Introduction
A Stationary process of strictly stationary process or strongly stationary
process is a stochastic process whose joint probability distribution does
not change when and variance if they are present also do not change
over the time and do not follow any trends.
Stationary is used as a tool in time series analysis where the raw data is
often transformed to become stationary.
Definition:
Formally, let
{
𝑥
𝑡
}
be a stochastic process and let 𝐹
𝑥
(𝑡
1
+ 𝜏........𝑡
𝑘
+ 𝜏)
represent the cummulative join distribution function of
{
𝑥
𝑡
}
at time
𝑡
1
+ 𝜏 to 𝑡
𝑘
+ 𝜏. Then
{
𝑥
𝑡
}
is said to be strictly or (strongly) stationary if
for all 𝑘, for all 𝜏 and for all (𝑡
1
, ........𝑡
𝑘
)
𝐹
𝑥
(𝑥
𝑡
1
+ 𝜏......𝑥
𝑡
𝑘
+ 𝜏) = 𝐹
𝑥
(𝑥
𝑡
1
, 𝑥
𝑡
2
......𝑥
𝑡
𝑘
)
since 𝜏 doesn’t affect 𝐹
𝑥
(....). This shows 𝐹
𝑥
is not function of time.
Econophysics, 4th Year Stationary and Time Coreelation
Let the stochastic process 𝑥(𝑡) is independent and identically
distinguished. The statistical observable characterizing a stochastic
process can be written in form of nth order statistical properties.
𝐸
{
𝑥
𝑛
(𝑡)
}
=
∞
−∞
𝑥
𝑛
𝑓 (𝑥)𝑑𝑥
In that case 𝑛 = 1 is sufficient to define the mean
𝐸
{
𝑥(𝑡)
}
=
∞
−∞
𝑓 (𝑥)𝑑𝑥
where 𝐹(𝑥) is probability density of observing random variable at time
’t’.
𝐸
{
𝑥(𝑡
1
), 𝑥(𝑡
2
)
}
=
∞
−∞
∞
−∞
𝑥
1
𝑥
2
𝑓 (𝑥
1
, 𝑥
2
, 𝑡
1
, 𝑡
2
)𝑑𝑥
1
𝑑𝑥
2
where 𝑓 (𝑥
1
, 𝑥
2
, 𝑡
1
, 𝑡
2
) be the joint probability density that is 𝑥
1
observed
at time 𝑡
1
and 𝑥
2
observed at time 𝑡
2
.
Econophysics, 4th Year Stationary and Time Coreelation
Types of Stationary
(a) A wide or weakly sense stationary stochastic process:
A wide-sense stationary stochastci process is formed by these condition
(mean, variance and co-variance remain unchanged under a time shift)
𝐸{𝑥(𝑡)} = 𝜇
𝐸{𝑥(𝑡
1
), 𝑥(𝑡
2
)} = 𝑅(𝑡
1
, 𝑡
2
)
where 𝑅(𝑡
1
, 𝑡
2
) = 𝑅(𝜏) is function of
′
𝜏
′
= 𝑡
2
− 𝑡
1
and 𝐸{𝑥
2
(𝑡)} = 𝑅(0)
Thus, the variance of this process 𝑅(0) ∼ 𝜇
2
is time independent.
Econophysics, 4th Year Stationary and Time Coreelation
(b) Asymptotically stationary stochastic process:
Asymptotically stationary stochastic process are observed when the
statistics of random variable 𝑥 (𝑡
1
+ 𝑐), 𝑥(𝑡
2
+ 𝑐)....𝑥(𝑡
𝑛
+ 𝑐) doesnotdepend
on 𝑐 if 𝑐 is large.
