The 22nd Annual Meeting in Mathematics (AMM 2017)
Department of Mathematics, Faculty of Science
Chiang Mai University, Chiang Mai, Thailand
Uniform random walk on unit interval
Kittisak Chaiyotha† , Thanawit Jeeruphan‡
Department of Mathematics Statistics and Computer, Faculty of Science
Ubon Ratchathani University, Ubon Ratchathani 34190, Thailand
Abstract
We study the first-passage properties of a uniform random walk in the unit interval.
Each step of a uniform random walk is uniformly distributed over [−a, na] where n is a
natural number greater than 1 and a > 1. In this project, we found that the exit probability
and mean exit time of uniform random walk are functions of the starting point.
Keywords: first-passage properties, random walk, uniformly distribution.
2010 MSC: 60G50.
1
Introduction
Random walk is a mathematical object known as random process that describes a path of
a succession of random steps on some mathematical space such as integer. There are several
types of random walk, for example, simple random walk [1], bursty random walk [2], symmetric
random walk [3] and uniform random walk [4]. In this work we will study some properties of
uniform random walk.
We consider a random walk where each step depend on uniform distribution. In 2006,
T. Antal and S. Redner [4] studied random walk which the length of each step is uniformly
distributed over the range [−a, a] where a is a real number. In this project, we consider a
random walk where each step size is drawn from uniform distribution on [−a, na] where n is a
natural number greater than 1 and a > 1. Uniform random walk is a random walk where each
step size is uniformly distributed. The walk begins at an arbitrary point in the unit interval,
when the walker leaves the unit interval by passing its end points, we can treat the walker as
being trapped exactly at that end point.
In this project, we study first passage properties of uniform random walk in the unit interval
[0, 1]. Let Ra,n (x) be right exit probability of uniform random walk, defined as uniform random
walk starting at arbitrary x to eventually cross the boundary at x = 1. We need only consider
uniform random walk exit the right boundary and we also consider the mean exit time of the
walk, ⟨ta,n ⟩, i.e., the average of time needed to exit the interval.
∗
This research was financially supported by The Science Achievement Scholarship of Thailand
Corresponding author
†
Speaker
E-mail address: [email protected] (K. Chaiyotha), [email protected] (T. Jeeruphan).
‡
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2
Preliminaries
Unlike the usual random walk which each step size is discrete, we study the random walk which
each step size is drawn from a continuous distribution. First, we give a definition of uniform
distribution.
Definition 2.1. A continuous random variable X is said to have a uniform distribution over
the interval [a, b], denoted by X ∼ unif orm(a, b), if its probability density function is given by
{
1
if x ∈ [a, b] ;
fX (x) = b−a
0
otherwise.
Next, we give a definition of random walk in which the length of each step is uniformly
distributed over the range [−a, na] where n is a natural number greater than 1.
Definition 2.2. Let Y1 , Y2 , Y3 , ... be a sequence of independent and identically distributed random variables, defined on probability space with uniform distribution on [−a, na] where n is a
natural number greater than 1. Define a sequence of random variables {Xk : k ≥ 0} by
Xk = x +
k
∑
Yi .
i=1
Then {Xk } is said to be a uniform random walk starting at x ∈ R.
The first passage properties, i.e., right exit probability, exit time and survival probability
are studied by T. Antal and S. Redner [4]. So, we give a definition of right exit probability, exit
time and survival probability, respectively.
Definition 2.3. Let {Xk } be uniform random walk starting at x as defined in Definition 2.2.
The right exit probability Ra,n (x) of {Xk } is the probability that Xk will reach the right side
of the unit interval before reaching its left side, that is,
Ra,n (x) = P (min{k|Xk > 1} < min{k|Xk < 0}).
Clearly, Ra,n (x) = 0 when x < 0 and Ra,n (x) = 1 when x > 1. It can also be shown that
Ra,n (x) satisfies the following equation
∫
Ra,n (x) = P (Xk+1 = y|Xk = x)Ra,n (y) dy
whenever x is between zero and one.
Definition 2.4. Let {Xk } be uniform random walk starting at x. ta,n (x), exit time, is defined
by
∫
ta,n (x) = P (Xk+1 = y|Xk = x) [ta,n (y) + 1] dy.
Definition 2.5. Let {Xk } be uniform random walk starting at x. Sa,n (k), survival probability,
is defined by
∫
Sa,n (k) = P (0 ≤ Xi ≤ 1, ∀i = 1, ..., k|X0 = x)dx
where n > 0.
