Capital Markets & Investments
Math Methods – Financial Price Analysis
Mathematics GR5360
Instructor: Alexei Chekhlov
Mathematics GR5360
Mathematics G4075
Instructor: Alexei Chekhlov
Stationarity, Correlation and Memory*
* – with some changes from “An Introduction to Econophysics” by Mantegna and Stanley, ref. B1.
Mathematics GR5360
Stationarity, Correlation and Memory*
* – with some changes from “An Introduction to Econophysics” by Mantegna and Stanley, ref. B1.
Mathematics GR5360
Stationarity, Correlation and Memory*
* – with some changes from “An Introduction to Econophysics” by Mantegna and Stanley, ref. B1.
Mathematics GR5360
Stationarity, Correlation and Memory*
* – with some changes from “An Introduction to Econophysics” by Mantegna and Stanley, ref. B1.
Mathematics GR5360
Stationarity, Correlation and Memory*
* – with some changes from “An Introduction to Econophysics” by Mantegna and Stanley, ref. B1.
Mathematics GR5360
Stationarity, Correlation and Memory*
* – with some changes from “An Introduction to Econophysics” by Mantegna and Stanley, ref. B1.
Mathematics GR5360
Stationarity, Correlation and Memory*
* – with some changes from “An Introduction to Econophysics” by Mantegna and Stanley, ref. B1.
Mathematics GR5360
Short-Range Memory Random Processes*
* – with some changes from “An Introduction to Econophysics” by Mantegna and Stanley, ref. B1.
Mathematics GR5360
Long-Range Memory Random Processes*
* – with some changes from “An Introduction to Econophysics” by Mantegna and Stanley, ref. B1.
Mathematics GR5360
Influence of Mean-Reversion on Variance
Mathematics GR5360
Influence of Mean-Reversion on Variance
Mathematics GR5360
Andrew Lo’s Variance Ratio Test
* – with changes from “The Economics of Financial Markets” by Campbell, Lo and MacKinlay, ref. B5.
Mathematics GR5360
Basic Behavioral Biases and Price Predictabilities
Over-Reaction or Mean-Reversion, when agents over-react to new information by overselling on new bad information with later correction and/or over-buying on good new information with later opposite correction.
Under-Reaction or Trend-Following, when agents under-react to new information, by establishing a partial position, waiting for confirmations to their actions from other agents. Once received, they continue to increase their position in the same direction – self-reinforcement. Thus, through delayed chain reactions, the new information is gradually priced into the market.
Mathematics GR5360
Price-Change Sign Counting Experiments
Data type used: 1-minute frequency, back-adjusted futures prices since inception (different for each market) until present.
In both of these experiments the frequency (1-minute) is chosen to: be small enough in order to reveal the self-similar statistical properties within the continuous price assumption p=p(t), and be large enough as compared to the so-called “bid-ask bounce” (“fake” mean-reversion). This can be easily verified by comparing the standard deviation of 1-minute price changes with the average ask-bid spread, the standard deviation has to be several times (5-10) larger.
Both experiments are inspired by some of the early experiments of Andrew W. Lo.
Mathematics GR5360
Experiment 1: Counting Continuations and Reversals
Mathematics GR5360
Experiment 1: Counting Continuations and Reversals
Mathematics GR5360
Experiment 1: Counting Continuations and Reversals
Mathematics GR5360
Experiment 1: Counting Continuations and Reversals
Mathematics GR5360
Experiment 1: Counting Continuations and Reversals
Evidence of short-term (up to a couple of hours) over-reaction or mean-reversion and longer-term (beyond a day) under-reaction or trend-following;
Short-term over-reaction or mean-reversion is quite strong and robust statistically.
Longer-term under-reaction or trend-following is weaker and less robust statistically.
The agreement with the RW model gets better as time-separation gets larger.
