Auburn University - Winter 2022 French 212, Chapter 1 Worksheet
Professor Paya, Section 6 March 10, 2022
In probability theory, the telegraph process is a memoryless continuous-time stochastic process that shows two distinct values. It models burst noise (also called popcorn noise or random telegraph signal). (Johnson, 2020)
Class Date: 4/2/2024
Professor’s Note: Avoid repetition of ideas in the main body.
GENERIC CONTENT:
## Discussion (List)
- If the two possible values that a random variable can take are
c
1
{\displaystyle c_{1}}
and
c
2
{\displaystyle c_{2}}
, then the process can be described by the following master equations:
∂
t
P (
c
1
, t
|
x ,
t
0
) = −
λ 1
P (
c
1
, t
|
x ,
t
0
) +
λ 2
P (
c
2
,
t
|
x ,
t
0
)
{\displaystyle \partial _{t}P(c_{1},t|x,t_{0})=-\lambda _{1}P(c_{1},t|x,t_{0})+\lambda _{2}P(c_{2},t|x,t_{0})}
and
∂
t
P (
c
2
, t
|
x ,
t
0
) =
λ 1
P (
c
1
, t
|
x ,
t
0
) −
λ 2
P (
c
2
, t
|
x ,
t
0
) .
- {\displaystyle \partial _{t}P(c_{2},t|x,t_{0})=\lambda _{1}P(c_{1},t|x,t_{0})-\lambda _{2}P(c_{2},t|x,t_{0}).}
- where
λ 1
{\displaystyle \lambda _{1}}
is the transition rate for going from state
c
1
{\displaystyle c_{1}}
to state
c
2
{\displaystyle c_{2}}
and
λ 2
{\displaystyle \lambda _{2}}
is the transition rate for going from going from state
c
2
{\displaystyle c_{2}}
to state
c
1
{\displaystyle c_{1}}
.
## Findings (List)
- The process is also known under the names Kac process (after mathematician Mark Kac), and dichotomous random process.
- == Solution ==
The master equation is compactly written in a matrix form by introducing a vector
P
= [ P (
c
1
, t
|
x ,
t
0
) , P (
c
2
, t
|
x
,
t
0
) ]
{\displaystyle \mathbf {P} =[P(c_{1},t|x,t_{0}),P(c_{2},t|x,t_{0})]}
,
d
P
d t
= W
P
{\displaystyle {\frac {d\mathbf {P} }{dt}}=W\mathbf {P} }
where
W =
(
−
λ 1
λ 2
λ 1
−
λ 2
)
{\displaystyle W={\begin{pmatrix}-\lambda _{1}&\lambda _{2}\\\lambda _{1}&-\
lambda _{2}\end{pmatrix}}}
is the transition rate matrix.
- The formal solution is constructed from the initial condition
P
( 0 )
{\displaystyle \mathbf {P} (0)}
(that defines that at
t =
t
0
{\displaystyle t=t_{0}}
, the state is
x
{\displaystyle x}
) by
P
(
t ) =
e
W t
P
( 0 )
{\displaystyle \mathbf {P} (t)=e^{Wt}\mathbf {P} (0)}
.
## Analysis
It can be shown that
e
W t
= I + W
( 1 −
e
−
2 λ t
)
2 λ
{\displaystyle e^{Wt}=I+W{\frac {(1-e^{-2\lambda t})}{2\lambda }}}
where
I
{\displaystyle I}
is the identity matrix and
λ = (
λ 1
+
λ 2
)
/
2
{\displaystyle \lambda =(\lambda _{1}+\lambda _{2})/2}
is the average transition rate. As
t → ∞
{\displaystyle t\rightarrow \infty }
, the solution approaches a stationary distribution
P
(
t → ∞ ) =
P
s
{\displaystyle \mathbf {P} (t\rightarrow \infty )=\mathbf {P} _{s}}
given by
P
s
=
1
2 λ
(
λ 2
λ 1
)
{\displaystyle \mathbf {P} _{s}={\frac {1}{2\lambda }}{\begin{pmatrix}\lambda _{2}\\\
lambda _{1}\end{pmatrix}}}
== Properties ==
Knowledge of an initial state decays exponentially.
## Background
Therefore, for a time
t ≫ ( 2 λ )
− 1
{\displaystyle t\gg (2\lambda )^{-1}}
, the process will reach the following stationary values, denoted by subscript s:
Mean:
⟨ X
⟩ s
=
c
1
λ 2
+
c
2
λ 1
λ 1
+
λ 2
. {\displaystyle \langle X\rangle _{s}={\frac {c_{1}\lambda _{2}+c_{2}\lambda _{1}}{\
lambda _{1}+\lambda _{2}}}.} Variance:
var { X
}
s
=
(
c
1
−
c
2
)
2
λ 1
λ 2
(
λ 1
+
λ
2
)
2
.
References / Works Cited:
1. Wikipedia (n.d.). Retrieved from https://wikipedia.org/
2. Random Book Title (2022). Academic Publishing House.