We conclude this section by given the theorem studied by T. Antal and S. Redner [4] where
a > 1.
Theorem 2.6. (T. Antal and S. Redner [4]) Let {Xk } be uniform random walk starting at x
in which the length of each step is uniformly distributed over the range [−a, a] where a ∈ R and
a > 1. Right exit probability is given by
Ra,1 (x) =
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x+a−1 1 1
2 (x + a) − 1
+
=
.
2a
2 2a
4a
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3
Main Results
We will only consider uniform random walk in [0, 1]. Each step of a uniform random walk is
uniformly distributed over [−a, na] where n is natural number greater than 1 and a > 1.
Theorem 3.1. Let {Xk } be uniform random walk starting at x in which the length of each step
is uniformly distributed over the range [−a, na] where n is natural number greater than 1 and
a > 1. Right exit probability is given by
Ra,n (x) =
na − 12
x + na − 1
+
.
(n + 1)a
((n + 1)a − 1)(n + 1)a
Figure 1: Probability distribution of the uniform random walk after single step.
Proof. Let {Xk } be uniform random walk starting at x. By definition of right exit probability,
we split [x − a, x + na] to [x − a, 0], [0, 1], [1, x + na] for integration right exit probability, so
∫
Ra,n (x) = P (Xk+1 = y|Xk = x)Ra,n (y) dy
∫ x+na
1
Ra,n (y)dy
=
(n
+
1)a
x−a
)
(∫ 0
∫ x+na
∫ 1
1
Ra,n (y)dy
Ra,n (y)dy +
=
Ra,n (y)dy +
(n + 1)a
1
0
x−a
(
)
∫ 1
1
=
0+
Ra,n (y)dy + (x + na − 1)
(n + 1)a
0
∫ 1
1
x + na − 1
.
(*)
=
Ra,n (y)dy +
(n + 1)a 0
(n + 1)a
Next, integrating both sides over the unit interval yields
∫
1
0
Since
∫1
0
Ra,n (y) =
∫
1 (∫ 1
1
x + na − 1
Ra,n (x)dx =
Ra,n (y)dydx +
dx
(n
+
1)a
(n + 1)a
0
0
∫ 1∫ 1
na − 12
1
=
Ra,n (y)dydx +
(n + 1)a 0 0
(n + 1)a
∫ 1
na − 21
1
=
Ra,n (x)dx +
.
(n + 1)a 0
(n + 1)a
∫1
0
)
Ra,n (x), we have
(
1−
)∫
na − 21
Ra,n (x)dx =
(n + 1)a
0
(
)∫ 1
na − 21
(n + 1)a − 1
Ra,n (x)dx =
(n + 1)a
(n + 1)a
0
∫ 1
na − 21
.
Ra,n (x)dx =
(n + 1)a − 1
0
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1
(n + 1)a
1
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We substitute the last equation in (*); therefore,
na − 12
x + na − 1
Ra,n (x) =
+
.
(n + 1)a
((n + 1)a − 1)(n + 1)a
As a → ∞ we found that exit probability is independent of the starting point, we get
n
n+1 .
Ra,n (x) =
When a → 1, we get Ra,n (1) −→
n2 +n− 12
n(n+1) .
Theorem 3.2. Let {Xk } be uniform random walk starting at x in which the length of each step
is uniformly distributed over the range [−a, na] where n is natural number greater than 1 and
a > 1. Mean exit time is given by
⟨ta,n ⟩ =
1
1−
1
(n+1)a
.
Proof. Let {Xk } be uniform random walk starting at x. The survival probability after k step
is Sa,n (k) = (1/(n + 1)a)k . This is come from probability function of unif orm(−a, na). By
definition of exit time and survival probability, we get
∫
⟨ta,n ⟩ =
1
ta,n (x)dx
∫ 1∫ 1
P (Xk+1 = y|Xk = x) [ta,n (y) + 1] dydx
=
0
0
[∫ 1
]
∫ 1∫ 1
=
P (Xk+1 = y|Xk = x) (
P (Xk+2 = s|Xk+1 = y)[ta,n (s) + 1]ds) + 1 dydx
0
0
0
∫ 1∫ 1
∫ 1
=
P (Xk+1 = y|Xk = x)(
P (Xk+2 = s|Xk+1 = y)[ta,n (s) + 1]ds)dydx
0
0
0
∫ 1∫ 1
+
P (Xk+1 = y|Xk = x)dydx
0
0
∫ 1∫ 1
∫ 1
=
[ta,n (s) + 1]
P (Xk+1 = y|Xk = x)P (Xn+2 = s|Xn+1 = y)dydsdx + Sa,n (1).