Mathematics GR5360
Experiment 2: Counting Up- and Down- Trends
Mathematics GR5360
Experiment 2: Counting Up- and Down- Trends
Mathematics GR5360
Experiment 2: 1-min
Mathematics GR5360
Experiment 2: 1-min
Mathematics GR5360
Experiment 2: 5-min
Mathematics GR5360
Experiment 2: 5-min
Mathematics GR5360
Experiment 2: 15-min
Mathematics GR5360
Experiment 2: 15-min
Mathematics GR5360
Experiment 2: 1-hour
Mathematics GR5360
Experiment 2: 1-hour
Mathematics GR5360
Experiment 2: Counting Up- and Down- Trends
Not only the numbers of trends above theoretical values indicate under-reaction or trend-following behavior, but conversely, the numbers of trends below theoretical values indicate possible over-reaction or mean-reversion;
There is a reasonable agreement between the two experiments, although this experiment provides further evidence on how weak the under-reaction or trend-following regime is;
Short length trends counts have better agreement with the RW formulas.
Mathematics GR5360
Variance Ratio Test
We will now transition from the signs under- and over-reaction to the price change under- and over-reaction studies;
Time series will be taken since inception to present at 1-min resolution with time-separation from 1 min to 90 trading hours, during most liquid session (pit session);
Mathematics GR5360
Variance Ratio Test
Mathematics GR5360
Push-Response Diagram Test
This test is free from the fat-tailed bias of the VR test – positive;
This test is quickly growing sample error as you increase the Δp – negative.
Mathematics GR5360
Push-Response Diagram Test
Mathematics GR5360
Push-Response Diagram Test
Mathematics GR5360
Random Walk Comparisons Tests Results
General inspection of the test results confirms the previous sign-tests results: a general pattern is statistically strong short-term over-reaction or mean-reversion, beyond which either inconclusive or statistically weaker, selective longer-term under-reaction or trend-following properties;
Beyond 10 trading days time-separation shows little predictability;
These tests are more general than the first two signs tests because they considers both the price change sign and its magnitude – positive;
These tests could be somewhat biased if the price-difference distributions function is “fat tailed” or the data sample is not large enough – negative.
Mathematics GR5360
Sheet1
1st Price Change 2nd Price Change Probability Type
+ + p^2 C
+ – p*q R
– + q*p R
– – q^2 C
+ 0 p*(1-p-q) 0
0 + (1-p-q)*p 0
– 0 q*(1-p-q) 0
0 – (1-p-q)*q 0
0 0 (1-p-q)^2 0
Sheet2
Sheet3
t.
independen
–
time
is
,
)
0
(
process,
such
of
variance
the
Therefore,
).
0
(
)
(
and
,
of
function
a
only
is
)
(
)
,
(
where
),
,
(
)
(
)
(
,
)
(
:
met
are
conditions
three
following
the
if
defined
is
process
stationary
sense
–
wide
a
example,
For
exist.
also
ty
stationari
of
s
definition
e
restrictiv
–
less
Some
.
:
shift
time
a
under
invariant
is
)
,
(
function
density
y
probabilit
its
if
stationary
called
is
)
(
process
stochastic
a
sense,
strict
a
In
2
2
1
2
2
1
2
1
2
1
m
t
t
m
–
=
–
=
=
=
×
=
D
+
®
R
R
t
x
t
t
R
t
t
R
t
t
R
t
x
t
x
t
x
t
t
t
t
x
P
t
x
.
have
we
of
values
large
for
Indeed,
.
correlated
range
–
short
or
dependent
weakly
are
variables
random
the
case
In this
.
lim
:
namely
,
of
values
large
for
finite
is
n terms
correlatio
of
sum
The
1.
:
cases
h two
distinguis
can
we
equation,
this
of
term
second
the
of
behavior
on the
Depending
.
2
:
get
we
),
(
of
values
large
for
and
0
variables
random
correlated
–
positively
only
assume
we
If
.
and
variables
e
between th
n
correlatio
–
auto
the
is
where
,
)
(
2
:
exactly
have
we
Indeed,
.