(c) Nth order stationary stochastic process arises when the joint
probability density,
𝑓 (𝑥
1
.....𝑥
𝑛
, 𝑡
1
....𝑡
𝑛
) = 𝑓 (𝑥
1
...𝑥
𝑛
, 𝑡
1
+ 𝑐, 𝑡
2
+ 𝑐, ....𝑡
𝑛
+ 𝑐)
holds not for every values on 𝑛 but only 𝑛 ≤ 𝑁
Econophysics, 4th Year Stationary and Time Coreelation
Correlation
If the change in one random variable affects a change in toher variable,
the variables are said to be cpreelated. if the two variable deviate in the
same direction i.e., if the increase (or decrease) in one results in a
corresponding increase (or decrease) in the other then the correlation
said to be direct or positive. However, if they are deviate in opposite
direction i.e., if increase or decrease in one results in corresponding
decrease or increase in the other, then the corresponding is said to be
diverse i.e., negative. For, example
(a) Height weight of a group of person
(b) Price and demand of goods etc.
The correlation coefficient between two random variable 𝑋 and 𝑌 is
denoted by 𝑐𝑜𝑟𝑟 (𝑋, 𝑌 ) or 𝑟
𝑋𝑌
i.e.,
𝑐𝑜𝑟𝑟 (𝑋, 𝑌 ) =
𝑐𝑜𝑣(𝑋, 𝑌 )
𝜎
𝑥
𝜎
𝑦
where. 𝑐𝑜𝑣 (𝑋, 𝑌 ) = 𝐸{(𝑋 − 𝐸 (𝑋)), (𝑌 − 𝐸 (𝑌 ))} → covariance
𝜎
2
𝑥
= 𝐸{𝑋 − 𝐸 (𝑋)}
2
, 𝜎
2
𝑦
= 𝐸{𝑌 − 𝐸 (𝑌)}
2
Econophysics, 4th Year Stationary and Time Coreelation
Auto correlation
Autocorrelation is a mathematical representation of the degree of
similarity between a given time series and a logged version of itself over
succesive time intervals. In statistics, the autocorrelation of a random
process describes the correlation between values of the process at
different coins as a function of time. It is also known as serial correlation.
Auto correlation function is a mathematical tool used for finding
patterns of distribution and measure the degree to which there is a
relationship between two variables.
The autocorrelation function for two random variable 𝑥 (𝑡
1
) and 𝑥(𝑡
2
) is
denoted by 𝑅(𝑡
1
, 𝑡
2
) and is given by
𝑅(𝑡
1
, 𝑡
2
) = 𝐸{𝑥 (𝑡
1
), 𝑥(𝑡
2
)}
𝐸{𝑥(𝑡
1
), 𝑥(𝑡
2
)} =
∞
−∞
∞
−∞
𝑥
1
𝑥
2
𝑓 (𝑥
1
, 𝑥
2
; 𝑡
1
, 𝑡
2
)𝑑𝑥
1
𝑑𝑥
2
Here 𝑓 (𝑥
1
, 𝑥
2
; 𝑡
1
, 𝑡
2
) represents the joint probability distribution for the
random variable 𝑥
𝑖
which is observed at time 𝑡
1
and 𝑡
2
which is observed
at time 𝑡
2
.
Econophysics, 4th Year Stationary and Time Coreelation
The autocorrelation function are (𝑡
1
, 𝑡
2
) for stochastic process which is
defined as the time series data is sensitive to the average value of the
process.
For stochastic process with average value different from zero. It is
considered as auto co-variance.
