal function Hk(t, z) = E[Ck(t, z)].
Now, let us consider the so-called Kac’s condition (or the hydrodynamic limit), i.e., let λ = ε−2,
v = cε−1, then as ε→ 0 (or λ→∞ and v →∞) we have that
v2
λ
→ c2.
It was proved in [5] that, under Kac’s condition, the telegraph process x(t) weakly converges to
the Wiener process W (t) which is normal distributed as N(0, ct).
Theorem 2. Under Kac’s condition we have
lim
λ→∞
=
Hk(t, 0)√
λ
= lim
v→∞
cHk(t, 0)
v
= lim
ε→0
εcHk(t, 0) = c
√
2
π
√
t.
Proof. The Laplace transform of the general renewal function will be used in this proof, see the
seminal book on this subject Cox [6]. It follows from (2) that the Laplace transform Ĥ1(s, 0) =
= L(H1(t, 0), t) of H1(t, 0) with respect to t has the form
Ĥ1(s, 0) =
1
s
∞∑
k=0
(
s+ λ−
√
s(s+ 2λ)
λ
)k
=
λ
s
√
s(s+ 2λ)− s2
.
It is not hard to verify that
E[C1(t, 0)] = L−1
(
λ
s
√
s(s+ 2λ)− s2
)
=
=
1
2
+
((
1
2
+ λt
)
I0(2λt) + λtI1(2λt)
)
e−λt, t ≥ 0.
Hence,
lim
λ→∞
E[C1(t, 0)]√
λ
=
√
2
π
√
t.
Theorem 2 is proved.
Taking into account λ = ε−2, v = cε−1, we have
H1(t, 0) =
√
λ
2
π
√
t = c
√
2
π
√
t v
as v →∞.
Therefore, for a fixed t > 0 the number of crossings of level 0 by the telegraph process, under
Kac’s condition, goes to∞ as the velocity v.
We should notice that the result of Theorem 2 is in correspondence with results of M. I. Portenko [7].
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 7
886 A. A. POGORUI, R. M. RODRIGUEZ-DAGNINO, T. KOLOMIIETS
Now, let us denote as
F1(x) =
x∫
0
f1(t, 0) dt =
x∫
0
e−λt
t
I1(λt) dt.
Theorem 3.
P
1−
λx∫
0
e−u
u
I1(u) du
C1(x, 0) ≥ 3
√
y
→ G1/2(y)
as λ → ∞. The cumulative distribution function (cdf) G1/2(y) is the one-sided stable distribution
satisfying the condition y1/2[1−G1/2(y)]→ 3 as y →∞ [8].
Proof. It is easily seen that
lim
x→∞
√
x(1− F1(x)) = lim
x→∞
√
x
1−
λx∫
0
e−2λt
t
I1(2λt) dt
=
= 2 lim
x→∞
e−2λxx1/2I1(2λx) =
1√
λπ
.
Therefore, L(x) =
√
x(1− F1(x)) is slowly varying and
1− F1(x) = x−1/2L(x).
Since F1(x) =
∫ x
0
e−λt
t
I1(λt) dt =
∫ λx
0
e−u
u
I1(u) du and λ → ∞ we have F1(x) = F (s) =
= s−1/2L(s), where for a fixed x > 0, s = λx→∞. By using a result in [8, p. 373] (Chapter XI.5),
we obtain
P
1−
λx∫
0
e−u
u
I1(u) du
C1(x, 0) ≥ 3
√
y
→ G1/2(y)
as λ → ∞, where the cdf G1/2(y) is the one-sided stable distribution satisfying the condition
y1/2[1−G1/2(y)]→ 3 as y →∞.
Theorem 3 is proved.
Therefore, under Kac’s condition the number of crossing of a level by the telegraph process, i.e.,
C1(x, 0), is of the order of magnitude
√
λ = v.
4. Estimation of the number of level-crossings in higher dimensions. Firstly, let us consider
the following modification of a telegraph process. Suppose {θk, k ≥ 1} is a sequence of independent
identically distributed exponential random variables with common rate λ > 0 and τn =
∑n
k=1
θk,
n ≥ 1. A particle starts its motion on a line from the origin and moves in one of two directions with
probability 1/2 during the random time θ1. At epoch τn, n ≥ 1, the particle chooses one of two
directions on the line with probability 1/2, and keeps moving along this direction with velocity v.
By using the notation stated in Section 1, for this process we have
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 7
THE FIRST PASSAGE TIME AND AN ESTIMATE OF THE NUMBER OF LEVEL-CROSSINGS . . . 887
ϕ0(s, z) = e−
s+λ
v z +
λ
2v
z∫
0
e−
s+λ
v u[ϕ0(s, z − u) + ϕ1(s, z − u)] du =
= e−
s+λ
v z
1 +
λ
2v
z∫
0
e
s+λ
v u[ϕ0(s, u) + ϕ1(s, u)] du
,
ϕ1(s, z) =
λ
2v
e
s+λ
v z
∞∫
z
e−
s+λ
v u[ϕ0(s, u) + ϕ1(s, u)] du.