:
variables
stochastic
of
sum
the
of
variance
of
bahavior
general
of
case
In the
2
1
1
n
1
1
2
2
1
1
2
2
1
n
S
n
const
x
x
n
x
x
x
n
S
n
n
x
x
x
x
x
x
x
x
k
n
x
n
S
x
S
x
n
n
n
n
n
i
n
k
k
i
i
n
i
n
k
k
i
i
i
k
i
i
k
i
i
k
i
i
n
i
n
k
k
i
i
i
n
i
i
n
i
µ
=
÷
ø
ö
ç
è
æ
×
+
®
+¥
®
>
þ
ý
ü
î
í
ì
–
×
+
=
=
å
å
å
å
å
å
å
=
=
+
+¥
®
=
=
+
+
+
+
=
=
+
=
(
)
.
,
1
0
for
:
)
R(
)
3
.
1
:
)
(
)
2
.
e
:
)
(
1)
:
examples
notable
Some
process.
the
of
n time
correlatio
the
called
memory
time
typical
a
exists
there
finite,
is
integral
When this
time.
of
dimension
physical
the
has
integral
This
memory.
long”
”
–
infinite
memory;
short”
”
–
finite
)
(
:
)
memory”
”
or time
scale
–
time
typical
(a
time
of
dimension
the
has
)
(
below
area
then the
process,
stationary
a
for
)
(
function
n
correlatio
–
auto
ess
dimensionl
the
of
integral
he
consider t
we
If
process.
stochastic
of
memory”
”
(time)
called
–
so
with the
do
to
has
n
correlatio
–
auto
the
of
meaning
physical
The
1
0
0
c
t
1
–
1
–
1
0
0
0
0
0
–
0
¥
=
£
<
µ
÷
ø
ö
ç
è
æ
G
×
=
=
=
-
×
=
×
=
=
î
í
ì
=
=
ò
ò
ò
ò
ò
¥
¥
-
-
¥
¥
¥
-
-
-
¥
+
t
t
h
t
t
n
n
t
t
t
t
t
t
t
t
t
t
t
t
t
h
h
n
t
t
t
t
t
t
t
t
n
n
d
d
e
e
R
e
e
dy
e
d
e
R
d
τ
R
R
R
c
c
y
c
c
c
c
.
1
)
0
(
:
nce
unit varia
and
,
0
mean
zero
with
processes
stochastic
consider
can
we
simplicity
For
.
for
0
)
(
to
off
falls
ly
continuous
and
,
)
0
(
at
starts
it
:
be
will
function
n
correlatio
-
auto
the
of
shape
typical
the
process
stochastic
correlated
positively
a
For
.
)
(
)
(
:
to
equal
is
covariance
-
auto
the
processes
stationary
For
).
(
)
(
)
,
(
)
,
(
:
covariance
-
auto
g
considerin
by
this
accomodate
to
needs
one
zero,
from
different
is
value
average
the
If
process.
stochastic
the
of
value
average
on the
dependent
is
above
introduced
function
n
correlatio
-
auto
The
2
2
2
2
1
2
1
2
1
=
=
=
+¥
®
+
®
=
-
=
×
-
º
s
m
t
t
s
m
t
t
m
m
R
C
C
R
C
t
t
t
t
R
t
t
C
memory.
with
processes
stochastic
correlated
-
auto
range
-
short
of
ts
fingerprin
the
are
1
resembling
spectra
energy
and
functions
n
correlatio
-
auto
decaying
-
fast
The
spectrum.
energy
or the
function
n
correlatio
-
auto
either the
ing
investigat
by
properties
l
statistica
their
respect to
with
zed
characteri
be
can
process
stochastic
correlated
range
-
short
summarize,
To
Walk.
Random
is
which
,
)
(2
1
for
,
1
2
;
noise"
white
"
t
independen
-
frequency
is
which
,
)
(2
1
for
,
2
)
(
:
behaviors
al
asymptotic
two
has
spectrum
energy
This
.
)
2
(
1
2
)
(
:
form
closed
in
doable
is
integral
this
above
function
n
correlatio
-
auto
For the
.
