x = 0.821339303278721..., N = 60000
x = 0.3866452376145986..., N = 60000
1.5
x = 0.4201141490785808..., N = 60000
0.4
1.5
1.0
0.2
0.0
0.5
-0.2
1.0
-0.4
0.0
0.2
0.4
0.6
0.8
1.0
0.0
-0.2
x = 0.9062394051094856..., N = 60000
0.0
0.2
0.4
0.6
x = 0.7504136536635329..., N = 60000
0.2
0.8
0.0
0.5
0.6
-0.2
-0.4
0.4
-0.6
0.2
-0.8
0.0
0.0
-1.0
-0.6
-0.4
-0.2
0.0
0.0
0.2
0.4
0.6
0.8
-0.4
Curlicues {XN (t)}0<t≤1 for five randomly chosen x, and α = c0 = 0, c1 =
-0.2
√
0.0
0.2
2. The color
ranges from red at t = 0 to blue at t = 1.
1
Re[SN (x)], N = 10000, sample size =310000
0.6
0.010
0.5
0.008
0.4
0.006
0.3
0.004
0.2
0.002
0.1
0.0
0.000
-3
-2
-1
0
1
2
3
-3
-2
-1
0
1
2
3
The value distribution for the real part of XN (1), N = 10000. The continuous curve is the tail
d
estimate for the limit density − dx
P{Re X(1) ≥ x} ∼
45
|x|−7
8π 2
as |x| → ∞.
2
1.0
0.5
t
0.0
7→
Re(XN (t))
for the five curlicues
{XN (t)}0<t≤1 shown in
Figure 1.
-0.5
0.0
0.2
0.4
0.6
0.8
1.0
0.5
Five sample paths for
0.0
the Wiener process.
-0.5
-1.0
0.0
0.2
0.4
0.6
0.8
1.0
3