We then differentiate these two equations to obtain the following system:
∂
∂z
ϕ0(s, z) = −s+ λ/2
v
ϕ0(s, z) +
λ
2v
ϕ1(s, u),
∂
∂z
ϕ1(s, z) =
s+ λ/2
v
ϕ1(s, z)−
λ
2v
ϕ0(s, u).
Much in the same manner as we obtained Eqs. (1), (2)
ϕ0(s, z) = e−
z
v
√
s(s+λ),
and
ϕ1(s, z) =
2s+ λ− 2
√
s(s+ λ)
λ
e−
z
v
√
s(s+λ).
Similarly to the developments in Section 3, we obtain
E[Ck(t, 0)] = L−1
(
λ/2
s
√
s(s+ λ)− s2
)
=
=
1
2
+
1
2
((1 + λt) I0(λt) + λtI1(λt)) e
−λt/2, t ≥ 0. (5)
Remark 1. We should note that E[Ck(t, 0)] does not depend on the particle’s velocity v.
It follows from Eq. (5) that
lim
λ→∞
E[Ck(t, 0)]√
λ
=
√
2
π
√
t.
In Kac’s condition we have λ = ε−2, v = ε−1, then
E[Ck(t, 0)] ∼
√
2
π
√
t v as ε→ 0.
Let {ν(t), t ≥ 0} be a renewal process such that ν(t) = max{m ≥ 0 : τm ≤ t}, where
τm =
∑m
k=1
θk, τ0 = 0 and θk ≥ 0, k = 1, 2, . . . , are nonnegative iid random variables denoting the
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 7
888 A. A. POGORUI, R. M. RODRIGUEZ-DAGNINO, T. KOLOMIIETS
interarrival times. We assume that these random variables have a cdf G(t) and that there exists the
pdf g(t) =
d
dt
G(t). In most of this paper, we have considered exponentially distributed interrenewal
times, i.e., g(t) = λe−λtI{t≥0}.
We will study the random motion of a particle that starts from the coordinate origin 0 =
= (0, 0, . . . , 0) of the space Rn, at time t = 0, and continues its motion with a velocity v > 0
along the direction η
(n)
0 , where η
(n)
i = (xi1, xi2, . . . , xin) = (x1, x2, . . . , xn), i = 0, 1, 2, . . . , are iid
random n-dimensional vectors uniformly distributed on the unit sphere Ωn−1
1 = {(x1, x2, . . . , xn) :
x21 + x22 + . . .+ x2n = 1}.
At instant τ1 the particle changes its direction to η
(n)
1 = (x11, x12, . . . , x1n), and the particle
continues its motion with a velocity v > 0 along the direction of η(n)
1 . Then, at instant τ2 the particle
changes its direction to η
(n)
2 = (x21, x22, . . . , x2n), and continues its motion with a velocity v along
the direction of η(n)
2 , and so on.
Denote by X(n)(t), t ≥ 0, the particle position at time t. We have that
X(n)(t) = v
ν(t)∑
j=1
η
(n)
j−1 (τj − τj−1) + v η
(n)
ν(t) (t− τν(t)). (6)
Basically, Eq. (6) determines the random evolution in the semi-Markov (or renewal) medium ν(t).
Thus, ν(t) denotes the number of velocity alternations occurred in the interval (0, t).
The probabilistic properties of the random vector X(n)(t) are completely determined by those of
its projection X(n)(t) = v
∑m
j=1 η
(n)
j−1(τj − τj−1) + v η
(n)
ν(t)(t− τν(t)) on a fixed line, where η(n)j is the
projection of η(n)
j on the line.
Indeed, let us consider the cdf FX(y) = P
(
X(n)(t) ≤ y
)
. Then, the characteristic function
(Fourier transform) H(t, α) = H(t) of X(n)(t), where α = ‖α‖ =
√
α2
1 + α2
2 + . . .+ α2
n, is given
by
H(t) = E
[
exp
{
i
(
α,X(n)(t)
)}]
= E
[
exp
{
i ‖α‖
(
e,X(n)(t)
)}]
=
= E
[
exp
{
i αX(n)(t)
}]
=
∞∫
0
exp {i αy} dFX(y),
where X(n)(t) is the projection of X(n)(t) onto the unit vector e and it has a cdf FX(y).
Let us denote by fη(n)(x) the pdf of the projection η(n)j of the vector η(n)
j onto a fixed line. It is
shown in [9] that fη(n)(x) is of the following form:
fη(n)(x) =
Γ
(n
2
)
√
π Γ
(
n− 1
2
) (1− x2)(n−3)/2, if x ∈ [−1, 1],
0, if x /∈ [−1, 1].