)
(
)
(
:
function
n
correlatio
-
auto
its
of
ansform
Fourier tr
the
is
process
random
stationary
sense
-
wide
a
of
spectrum
energy
the
Theorem,
Khinchin
the
Recalling
domain.
frequency
-
Fourier
in the
properties
l
statistica
same
the
e
investigat
now
can
we
function,
n
correlatio
-
auto
of
in terms
properties
l
statistica
point
-
two
the
of
zation
characteri
the
o
addition t
In
.
for
0
)
(
assume
can
one
as
,
easy"
"
are
processes
Such
).
(
of
function
n
correlatio
-
auto
the
as
particle,
Brownian
a
of
)
(
velocity
the
of
memory
l
statistica
the
describes
example,
for
function,
n
correlatio
-
auto
This
.
)
(
:
function
n
correlatio
-
auto
decaying
lly
exponentia
h
namely wit
us,
by
considered
already
was
example
One
memory.
finite
or
scale
time
typical
a
by
zed
characteri
are
processes
random
correlated
range
-
short
that
noted
have
We
2
c
2
2
2
c
2
2
2
2
|
|
2
w
pt
w
w
t
p
s
pt
w
t
s
w
pwt
t
s
w
t
t
w
t
t
t
s
t
t
wt
p
t
t
ï
ï
î
ï
ï
í
ì
>>
×
<<
®
+
=
×
×
=
>
=
×
=
ò
¥
+
-¥
=
–
–
c
c
c
c
i
c
E
E
d
e
R
E
R
t
v
t
v
e
R
c
.
correlated
range
–
long
be
to
called
are
(3)
case
in
as
such
function
n
correlatio
–
auto
an
by
zed
characteri
variables
Random
ce.
independen
pairwise
of
regime
a
from
ns
correlatio
temporal
of
regime
a
separate
can
that
scale
time
a
select
to
impossible
is
it
above,
(3)
case
in
example,
For
finite!
are
functions
decreasing
lly
monotonica
the
of
integrals
all
not
Obviously,
function.
n
correlatio
–
auto
under the
area
the
is
where
,
for
present
is
0)
(
n
correlatio
no
and
,
scales
to
up
smallest
the
from
present
is
n
correlatio
1)
(
full
that the
assuming
by
system
the
model
to
possible
is
it
ion,
approximat
order
–
zero
a
as
fact,
In
process.
the
of
memory
the
of
scale
time
typical
about the
n
informatio
carries
function
n
correlatio
–
auto
under the
area
the
of
finiteness
The
*
*
*
t
t
t
t
>
=
+
=
(
)
(
)
(
)
(
)
move.
price
unchanged
an
observe
y to
probabilit
the
be
will
1
then
move;
negative
a
observe
y to
probabilit
–
move;
positive
a
observe
y to
probabilit
–
:
measure
also
can
we
separation
any
For
.
0
zero,
to
equal
is
change
price
one
least
at
where
changes
price
e
consecutiv
of
pairs
of
number
the
calculate
also
will
We
).
(Reversals
direction
opposite
in the
changes
price
ve
consequiti
of
pairs
of
number
the
ions);
(Continuat
direction
same
in the
changes
price
ve
consequiti
of
pairs
of
number
the
:
calculate
will
we
minutes,
1,000
of
maximal
the
to
minute
1
of
minimal
the
from
starting
,
separation
–
time
particular
any
For
q
p
q
p
N
N
N
–
–
–
–
–
+
t
t
t
t
t
(
)
memory.
range
–
long
and
range
–
short
with
processes
between
h
distinguis
to
used
be
can
function
n
correlatio
–
auto
the
of
integral
the
series
price
financial
a
for
,
therefore
,
where
,
)
(
)
(
)
(
)
(
)
(
:
identity
simple
very
a
recall
us
Let
above.
(3)
case
in
as
function,
n
correlatio
–
auto
law
–
power
a
by
zed
characteri
processes
stochastic
in
observed
is
behavior
This
scale.
temporal
typical
a
of
lack
by the
zed
characteri
are
variables
random
correlated
range
–
Long
.
correlated
range
–
long
or
dependent
strongly
be
to
said
are
variables
random
the
case
In this
.
lim
:
or
diverges,
n terms
correlatio
of
sum
The
2.
