19
- p. 1/19 Stochastic Processes: Examples STATS116 – Dec. 29, 2004 Jonathan Taylor
- p. 1/19
Stochastic Processes: Examples
STATS116 – Dec. 29, 2004
Jonathan Taylor
Outline
● Outline
Convergence
Stochastic Processes
Conclusions
- p. 2/19
Outline
■ Illustration of CLT, WLLN, SLLN.■ Stochastic processes.■ Poisson process.■ Smooth processes in 1D.■ Fractal and smooth processes in 2+D.
Outline
Convergence
● Central Limit Theorem I
● Central Limit Theorem II
● Weak Law of Large Numbers
● Strong Law of Large Numbers
Stochastic Processes
Conclusions
- p. 3/19
Central Limit Theorem I
−2 −1 0 1 2
0.0
0.4
0.8
xval
CD
F(x
val)
−2 −1 0 1 2
0.0
0.4
0.8
xval
CD
F(x
val)
−2 −1 0 1 2
0.0
0.4
0.8
xval
CD
F(x
val)
−2 −1 0 1 2
0.0
0.4
0.8
xval
CD
F(x
val)
Outline
Convergence
● Central Limit Theorem I
● Central Limit Theorem II
● Weak Law of Large Numbers
● Strong Law of Large Numbers
Stochastic Processes
Conclusions
- p. 4/19
Central Limit Theorem II
−4 −2 0 2 4
02
46
8
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
−4 −2 0 2 4
−2
02
4
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
−4 −2 0 2 4
−2
02
4
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
−4 −2 0 2 4
−2
02
4
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
Outline
Convergence
● Central Limit Theorem I
● Central Limit Theorem II
● Weak Law of Large Numbers
● Strong Law of Large Numbers
Stochastic Processes
Conclusions
- p. 5/19
Weak Law of Large Numbers
0 200 400 600 800 1000
0.0
0.1
0.2
0.3
0.4
n
P(|
Xba
r(n)
| > d
elta
)
Outline
Convergence
● Central Limit Theorem I
● Central Limit Theorem II
● Weak Law of Large Numbers
● Strong Law of Large Numbers
Stochastic Processes
Conclusions
- p. 6/19
Strong Law of Large Numbers
0e+00 2e+05 4e+05 6e+05 8e+05 1e+06
−0.
04−
0.02
0.00
0.02
0.04
n
Xba
r[n]
Outline
Convergence
Stochastic Processes
● Stochastic Processes
● Poisson Process
● Brownian Motion I
● Brownian Motion II
● Brownian Motion III
● Brownian Motion IV
● Smooth processes I
● Smooth processes II
● Fractal process in the plane
● Smooth process in the plane
● Intersections in the plane
Conclusions
- p. 7/19
Stochastic Processes
■ A sequence
� � � � � �� is just a function �� � � .
■ A sequence of random variables
� � ��� � � � � � � � �� istherefore a random function from
� � .
■ No reason to only consider functions defined on
�
: whatabout functions
�
?■ Example: Poisson process, rate
�.
Outline
Convergence
Stochastic Processes
● Stochastic Processes
● Poisson Process
● Brownian Motion I
● Brownian Motion II
● Brownian Motion III
● Brownian Motion IV
● Smooth processes I
● Smooth processes II
● Fractal process in the plane
● Smooth process in the plane
● Intersections in the plane
Conclusions
- p. 8/19
Poisson Process
0 1 2 3 4
01
23
4
t
N(t
)
Outline
Convergence
Stochastic Processes
● Stochastic Processes
● Poisson Process
● Brownian Motion I
● Brownian Motion II
● Brownian Motion III
● Brownian Motion IV
● Smooth processes I
● Smooth processes II
● Fractal process in the plane
● Smooth process in the plane
● Intersections in the plane
Conclusions
- p. 9/19
Brownian Motion I
■ Let
� � � � � � � be a sequence of IID random variableswith mean 0, variance � � .
■
� ��� �� ��� � �
� �� �
■ For each
�
,
� ��� � is approximately a
� ���� � �
for large � .■ For each � � �
,
� ��� �� � ��� � is approximately a
� � ��� �� � �
random variable for large � .■ As � � � the random function
� ��� � converges indistribution to something called Brownian motion.
■ Model comes up in physics (studied by Einstein), finance(used in Black-Scholes options pricing), random walk.
Outline
Convergence
Stochastic Processes
● Stochastic Processes
● Poisson Process
● Brownian Motion I
● Brownian Motion II
● Brownian Motion III
● Brownian Motion IV
● Smooth processes I
● Smooth processes II
● Fractal process in the plane
● Smooth process in the plane
● Intersections in the plane
Conclusions
- p. 10/19
Brownian Motion II
0.0 0.2 0.4 0.6 0.8 1.0
−1.