i.e., 𝑐𝑜𝑟𝑟 (𝑡
1
, 𝑡
2
) = 𝑐𝑜𝑣{𝑥(𝑡
1
), 𝑥(𝑡
2
)}
= 𝐸{𝑥(𝑡
1
) − 𝐸 (𝑥(𝑡
1
)), 𝑥(𝑡
2
) − 𝐸 (𝑥(𝑡
2
))}
= 𝐸{𝑥(𝑡
1
), 𝑥(𝑡
2
)} − 𝜇(𝑡
1
)𝜇(𝑡
2
) ...(1)
for stationary process;
𝜏 = 𝑡
2
− 𝑡
1
and 𝜇(𝑡
1
) = 𝜇(𝑡
2
) = 𝜇 = constant
then equantion(1) becomes
𝑐(𝜏) = 𝑅(𝑡
1
, 𝑡
2
) − 𝜇
2
or, 𝑐(𝜏) = 𝑅(𝜏) − 𝜇
2
...(2)
where, [𝐸{𝑥 (𝑡
1
), 𝑥(𝑡
2
)} = 𝑅(𝑡
1
, 𝑡
2
)]
Econophysics, 4th Year Stationary and Time Coreelation
Again for stationary process,
𝐸{𝑥(𝑡)} = 𝜇 = constant
and 𝐸{𝑥(𝑡
1
), 𝑥(𝑡
2
)} = 𝑅(𝑡
1
, 𝑡
2
) is independent only in
time length 𝜏 = 𝑡
2
− 𝑡
1
then, 𝐸{𝑥(𝑡)
2
} = 𝑅(0) ...(3)
and, variance 𝜎
2
= 𝐸{𝑥(𝑡)
2
} − [𝐸{𝑥(𝑡)}]
2
𝜎
2
= 𝑅(0) − 𝜇
2
...(4)
Let us consider a stochastic process with 0 mean and unit variance i.e.,
𝜎
2
= 1
then equation (4) becomes
𝑅(0) = 1
=⇒ 𝜎
2
≃ 𝑅(0) = 1
This clearly shows that from above condition the auto-correlation
function 𝑅(𝜏) and auto co-variance function are same.
Econophysics, 4th Year Stationary and Time Coreelation
Use of Autocorrelation
The autocorrelation function can be used in finance as:
(i) To determine non-correlation in data i.e., to determine randomness or
non-randomness of the data.
(ii) To identify the appropriate time series model to describe economic
an financial activities.
For stochastic process integral of 𝑅(𝑡) becomes different the area below
𝑅(𝑡) can be taken of the possible values.
∞
0
𝑅(𝜏)𝑑𝜏 =
𝑓 𝑖𝑛𝑖𝑡𝑒
𝑖𝑛 𝑓 𝑖𝑛𝑖𝑡𝑒
𝑖𝑛𝑑𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑎𝑛𝑎𝑡𝑒
where
∞
0
𝑅(𝜏)𝑑𝜏 is finite there exist a typical time memory 𝜏
𝑐
is called
the correlation time of the process.
Econophysics, 4th Year Stationary and Time Coreelation
Short Range correlated random process
Short range correlated random processes are characterized by typical
memory on simple example is given by a stochastic process having an
exponential decaying auto correlation function
𝑅(𝜏) = 𝑒𝑥 𝑝
−
𝜏
𝜏
𝑐
This form describes for example the statistical memory of the velocity
𝑣(𝑡) of a Brownian particle, as the autocorrelation function of is
𝑅(𝜏) = 𝜎
2
𝑒𝑥 𝑝
−
|𝜏|
𝜏
𝑐
Short range correlated stochastic processes can be characterized with
respect to their second order statistical properties by investigating the
autocorrelation function and/or the power spectrum.
Econophysics, 4th Year Stationary and Time Coreelation
Power spectrum of Random variable
The power spectrum of a wide-sense stationary random process is the
fourier transform of its auto - correlation fucntion.