1
1
0
1
0
1
1
n
t
i
t
t
P
t
P
t
P
P
t
P
t
P
x
x
i
n
i
i
i
n
i
i
n
n
i
n
k
k
i
i
D
×
=
–
+
=
D
+
=
¥
=
å
å
å
å
=
+
=
=
=
+
+¥
®
(
)
(
)
(
)
.
shifts
–
time
small
near
)
(
test
the
of
slope
by the
series
time
a
of
properties
l
statistica
reverting
–
mean
and
following
–
end
between tr
h
distinguis
easily
can
we
test,
this
using
Therefore,
.
1
2
1
)
(
:
that
show
can
one
integer
any
for
Further,
)
3
.
3
2
3
4
1
)
3
(
that
follows
which
from
,
2
4
3
:
have
we
3
case
in the
Similarly,
)
2
.
1
)
2
(
have
we
Therefore,
.
1
2
:
have
we
2
case
In the
1)
.
)
1
(
)
(
)
(
:
test
Ratio
Variance
following
he
consider t
us
Let
).
(
)
2
(
:
have
obviously,
Then we,
).
(
)
2
(
and
),
(
)
(
:
shift
time
same
over the
changes
price
successive
wo
consider t
us
Let
Walk.
Random
a
of
those
to
series
time
random
a
of
properties
l
statistica
compare
to
1987
around
al.
et.
Lo
Andrew
by
proposed
test was
simple
A very
1
1
2
1
2
2
1
2
2
2
3
2
1
1
1
2
2
2
1
2
1
2
1
q
q
VR
q
k
q
VR
q
VR
p
p
p
q
VR
p
p
q
Var
q
q
Var
q
VR
t
p
t
p
p
p
t
p
t
p
p
t
p
t
p
p
k
q
k
r
r
r
r
s
r
s
s
r
r
s
t
t
t
t
t
×
÷
÷
ø
ö
ç
ç
è
æ
–
×
+
=
+
+
=
×
×
+
×
×
+
×
=
D
+
D
+
D
=
+
=
+
×
×
=
D
+
D
=
×
=
–
+
=
D
+
D
+
–
+
=
D
–
+
=
D
å
–
=
(
)
(
)
:
form
graphical
In
.
dominating
reversion
–
mean
to
s
correspond
which
,
1
for
,
;
dominating
Walk
Random
to
s
correspond
which
,
1
for
,
)
(
:
cases
limiting
wo
consider t
can
we
Therefore
.
1
0
for
1
if
1
1
1
2
)
(
:
ce
for varian
that
show
can
one
it,
Using
.
:
is
it
of
on)
substituti
recursive
(by
solution
The
.
0
;
variables
random
i.i.d.
–
and
,
1
0
for
,
:
process
reverting
–
mean
discrete
following
he
consider t
us
let
Now,
.
)
(
)
(
or
,
)
(
)
(
)
(
)
(
:
have
Walk we
Random
a
for
that
recall
us
Let
it.
of
version
continuous
then
and
discrete
a
first
solving
by
processes,
reverting
–
mean
such
of
details
he
consider t
now
will
we
follows
In what
process.
Uhlenbeck
–
Ornstein
an
by
modeled
as
reversion
–
mean
of
effect
the
simulated
and
introduced
already
have
Earlier we
2
2
2
2
2
1
0
1
0
k
1
2
2
ï
ï
î
ï
ï
í
ì
>>
<<
®
<<
<
-
=
-
×
®
-
-
×
=
×
=
î
í
ì
=
£
<
+
×
=
µ
=
µ
-
+
=
D
º
-
-
=
-
-
+
å
e
t
e
s
e
t
t
s
t
e
e
a
e
s
a
a
s
t
x
a
x
a
x
a
t
t
t
t
t
t
et
t
V
e
V
x
x
x
x
V
S
t
p
t
p
p
V
k
l
l
l
k
k
k
k
k
(
)
(
)
(
)
(
)
move.
price
unchanged
an
observe
y to
probabilit
the
be
will
1
then
move;
negative
a
observe
y to
probabilit
-
move;
positive
a
observe
y to
probabilit
-
:
measure
also
will
again,
we,
separation
any
For
.