2−
1.0
−0.
8−
0.6
−0.
4−
0.2
0.0
0.2
t
W
Outline
Convergence
Stochastic Processes
● Stochastic Processes
● Poisson Process
● Brownian Motion I
● Brownian Motion II
● Brownian Motion III
● Brownian Motion IV
● Smooth processes I
● Smooth processes II
● Fractal process in the plane
● Smooth process in the plane
● Intersections in the plane
Conclusions
- p. 11/19
Brownian Motion III
■ If
� � � � � �� is a Poisson process with rate
�
, then, for large
�
� ��� � �� � � � � �� � � �� � ���� � ���
■ As
� � � � � � � � also converges to Brownian motion.
Outline
Convergence
Stochastic Processes
● Stochastic Processes
● Poisson Process
● Brownian Motion I
● Brownian Motion II
● Brownian Motion III
● Brownian Motion IV
● Smooth processes I
● Smooth processes II
● Fractal process in the plane
● Smooth process in the plane
● Intersections in the plane
Conclusions
- p. 12/19
Brownian Motion IV
0.0 0.2 0.4 0.6 0.8 1.0
−0.
20.
00.
20.
40.
60.
8
t
N(t
)
Outline
Convergence
Stochastic Processes
● Stochastic Processes
● Poisson Process
● Brownian Motion I
● Brownian Motion II
● Brownian Motion III
● Brownian Motion IV
● Smooth processes I
● Smooth processes II
● Fractal process in the plane
● Smooth process in the plane
● Intersections in the plane
Conclusions
- p. 13/19
Smooth processes I
■ A smooth process: fix
�
“frequencies” ��� � � �� � ■
� ��
� ��� � �� ��� � � �� � � � � �� � � � � �� ��
where
� ��� � � �� � � � are IID
� ���� � �random variables.
Outline
Convergence
Stochastic Processes
● Stochastic Processes
● Poisson Process
● Brownian Motion I
● Brownian Motion II
● Brownian Motion III
● Brownian Motion IV
● Smooth processes I
● Smooth processes II
● Fractal process in the plane
● Smooth process in the plane
● Intersections in the plane
Conclusions
- p. 14/19
Smooth processes II
0.0 0.2 0.4 0.6 0.8 1.0
−0.
20.
00.
20.
40.
60.
8
t
N(t
)
Outline
Convergence
Stochastic Processes
● Stochastic Processes
● Poisson Process
● Brownian Motion I
● Brownian Motion II
● Brownian Motion III
● Brownian Motion IV
● Smooth processes I
● Smooth processes II
● Fractal process in the plane
● Smooth process in the plane
● Intersections in the plane
Conclusions
- p. 15/19
Fractal process in the plane
0.0
0.5
1.0
1.5
2.0
0.00.51.01.52.0
X
Y
Outline
Convergence
Stochastic Processes
● Stochastic Processes
● Poisson Process
● Brownian Motion I
● Brownian Motion II
● Brownian Motion III
● Brownian Motion IV
● Smooth processes I
● Smooth processes II
● Fractal process in the plane
● Smooth process in the plane
● Intersections in the plane
Conclusions
- p. 16/19
Smooth process in the plane
02
46
810
0.00.51.01.52.02.53.0
X
Y
Outline
Convergence
Stochastic Processes
● Stochastic Processes
● Poisson Process
● Brownian Motion I
● Brownian Motion II
● Brownian Motion III
● Brownian Motion IV
● Smooth processes I
● Smooth processes II
● Fractal process in the plane
● Smooth process in the plane
● Intersections in the plane
Conclusions
- p. 17/19
Intersections in the plane
02
46
810
0246810
X
Y
Outline
Convergence
Stochastic Processes
Conclusions
● Conclusions
● Trying these examples out
- p. 18/19
Conclusions
■ Stochastic processes are natural generalizations ofsequences of random variables.
■ This is what probabilists do.■ Some neat pictures.
Outline
Convergence
Stochastic Processes
Conclusions
● Conclusions
● Trying these examples out
- p. 19/19
Trying these examples out
■ Download R,http://cran.r-project.org/bin/windows/base/.Package used by statisticians (more on R in STATS 191 ifyou take it with me).
■ In the GUI, assuming you are on a network, try typinginstall.packages(’RandomFields’) [Enter]
source(’http://www-stat.stanford.edu/˜jtaylo/courses/stats116/simulation/brownian-motion.R’)[Enter]
■ Other examples in same directory.