Power Spectrum 𝑠( 𝑓 ) =
∞
−∞
𝑅(𝜏)𝑒
−2 𝜋 𝑓 𝜏
𝑑𝜏
for the Brownian particle autocorrelation is defined by
𝑅(𝜏) = 𝜎
2
𝑒
−
|𝜏 |
𝜏
𝑐
then, 𝑠( 𝑓 ) = 𝜎
2
∞
−∞
𝑒
−
|𝜏 |
𝜏
𝑐
𝑒
−2𝑖 𝜋 𝑓 𝜏
𝑑𝜏
= 𝜎
2
0
−∞
𝑒
−
|𝜏 |
𝜏
𝑐
𝑒
−2𝑖 𝜋 𝑓 𝜏
𝑑𝜏 +
∞
0
𝑒
−
|𝜏 |
𝜏
𝑐
𝑒
−2𝑖 𝜋 𝑓 𝜏
𝑑𝜏
Econophysics, 4th Year Stationary and Time Coreelation
Let us define 𝜏 = −𝜏 in first integral and in 2nd 𝜏 = 𝜏
= 𝜎
2
𝑒
−
𝜏
𝜏
𝑐
𝑒
2𝑖 𝜋 𝑓 𝜏
+
∞
0
𝑒
−
𝜏
𝜏
𝑐
𝑒
−2𝑖 𝜋 𝑓 𝜏
𝑑𝜏
= 2𝜎
2
∞
0
𝑒
−
𝜏
𝜏
𝑐
𝑒
2𝑖 𝜋 𝑓 𝜏
+ 𝑒
−2𝑖 𝜋 𝑓 𝜏
2
𝑑𝜏
= 2𝜎
2
∞
0
𝑒
−
𝜏
𝜏
𝑐
cos
(
2𝜋 𝑓 𝜏
)
𝑑𝜏
= 2𝜎
2
𝑒
−
𝜏
𝜏
𝑐
−1/𝜏
𝑐
∞
0
+
∞
0
𝑒
−
𝜏
𝜏
𝑐
−1/𝜏
𝑐
2𝜋 𝑓 {− sin
(
2𝜋 𝑓 𝜏
)
}𝑑𝜏
= 2𝜎
2
𝜏
𝑐
+ (2𝜋 𝑓 𝜏
𝑐
)
∞
0
𝑒
−
𝜏
𝜏
𝑐
sin
(
2𝜋 𝑓 𝜏
)
𝑑𝜏
= 2𝜎
2
𝜏
𝑐
+ (2𝜋 𝑓 𝜏
𝑐
)
𝑒
−
𝜏
𝜏
𝑐
−1/𝜏
𝑐
sin
(
2𝜋 𝑓 𝜏
)
∞
0
+
∞
0
𝑒
−
𝜏
𝜏
𝑐
−1/𝜏
𝑐
2𝜋 𝑓 cos
(
2𝜋 𝑓 𝜏
)
𝑑𝜏
Econophysics, 4th Year Stationary and Time Coreelation
or, 𝑠( 𝑓 ) = 2𝜎
2
[𝜏
𝑐
− (2𝜋 𝑓 𝜏
𝑐
)
2
𝑠( 𝑓 )]
or, 𝑠( 𝑓 ) [𝜏
𝑐
− (2𝜋 𝑓 𝜏
𝑐
)
2
] = 2𝜎
2
𝜏
=⇒ 𝑠( 𝑓 ) =
2𝜎
2
𝜏
𝑐
1 + (2𝜋 𝑓 𝜏
𝑐
)
2
This is power spectrum of random variable.
When, 𝑓 <<
1
2 𝜋 𝜏
𝑐
essentially power spectrum is independent of
frequency. For a time window much longer than 𝜏,
𝑠( 𝑓 ) ∼
1
𝑓
2
This is called Wiener process.
Econophysics, 4th Year Stationary and Time Coreelation
Long range correlated random process
Stochastic process characterized by a power auto-correlation fucntion
𝑅(𝜏) = |𝜏|
𝑛−1
are long range correlated. Power law autocorrelation is
observed in mnay systems physical, biological and economic.
Let us consider a stochastic process with power spectrum of the form,
𝑠( 𝑓 ) =
constant
| 𝑓 |
𝜂
...(1) with 0 < 𝜂 < 2
𝜂 = 𝑜 for white noise
𝜂 = 2 for Weiner process (continous limit)
𝜂 ≈ 1 for a stochastic process — stochastic spectral density
the stochastic process or non-stationary in this system for 0 < 𝜂 < 1 and
𝜏 > 0, the autocorrelation function defined by
𝑅(𝜏) ∼ |𝜏|
𝑛−1
for 1 < 𝜂 < 2
𝑅(𝑡
2
, 𝜏) ∼ 𝑡
𝑛−1
2
− 𝑐(𝜏)
𝑛−1
Econophysics, 4th Year Stationary and Time Coreelation
For the borderline case 𝜂 = 1 and 0 < 𝜏 << 𝑡
2
𝑅(𝑡
2
, 𝜏) ∼ ln
(
4𝑡
2
)
− ln |𝜏|
The typical shape of these autocorrelation fucntions are shown in fig
below. The autocorrelation fucntion for 1/ 𝑓 noise lacks a typical time
scale, so 1/ 𝑓 noise is a long-range correlated stochastic process.
