Ukrainian Mathematical Journal, Vol. 61, No. 12, 2009
LIMITING DISTRIBUTION OF FADING EVOLUTION IN SOME SEMI-MARKOV MEDIA A. Pogorui 1 and R. M. Rodriguez-Dagnino2
UDC 519.21
We consider fading evolutions on a line. The limiting distribution of a fading evolution in the case where the driving process has the uniform distribution of sojourn times is obtained. In addition, we study the limiting distributions for stable distributed sojourn times of the driving renewal process.
Introduction In 1951, in his seminal paper [1], Goldstein introduced the telegraph stochastic process, which is a random motion driven by a homogeneous Poisson process. This model describes the evolution of a particle on a line. The telegraph process has been extended in many ways since then (see [2–4] and references therein). In this paper, we study the fading evolution, which was introduced in [5]. In contrast to the Goldstein – Kac model, in the fading evolution a particle moves along a straight line under the action of external forces, say, gravity or magnetic fields. As a result, the particle drifts to a certain point where it freezes as t → + ∞. In [5], the Markov fading evolution (the case where the fading random motion is driven by a homogeneous Poisson process) was investigated. In [6], the 2-Erlang semi-Markov case was investigated. In the present paper, we study the fading evolution in a special semi-Markov medium where sojourn times have the uniform or stable distribution. 1. Limiting Distribution of Fading Evolution in the Uniform Case Let ξ ( t ) be the following renewal process: ξ ( t ) = max { n ≥ 0 : τn ≤ t }, t > 0, where n
τn =
∑ θk
k=0
and θk ≥ 0, k = 0, 1, 2, … , are independent identically distributed random variables with general cumulative probability function Gθ (t ) . Suppose that the following density exists: gθ ( t ) =
dGθ (t ) . dt
In this case, the process 1 Zhitomir University, Zhitomir, Ukraine. 2 Monterrey Institute of Technology, Monterrey, Mexico.
Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 61, No. 12, pp. 1720 – 1724, December, 2009. Original article submitted April 17, 2009. 2016
0041–5995/09/6112–2016
© 2009
Springer Science+Business Media, Inc.
L IMITING DISTRIBUTION OF FADING EVOLUTION
IN
SOME SEMI-MARKOV MEDIA t
x (t ) =
∫ (− a)
ξ( s )
ds ,
2017
0 < a < 1,
0
is a random evolution in the semi-Markov medium ξ ( t ). Our purpose is to calculate the limiting distribution of x ( t ) for some special cases of the driving process. Consider the following random variable: ∞
σ =
∫ (− a)
ξ( s )
ds .
0
Since the random variables θk are identically distributed and σ = (θ1 + a 2θ3 + …) − a(θ 2 + a 2θ 4 + …) , for the determination of the cumulative distribution function Fσ (x ) = P { σ ≤ x } it is sufficient to obtain the distribution of the random variable η = θ1 + a 2θ3 + … . Indeed, it is easy to see that Fσ (x ) = P { σ ≤ x } = P{(θ1 + a 2θ3 + …) − a(θ 2 + a 2θ 4 + …) ≤ x} (θ + a 2θ3 + …) − x ⎪⎧ = P ⎨ (θ 2 + a 2θ 4 + …) ≥ 1 a ⎪⎩
⎪⎫ ⎬ = ⎪⎭
∞
⎡
⎛ y−x⎞ ⎤ . a ⎟⎠ ⎥⎦
∫ dFη( y)) ⎢⎣ 1 − Fη ⎜⎝ x
For exponential Gθ ( t ), the function Fσ (x ) was studied in [5], and in the case of 2-Erlang Gθ ( t ) the function Fσ (x ) was studied in [6]. Denote Fη (x ) = P { η ≤ x }. Then Fη (x ) = P{θ1 + a 2 η′ ≤ x } =
∞
2 ∫ gθ (u) P{u + a η′ ≤ x} du = 0
∞
⎧
∫ gθ (u) P ⎨⎩ η′ 0
≤
x−u⎫ ⎬ du , a2 ⎭
where η′ and η are identically distributed. Since η ≥ 0, we have x
Fη ( x ) =
⎧x−u⎫ ⎬ du . a2 ⎭
∫ gθ (u)Fη ⎨⎩ 0
(1)
Consider the case where θ is uniformly distributed on [ 0, 1 ] . In this case, Eq. (l) takes the form x
Fη ( x ) =
⎧x−u⎫ ⎬ du , a2 ⎭
∫ Fη ⎨⎩ 0
0 ≤ x ≤ 1, (2)
1
Fη ( x ) =
⎧x−u⎫ ⎬ du , a2 ⎭
∫ Fη ⎨⎩ 0
x > 1.