Hence, it is not hard to verify that the cdf Gn(t) = P [vη
(n)
i θi ≤ t] is of the form
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 7
THE FIRST PASSAGE TIME AND AN ESTIMATE OF THE NUMBER OF LEVEL-CROSSINGS . . . 889
Gn(t) =
1
2
+
Γ
(n
2
)
√
π Γ
(
n− 1
2
) ∫ 1
0
λe−
λt
vx (1− x2)(n−3)/2dx, if t ≥ 0,
1
2
−
Γ
(n
2
)
√
π Γ
(
n− 1
2
) ∫ 1
0
λe
λt
vx (1− x2)(n−3)/2dx, if t < 0.
Denote by Ck(t, 0) the number of crossing the hyperplane H = {x1 = c = constant} of the
space Rn = {(x1, x2, . . . , xn)}, xi ∈ R, under the condition that the stochastic process starts from
the hyperplane H. The number Ck(t, 0) is also equal to the number of crossing of the level x1 = c
by the projection X(n)(t) of X(n)(t) on the line ` = {(t, 0, . . . , 0)} such that t ∈ R.
In accordance with Remark 1 the mean value E[Ck(t, 0)] does not depend on the particle’s
velocity v.
Therefore,
E[Ck(t, 0)] =
1
2
+
1
2
((1 + λt)I0(λt) + λt I1(λt)) e
−λ2 t.
Under Kac’s condition we have λ = ε−2, v = ε−1, then
E[Ck(t, 0)] ∼
√
2
π
√
t v as ε→ 0.
It is well-known that under Kac’s condition X(n)(t) weakly converges to an n-dimensional Wiener
process W(t) [10].
Acknowledgments. We thank ITESM for the support provided in the development of this work.
The authors also warmly thank Professor M. I. Portenko for having posed the problem.
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1961.
3. Bateman H., Erdélyi A. Higher transcendental functions. – New York: McGraw-Hill Book Co., 1953 – 1955.
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Received 13.08.14
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 7
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| id | umjimathkievua-article-2031 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:17:22Z |
| publishDate | 2015 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/a3/916e5d6c371b1256bdd176253b49c4a3.pdf |
| spelling | umjimathkievua-article-20312019-12-05T09:49:13Z The First Passage Time and Estimation of the Number of Level-Crossings for a Telegraph Process Час першого досягнення та оцінка числа перетинів рівня для телеграфних процесів Kolomiets, T. Pogorui, A. О. Rodriguez-Dagnino, R. M. Коломиєць, Т. Погоруй, А. О. Родріжес-Дагніно, Р.М. It is a well-known result that almost all sample paths of a Brownian motion or Wiener process {W(t)} have infinitely many zero-crossings in the interval (0, δ) for δ > 0. Under the Kac condition, the telegraph process weakly converges to the Wiener process. We estimate the number of intersections of a level or the number of level-crossings for the telegraph process. Passing to the limit under the Kac condition, we also obtain an estimate of the level-crossings for the Wiener process. Відомо, що майже всі ви6іркові траєкторії броунівського руху чи вінєрівського процесу {W(t) мають нескінченно багато нульових перетинів в інтервалі (0, δ) при δ > 0. За умови Каца телеграфний процес слабко збігається до вінерівського процесу. В роботі оцінюється число перетинів рівня для телеграфного процесу. Переходячи до границі за умови Каца, ми також отримуємо оцінку перетинів рівня для вінерівського процесу. Institute of Mathematics, NAS of Ukraine 2015-07-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2031 Ukrains’kyi Matematychnyi Zhurnal; Vol. 67 No. 7 (2015); 882-889 Український математичний журнал; Том 67 № 7 (2015); 882-889 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2031/1080 https://umj.imath.kiev.ua/index.php/umj/article/view/2031/1081 Copyright (c) 2015 Kolomiets T.; Pogorui A. О.; Rodriguez-Dagnino R. M. |
| spellingShingle | Kolomiets, T. Pogorui, A. О. Rodriguez-Dagnino, R. M. Коломиєць, Т. Погоруй, А. О. Родріжес-Дагніно, Р.М. The First Passage Time and Estimation of the Number of Level-Crossings for a Telegraph Process |
| title | The First Passage Time and Estimation of the Number of Level-Crossings for a Telegraph Process |
| title_alt | Час першого досягнення та оцінка числа перетинів рівня для телеграфних процесів |
| title_full | The First Passage Time and Estimation of the Number of Level-Crossings for a Telegraph Process |
| title_fullStr | The First Passage Time and Estimation of the Number of Level-Crossings for a Telegraph Process |
| title_full_unstemmed | The First Passage Time and Estimation of the Number of Level-Crossings for a Telegraph Process |
| title_short | The First Passage Time and Estimation of the Number of Level-Crossings for a Telegraph Process |
| title_sort | first passage time and estimation of the number of level-crossings for a telegraph process |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2031 |
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