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Random Processes Random Processes Introduction Introduction (2)(2)
Professor Ke-Sheng Cheng Professor Ke-Sheng Cheng
Department of Bioenvironmental Systems Department of Bioenvironmental Systems
Engineering Engineering
E-mail: [email protected]
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Stochastic continuity
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Stochastic Convergence
A random sequence or a discrete-time random
process is a sequence of random variables
{ 1(ω ) 2(ω ) ! n(ω )!" # { n(ω )" ω ∈ .
$or a specific ω { n(ω )" is a sequence of
numbers t%at mi&%t or mi&%t not conver&e.
'%e notion of conver&ence of a random
sequence can be &iven several interpretations.
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Sure convergence
(convergence everywhere)'%e sequence of random variables
{ n(ω )" conver&es surel to t%e random
variable (ω ) if t%e sequence offunctions n(ω ) conver&es to (ω ) as n
→
∞
for all ω ∈
i.e.
n(ω ) → (ω ) as n → ∞ for all ω ∈ .
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Almost-sure convergence
(Convergence with probability 1)
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Mean-square convergence
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Convergence in probability
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Convergence in distribution
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Remars
onver&ence wit% probabilit one appliesto t%e individual reali*ations of t%erandom process. onver&ence in
probabilit does not. '%e wea+ law of lar&e numbers is ane,ample of conver&ence in probabilit.
'%e stron& law of lar&e numbers is an
e,ample of conver&ence wit% probabilit1.
'%e central limit t%eorem is an e,ampleof conver&ence in distribution.
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Weak Law of Large Numbers
(WLLN)
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Strong Law of Large Numbers
(SLLN)
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The Central Limit Theorem
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!enn diagram o" relation o"
types o" convergence
Note that even
sure convergencemay not implymean squareconvergence.
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#$ample
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Ergodic Theorem
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%he Mean-Square #rgodic
%heorem
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'%e above t%eorem s%ows t%at one can
e,pect a sample avera&e to conver&e to a
constant in mean square sense if andonl if t%e avera&e of t%e means
conver&es and if t%e memor dies out
asmptoticall t%at is if t%e covariancedecreases as t%e la& increases.
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Mean-#rgodic &rocesses
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Strong or 'ndividual #rgodic
%heorem
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#$amples o" Stochastic
&rocessesiid random process
A discrete time random process { (t ) t #
1 2 !" is said to be independent andidenticall distributed (iid ) if an finitenumber sa ! of random variables (t 1)
(t 2) ! (t ! ) are mutuall independent
and %ave a common cumulativedistribution function " (⋅) .
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'%e oint cdf for (t 1) (t 2) ! (t ! ) is
&iven b
t also ields
w%ere p( # ) represents t%e commonprobabilit mass function.
( )
)()()(
,,,),,,(
21
221121,,, 21
k X X X
k k k X X X
x F x F x F
x X x X x X P x x x F k
=
≤≤≤=
)()()(),,,( 2121,,, 21 k X X X k X X X x p x p x p x x x p k =
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Random walk rocess
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/et 0 denote t%e probabilit mass
function of 0. '%e oint probabilit of
0 1…
n is
( )
)|()|()(
)()()(
)()()(
,,,
),,,(
10100
10100
101100
101100
1100
−
−
−
−
=−−=
−=−===
−=−======
nn
nn
nnn
nnn
nn
x x P x x P x
x x f x x f x
x x P x x P x X P
x x x x x X P
x X x X x X P
π
π ξ ξ
ξ ξ
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)|()|()|()(
)|()|()|()(
),,,(),,,,(
),,,|(
1
10100
110100
1100
111100
110011
nn
nn
nnnn
nn
nnnn
nnnn
x x P x x P x x P x
x x P x x P x x P x
x X x X x X P x X x X x X x X P
x X x X x X x X P
+
−
+−
++
++
=
⋅=
=== =====
====
π
π
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'%e propert
is +nown as t%e ar+ov propert.
A special case of random wal+: t%e
rownian motion.
)|(),,,|( 1110011 nnnnnnnn x X x X P x X x X x X x X P ======= +++
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!aussian rocess
A random process { (t )" is said to be a3aussian random process if all finitecollections of t%e random process 1# (t 1) 2# (t 2) ! ! # (t ! ) are
ointl 3aussian random variables for all! and all c%oices of t 1 t 2 ! t ! .
4oint pdf of ointl 3aussian randomvariables 1 2 ! ! :
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Time series " #R random
rocess
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%he rownian motion
(one-dimensional also nown as random wal) onsider a particle randoml moves on a
real line.
5uppose at small time intervals τ t%e particle
umps a small distance randoml and
equall li+el to t%e left or to t%e ri&%t.
/et be t%e position of t%e particle on
t%e real line at time t .
)(t X τ
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Assume t%e initial position of t%e
particle is at t%e ori&in i.e.6osition of t%e particle at time t can be
e,pressed as
w%ere are independent random
variables eac% %avin& probabilit 172 of
equatin& 1 and 1.
( represents t%e lar&est inte&er not
e,ceedin& .)
0)0( =τ X
( )]/[21)( τ τ δ t Y Y Y t X +++=
,, 21 Y Y
[ ]τ /t
τ /t
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8istribution of τ (t )
/et t%e step len&t% equal t%en
$or fi,ed t if τ is small t%en t%edistribution of is appro,imatel
normal wit% mean 0 and variance t i.e.
.
δ τ
( )]/[21)( τ τ τ t Y Y Y t X +++=
)(t X τ
( )t N t X ,0~)(τ
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*raphical illustration o"
+istribution o" τ (t )
Time, t
PDF of X (t )
X (t )
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f t and h are fi,ed and τ is sufficientl
small t%en
( ) ( )[ ]
( )
[ ] [ ] [ ] +++=
+++=
+++−+++=−+
+++
+++
+
τ τ τ
τ τ
τ τ τ
τ τ τ τ
τ
τ
τ
ht t t
ht t t
t ht
Y Y Y
Y Y Y
Y Y Y Y Y Y t X ht X
2
]/)[(2]/[1]/[
]/[21]/)[(21)()(
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+istribution o" the
displacement
'%e random variable
is normall distributed wit% mean 0
and variance h i.e.
)()( t X ht X τ τ −+
)()( t X ht X τ τ −+
( )[ ] duh
u
h xt X ht X P
x
∫ ∞−
−=≤−+
2exp
2
1)()(
2
π τ τ
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9ariance of is dependent on t
w%ile variance of is not.
f t%en
are independent random variables.
)(t X τ
)()( t X ht X τ τ −+
mt t t 2210 ≤≤≤≤ )()( 12 t X t X τ τ −
,),()( 34 t X t X τ τ − )()( 122 −− mm t X t X τ τ
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t
X
C C
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Covariance and Correlation
"unctions o" )(t X τ
[ ] [ ]
[ ] [ ]
[ ] [ ] [ ] [ ] [ ]
[ ]
t
Y Y Y E
Y Y Y Y Y Y Y Y Y E
Y Y Y Y Y Y E
ht X t X E ht X t X Cov
t
ht t t t t
ht t
=
+++=
+++⋅
++++
+++=
+++⋅
+++=
+=+
+++
+
2
21
2121
2
21
2121
)()()(),(
τ
τ τ τ
τ τ
τ τ
τ τ τ τ
τ
τ
τ
[ ][ ]
( ) ( )ht t
t
ht t
ht X t X Cov
ht X t X Correl
+⋅
=
+⋅
+=
+
)(),()(),(
τ τ
τ τ