0
changes,
price
zero
e
consecutiv
of
chains
of
number
the
calculate
also
will
We
ds).
(down tren
direction
negative
in the
length
of
changes
price
ve
consequiti
of
chains
of
number
the
,
trends);
(up
direction
positive
in the
length
of
changes
price
ve
consequiti
of
chains
of
number
the
,
:
calculate
will
we
1hr},
15min,
5min,
min,
{1
set
in the
separation
-
time
particular
any
for
Here
q
p
q
p
N
l
l
N
l
l
N
-
-
-
-
-
+
t
t
t
t
t
:
hold
formulas
simple
following
the
separation
any time
and
RW
symmetric
-
non
For
t
1st Price Change2nd Price ChangeProbabilityType
++p^2C
+-p*qR
-+q*pR
--q^2C
+0p*(1-p-q)0
0+(1-p-q)*p0
-0q*(1-p-q)0
0-(1-p-q)*q0
00(1-p-q)^20
(
)
(
)
(
)
(
)
.
1
0
;
2
;
2
2
2
q
p
P
pq
R
P
q
p
C
P
+
-
=
=
+
=
(
)
(
)
(
)
(
)
(
)
(
)
(
)
ons.)
distributi
l
Exponentia
or
Poisson
called
are
ons
distributi
(similar
.
length
of
periods
flat
of
number
for the
-
L
q
-
p
-
1
1
-
L
-
M
,
0
and
;
length
of
trends
-
down
of
number
for the
-
L
q
1
-
L
-
M
,
;
length
of
trends
-
up
of
number
for the
-
L
p
1
-
L
-
M
,
:
RW
asymmetric
an
for
chains
of
numbers
for those
solutions
exact
the
are
length
chain
the
and
,
separation
time
,
length
series
for time
Here,
L
l
N
L
L
N
L
L
N
l
M
×
=
×
=
-
×
=
+
t
t
t
t
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
.
2
,
,
,
and
,
2
,
,
,
:
,
,
,
:
response
the
influences
part
asymmetric
only the
which
of
parts,
asymmetric
and
symmetric
a
into
decomposed
be
can
It
.
of
change
price
preceding
y
immediatel
the
subject to
of
change
price
of
function
density
y
probabilit
a
is
,
Here
.
push"
"
a
to
response
mean
l
conditiona
the
as
,
:
as
defined
is
response"
"
al,
et.
Trainin
V.
by
papers
in
defined
As
y
x
P
y
x
P
y
x
P
y
x
P
y
x
P
y
x
P
y
x
P
y
x
P
y
x
P
x
y
x
P
y
x
P
x
y
P
x
dy
x
y
P
y
y
y
a
s
a
s
x
-
-
=
-
+
=
+
=
=
×
×
=
ò
¥
+
¥
-
(
)
(
)
axys.
-
x
same
on the
all
put them
to
us
allows
which
hours,
in trading
measured
here
is
separation
Time
minutes.
by
separated
changes
price
two
of
t
coefficien
n
correlatio
-
auto
is
and
minutes,
in
separation
time
discrete
a
is
where
,
1
2
1
1
)
(
:
is
ratio
variance
MacKinlay,
Lo,
Campbell,
in
defined
As
k
1
1
q
k
q
q
k
q
Var
q
Var
q
VR
q
k
k
r
r
å
-
=
×
÷
÷
ø
ö
ç
ç
è
æ
-
×
+
=
×
º
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
.
if
,
2
1
2
)
(
:
have
we
variance
for the
Lastly,
.
1
2
)
(
)
(
:
have
we
covariance
For the
.
if
,
)
(
:
have
mean we
for the
that
see
can
e
solution w
general
this
Using
.
)
(
1
)
0
(
)
(
:
get
we
)
(
to
reverting
After
.
)
(
1
)
0
(
)
(
:
solution
general
the
has
which
,
:
get
on we
substituti
after
Then
.
:
form
in the
solution
seek the
us
Let
.