2018
A. POGORUI
AND
R. M. RODRIGUEZ -D AGNINO
We seek a solution of this equation in the following form:
Fη (x ) =
+∞
∑ cn e − x / a
2n
+
n=0
+∞
∑ bme − x a
2m
.
(3)
m =1
Inserting (3) into Eq. (2), for 0 ≤ x ≤ 1 we get +∞
∑ cn e − x / a
n=0
2n
+
+∞
∑
m =1
bm e − x a
2m
+∞
=
x
∑ cn ∫ e −( x − u )/ a
n=0
2n+2
+∞
∑
m =1
0
+∞
=
du +
∑ cn a 2n + 2 (1 − e − x / a
2n+2
n=0
)+
x
bm
∫e
du
0
+∞
∑
−( x − u )a 2 m − 2
m =1 a
bm 2m − 2
(1 − e − x a
2m−2
).
Equating the coefficients of the corresponding exponents, we get cn a 2 n + 2 = − cn +1 , n = 0, 1, 2, … , b1 = – c0 , bm = − a 2 m − 2 bm −1 , m = 2, 3, … . Therefore, cn = (− 1)n a n( n +1)c0 and bm +1 = (− 1)m +1 a m( m +1)c0 . It is easy to verify that +∞
∑ cn a 2 n + 2 +
n=0
+∞
∑
m =1 a
bm 2m − 2
= 0.
(4)
Since θ is an absolutely continuous random variable, the distribution Fη (x ) is also absolutely continuous [7]. In addition, it is easy to verify that the support of the density function fη ( x ) =
d Fη (x ) dx
is the segment 1 ⎡ ⎢ 0, ⎣ 1 − a2
⎤ ⎥, ⎦
and, therefore, ⎛ 1 ⎞ = 1. Fη ⎜ ⎝ 1 − a 2 ⎟⎠ Hence, +∞
+∞
2n 2m ⎛ Fη (x ) = c0 ⎜ ∑ (− 1)n a n( n +1)e − x / a + ∑ (− 1)m +1 a m( m +1)e − x a ⎝ n=0 m =1
⎞ ⎟⎠ ,
0 ≤ x ≤ 1.
L IMITING DISTRIBUTION OF FADING EVOLUTION
IN
SOME SEMI-MARKOV MEDIA
2019
It is easy to see that Fη (0 ) = 0. It follows from (2) that, for 1 < x ≤
1 1 − a2
,
we have ⎛ +∞ 2n 2n Fη (x ) = c0 ⎜ ∑ (− 1)n −1 a n( n +1) (e(1− x ) / a − e − x / a ) + ⎝ n=0
+∞
∑ (− 1)m a m( m +1)(e(1− x )a
2m
m =1
2m ⎞ − e− x a ) ⎟ . ⎠
Taking into account that ⎛ 1 ⎞ = 1, Fη ⎜ ⎝ 1 − a 2 ⎟⎠ we get ⎛ c0 = ⎜ ⎝
+∞
∑ (− 1)n −1 a n( n +1)(e a
2
/ ( a 2 n ( a 2 −1))
− e1/ ( a
2n
( a 2 −1))
n=0
+∞
+
∑ (− 1)
m m( m +1)
a
m =1
(e
a 2 m + 2 / ( a 2 −1)
−e
)
a 2 m / ( a 2 −1)
⎞ )⎟ ⎠
−1
.
It follows from Eq. (2) that its solution is a monotone function. Hence, Fη ( x ) is a distribution function. There is no other distribution function that is a solution of Eq. (2). Indeed, if a distribution function F ( x ) is a soluη
tion of Eq. (2), then the function G ( x ) = Fη (x ) – Fη ( x ) is also a solution of Eq. (2), which is impossible because G ( x ) is not monotone. 2. Limiting Distribution of Fading Evolution in the Stable Case Passing to the Fourier transform in Eq. (l), we get ϕη (λ ) = ϕθ ( λ ) ϕη (a2 λ ),
(5)
where ϕη and ϕθ are the characteristic functions of η and θk , respectively. By using recursion, we get
ϕη ( λ ) =
+∞
∏ ϕ θ (a 2n λ ) .
