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Unformatted text preview: . 2N +1
2n = n=1 In the limit, as N → ∞, we obtain R∞ N � In
= . 2n
n=1 Thus, the binary expansion of R∞ is
R∞ = 0.I1 I2 I3 I4 I5 I6 . . .
From this form, it is clear that every possible binary expansion between 0 and 1 can be achieved,
each with equal probability. Thus R∞ is uniformly distributed on the interval [0, 1]. Since the same
is true for S∞ , we know that the joint PDF is given by
� 1
for 0 < r < 1, 0 < s < 1
P∞ (r, s) =
0
otherwise.
Thus the PDF of X∞ is given by
P∞ (r, s) = � 1
2 0 for x − y  < 1, x + y  < 1
otherwise. M. Z. Bazant – 18.366 Random Walks and Diﬀusion – Problem Set 3 Solutions 1 1 0.8
C∞ (q ) C∞ (x) 0.8
0.6
0.4 0.6
0.4
0.2 0.2 0 0
0.4 0.2 0 0.2 0 0.4 0.2 0.4 0.6 0.8 1 q x
Figure 3: The cumulative distribution func�
tion of the x component of X∞ for a = 1/3. 2.3 5 Figure 4: The cumulative distribution function of the rotated variable q = 1/2 + x − y
for a = 1/3. �
The CDF of X∞ for a = 1/3 �
To compute the CDF of x component of X∞ for a = 1/3, we ﬁrst consider the possible values that
it can take. The maximum that x can increase by at the nth step is 1/3n and thus the maximum
value that x∞ can take is
∞
�1
1
1
=
1 = 2.
n
3
3(1 − 3 )
n=1
By symmetry, we therefore know that X∞ can take values in the range [−1/2, 1/2]. We also note
that the sum of steps starting at n = 2 follows the same distribution as X∞ , but scaled by 1/3. If
the ﬁrst step is ΔX1 , then X∞ lies in the interval [ΔX1 − 1/6, ΔX1 + 1/6]. For the three possible
choices ΔX1 = −1/3, 0, 1/3 these intervals are mutually exclusive, and this allows us to write the
CDF C∞ (x) recursively as
⎧
for x < − 1
⎨ C∞ (3x + 1)/4
6
1/4 + C∞ (3x)/2
for − 1 < x < 1
C∞ (x) =
6
6
⎩
3/4 + C∞ (3x − 1)/4
for 1 < x
6
This can be easily calculated using a recursive function, and a graph is shown in ﬁgure 3. It is also
�
interesting to consider the CDF of X∞ in a coordinate rotated by 45◦ . Suppose we consider the
coordinate transformation
p=
q= 1
+x+y
2
1
+x−y
2 Then we ﬁnd that the walk from starts fro...
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 Fall '06
 MartinBazant

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