Econophysics, 4th Year Stationary and Time Coreelation
Short-range compared with long-range correlated noise
If a time scale 𝜏
𝑐
characterizes the memory of a stochastic process, then
for time intervals longer than 𝜏
𝑐
the conditional probability densities
verify the equation
𝑓 (𝑥
1
, 𝑥
2
, ...., 𝑥
𝑛−1
; 𝑡
1
, 𝑡
2
, ..., 𝑡
𝑛−1
|𝑥
𝑛
; 𝑡
𝑛
) = 𝑓 (𝑥
𝑛−1
; 𝑡
𝑛−1
|𝑥
𝑛
; 𝑡
𝑛
)
Stochastic processes with the above form for their conditional probability
density are called Markov processes. For the simplest Markov process,
𝑓 (𝑥
1
, 𝑥
2
, 𝑥
3
; 𝑡
1
, 𝑡
2
, 𝑡
3
) = 𝑓 (𝑥
1
, 𝑡
1
) 𝑓 (𝑥
1
; 𝑡
1
|𝑥
2
; 𝑡
2
) 𝑓 (𝑥
2
; 𝑡
2
|𝑥
3
; 𝑡
3
)
Thus only the first- and second-order conditional probability densities
𝑓 (𝑥
1
, 𝑡
1
) and 𝑓 (𝑥
𝑛
; 𝑡
𝑛
|𝑥
𝑛+1
; 𝑡
𝑛+1
) are needed to fully characterize the
stochastic process. Stochastic processes lacking a typical time scale, such
as 1/ 𝑓 noise, are not Markov processes.
The knowledge of the first- and second-order conditional probability
densities fully characterizes a Markov process since any higher-order
joint probability density can be determined from them. For a
non-Markovian process, this knowledge is not sufficient to fully
characterize the stochastic process.
Econophysics, 4th Year Stationary and Time Coreelation
Non-Markovian stochastic processes with the same first-order and
second-order conditional probability dnsities are, in general, different
because the joint probability densities of all orders are required to fully
characterize long-range correlated stochastic processes. Thus, different
1/ 𝑓 noise signals cannot be considered to be the same stochastic process,
unless information about higher-order joint probability densities is also
known.
Note: Auto-correlation for non-stationary stochastic process: Stochastic
process with
1
𝑓
𝑛
noise are nonstationary provided that one observes the
noise at times and such that observation time 𝑇
𝑎𝑏𝑠
is short compared
with the time elapsed since the process began (𝑇
𝑎𝑏𝑠
< 𝑡
1
), one can
evaluate the autocorrelation function.
Econophysics, 4th Year Stationary and Time Coreelation
Stationary of price changes
From the empiria investigations, we conclude that the stochastic
dynamics of price of a financial good can be approximately described by
a random walk characterized bya short range pairwise correlation. Then
a question arises, can we describe price changes in terms of stationary
process? Empirical analysis of financial data - show that price changes
cannot be discribed by a strict - sense statioanry stochastic process, since
the standard deviation of price changes, n amely the volatility is time
dependent in real markets. Hence the form of stationarity that is present
in financialmarkets is at best asymtotic stationarity. By analyzing a
sufficiently lond time series, the asymptotic probability density function
of price changes is obtained. The asymptotic probability density function
gives the large time statistical properties of the stochastic process.
Econophysics, 4th Year Stationary and Time Coreelation