:
case
model
Uhlenbeck
-
Ornstein
linear
continuous
with the
proceed
us
let
Now
2
2
2
2
2
)
,
min(
2
)
(
2
0
)
(
0
+¥
®
®
-
×
×
=
-
×
×
=
-
-
+¥
®
®
×
×
+
-
×
+
×
=
×
×
+
-
×
+
=
×
+
×
×
=
×
×
=
×
+
×
-
×
-
=
-
×
+
-
=
-
-
-
=
-
-
ò
ò
t
e
e
x
-
t
x
e
e
x
s
x
x
t
x
t
x
t
x
s
dW
e
e
x
e
x
t
x
t
x
s
dW
e
e
x
y
t
y
dW
dt
x
e
dy
e
y
x
dW
dt
x
x
dx
t
t
t
s
t
s
t
s
t
s
t
t
t
s
s
t
t
t
a
s
a
s
a
s
s
s
s
a
s
a
a
a
a
a
a
a
a
a
a
a
a
g.
forecastin
some
for
allow
do
ns
applicatio
financial
of
case
in the
which
processes
g
interestin
most
the
are
These
scales.
time
stic
characteri
many
with
process
a
from
scale
time
stic
characteri
no
has
which
noise
-
1
h
distinguis
to
diffucult
is
It
.
stationary
-
non
are
processes
Such
etc.
highway,
a
on
low
in traffic
ns
fluctuatio
e;
turbulenc
strong
in
velocity
of
ns
fluctuatio
rs;
transisto
and
diodes
in
ns
fluctuatio
current
:
phenomena
of
variety
in wide
observed
were
processes
Such
noise".
-
1
"
called
often
are
and
g
interestin
very
are
2
0
cases
between
-
in
the
All
Walk.
Random
to
s
correspond
2
case
the
and
noise
white
to
s
correspond
0
case
that the
concluded
we
page
previous
On the
.
and
2
0
where
,
)
(
:
form
the
of
spectrum
energy
a
with
process
stochastic
a
consider
us
Let
economic.
and
,
biological
physical,
-
systems
many
in
observed
are
functions
n
correlatio
-
auto
law
-
Power
.
correlated
range
-
long
are
function
n
correlatio
-
auto
law
-
power
a
by
zed
characteri
processes
stochastic
seen that
already
have
We
w
w
h
h
h
h
w
w
h
h
<
<
=
=
=
<
<
=
const
C
C
E
.
statistics
point
-
two
only the
know
to
sufficient
is
it
studies
most
in
but
,
and
,
every
for
)
,
,
,
;
,
,
,
(
function
density
y
probabilit
joint
the
know
to
need
we
process,
stochastic
a
of
properties
l
statistica
the
describe
fully
o
Although t
.
at time
variable
random
the
and
at time
variable
random
the
observing
of
function
density
y
probabilit
joint
the
is
)
,
;
,
(
where
,
)
,
;
,
(
)
(
)
(
:
function
n
correlatio
-
auto
the
define
to
sufficient
is
it
case,
order
nd
-
2
In the
.
at time
variable
random
the
observing
of
function
density
y
probabilit
the
is
)
,
(
where
,
)
,
(
)
(
:
mean
the
define
to
sufficient
is
it
case
order
st
-
1
in the
example,
For
.
properties
l
statistica
order
th
-
of
in terms
written
be
can
process
stochastic
a
zing
characteri
es)
(measurabl
s
observable
l
statistica
The
(iid).
d
distribute
y
identicall
t
independen
is
)
(
process
stochastic
that the
implies
ty
stationari
t,
independen
are
variables
stochastic
When the
first.
changes,
price
of
ty
stationari
of
property
the
discuss
us
Let
2
1
2
1
2
2
1
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
n
t
x
t
t
t
x
x
x
P
t
x
t
x
t
t
x
x
P
dx
dx
t
t
x
x
P
x
x
t
x
t
x
t
x
t
x
P
dx
t
x
P
x
t
x
n
t
x
i
i
n
n
K
K
ò
ò
ò
¥
¥
-
¥
¥
-
¥
¥
-
×
×
×
×
=
×
×
×
=