n=0
(6)
2020
A. POGORUI
AND
R. M. RODRIGUEZ -D AGNINO
It is not easy to obtain the limiting distribution if the distribution Gθ ( t ) is not stable even in the exponential [5] and 2-Erlang [6] cases. In the rest of the paper, we consider the case where Gθ ( t ) is stable distributed. The Levy–Khinchin formulas for the characteristic functions of the stable distribution Gθ ( t ) are well known and are as follows: ⎛ ϕθ ( λ ) = exp ⎜ i γλ − c λ ⎝
α
⎧ ⎫⎞ λ ω( λ , α ) ⎬ ⎟ , ⎨ 1 + iβ λ ⎩ ⎭⎠
where γ ∈ R, 0 < α ≤ 2, – 1 ≤ β ≤ 1, and c ≥ 0 are constants and ⎪⎧ tan απ / 2, ω ( λ, α ) = ⎨ ⎩⎪ 2 ln λ / π,
α ≠ 1, α = 1.
It is easy to verify that, for α ≠ 1, one has ⎛ γ c λ− λ ϕη (λ ) = exp ⎜ i 2 ⎝ 1− a 1 − a 2α
α
⎧ ⎫⎞ λ ω( λ , α ) ⎬ ⎟ . ⎨ 1 + iβ λ ⎩ ⎭⎠
(7)
Therefore, in this case, Fη (x ) is a stable distribution and can be obtained from (7) by the inverse Fourier transformation. Since θk ≥ 0, it is convenient to use the Laplace transform
fˆη ( λ ) =
∞
∫e
−λ x
dFη ( x ) dx ,
λ > 0.
0
It follows from (1) that fˆη ( λ ) =
+∞
∏ gˆθ (a 2n λ ) .
(8)
n=0
Example. For the stable distribution with exponent 1 2 defined in [8, p. 436], we have ⎡ 1 Gθ ( x ) = 2 ⎢ 1 − 2π ⎢⎣
1/ x
∫
−∞
e −t
2
/2
⎤ dt ⎥ , ⎥⎦
x > 0,
and ∞
gˆ θ ( λ ) =
∫e 0
−λ x
dGθ ( x ) = e
2λ
.
L IMITING DISTRIBUTION OF FADING EVOLUTION
IN
SOME SEMI-MARKOV MEDIA
2021
Therefore,
fˆη ( λ ) = e
2 λ (1+ a + a 2 + a 3 +…)
= e
⎛ 1 ⎞ 2λ ⎜ ⎝ 1− a⎟⎠
⎛ λ ⎞ = gˆ θ ⎜ . ⎝ (1 − a )2 ⎟⎠
Passing to the inverse Laplace transform, we get fη (x ) =
4 d 1 1 Fη ( x ) = (1 − a )2 gθ ((1 − a )2 x ) = e −1/ ( 2(1− a ) x ) , 2 dx (1 − a ) 2π x3
x > 0.
Obviously, no expectation exists. Hence, ∞
Fσ ( x ) =
⎡
⎛ y−x⎞ ⎤ a ⎟⎠ ⎥⎦
∫ dFη( y) ⎢⎣ 1 − Fη ⎜⎝ 0
= 1−
1 (1 − a )4 2π
+∞
∫ x
e
−1/ ( 2(1− a )4 y )
( y − x )/ a ⎤ 4 1 ⎡ 1 dt ⎥ dy . ⎢ ∫ e −1/ ( 2(1− a ) t ) t 3 ⎥⎦ y 3 ⎢⎣ 0
We have obtained interesting cases where a random evolution in a semi-Markov medium can be investigated easier than in a Markov medium. REFERENCES 1. S. Goldstein, “On diffusion by discontinuous movements and on the telegraph equation,” Quart. J. Math. Mech., 4, 129–156 (1951). 2. M. Kac, “A stochastic model related to the telegrapher’s equation,” Rocky Mountain J. Math., 4., 497–509 (1974). 3. A. A. Pogorui and R. M. Rodriguez-Dagnino, “One-dimensional semi-Markov evolution with general Erlang sojourn times,” Random Oper. Stochast. Equat., 13, No. 4, 399–405 (2005). 4. E. Orsingher, “Hyperbolic equations arising in random models,” Stochast. Process. Appl., 21, 93–106 (1985). 5. I. V. Samoilenko, “Fading Markov random evolution,” Ukr. Math. J., 54, No. 3, 448–459 (2002). 6. A. O. Pogorui, “Stationary distributions of fading evolutions” Ukr. Mat. Zh., 61, No. 3, 425–431 (2009). 7. A. F. Turbin and N. V. Pratsevityj, Fractal Sets. Functions and Distributions [in Russian], Naukova Dumka, Kiev (1992). 8. W. Feller, Introduction to Probability Theory and Its Applications, Vol. 2, Wiley, New York (1970).