0
0
0
0
(**)
By transition probability, we get
∫
∫
1
P (Xk+1 = y|Xk = x)P (Xk+2 = s|Xk+1 = y)dy = P (Xk+2 = s|Xk = x)
0
1
P (Xk+1 = y)dy
0
= P (Xk+2 = s|Xk = x).
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We substitute the last equation in (**); therefore,
∫
⟨ta,n ⟩ =
1∫ 1
P (Xk+2 = s|Xk = x)[ta,n (s) + 1]dsdx + Sa,n (1)
∫ 1
∫ 1∫ 1
P (Xk+2 = s|Xk = x)[(
P (Xk+3 = z|Xk+2 = s)[ta,n (z) + 1]dz) + 1]dsdx + Sa,n (1)
=
0
0
0
∫ 1∫ 1
∫ 1
=
P (Xk+2 = s|Xk = x)(
P (Xk+3 = z|Xk+2 = s)[ta,n (z) + 1]dz)dsdx
0
0
0
∫ 1∫ 1
+
P (Xk+2 = s|Xk = x)dsdx + Sa,n (1)
0
0
∫ 1∫ 1
∫ 1
=
[ta,n (z) + 1]
P (Xk+2 = s|Xk = x)P (Xk+3 = z|Xk+2 = s)dsdzdx + Sa,n (2) + Sa,n (1)
0
0
0
∫ 1∫ 1
= Sa,n (1) + Sa,n (2) +
P (Xk+3 = z|Xk = x)[ta,n (z) + 1]dzdx
0
0
0
0
= Sa,n (1) + Sa,n (2) + Sa,n (3) + . . .
∞
∑
=
Sa,n (k)
k=1
1
+
=1+
(n + 1)a
1
=
.
1
1 − (n+1)a
(
1
(n + 1)a
)2
(
+
1
(n + 1)a
)3
+ ...
When a → ∞, ⟨ta,n ⟩ → 1, while a → 1 from above, ⟨ta,n ⟩ → n+1
n .
We study behavior of exit probability of uniform random walk using Scilab. In case
n = 2, we plot the theoretical result of right exit probability which step size is drawn from
unif orm(−a, 2a).
We have a simulation using Scilab. We let it run for 10000 times where length of uniform
distribution a = 1, 1.1, 1.5, 1.9, 2, 3, 4, 5, 10, 100 and we vary starting point from 0.1 − 0.9. We
get the graph of numerical result.
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Also, case n = 3, we plot the theoretical result of right exit probability which step size is
drawn from unif orm(−a, 3a).
We have a simulation using Scilab in case n = 3. We let it run for 10000 times where
length of uniform distribution a = 1, 1.1, 1.5, 1.9, 2, 3, 4, 5, 10, 100 and we vary starting point
from 0.1 − 0.9. We get the graph of numerical result.
We found that the graph of numerical result is very similar to the graph of theoretical
result. Furthermore, if we increase the number of trials, then the graph of numerical result will
be smoother.
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4
Conclusion
We can see that when we change the distribution of step size, the walk becomes asymmetric
random walk. The right exit probability depends on both the a, length of step size, and n,
asymmetric factor. This it very similar to the work of T. Antal and S. Redner [4] but, this is
not quite the same since our walk is asymmetric. If we substitute n = 1 in the mean exit time,
we recover their result. Also, we have the simulate of the walk in case n = 2, 3. We found that
graph of numerical result is similar to graph of theoretical result.
5
Appendix
This is an example of the code in Scilab which we used to simulate the uniform random walk.
n=10000;//number of experiment
R=0;//right exit probability
L=0;//left exit probability
S=0;//survival rate in unit interval after t=1
a=0.4;//length of uniformly
x=0.9;//starting point
t=0; //dummy index for time
m=0; //Mean exit time
for i=1:n do //loop for number experiment
while (0<=x & x<=1 & t<=100)
//loop for check condition of random walk
u=-a+(3*a)*rand(1,1,"uniform") //step of uniformly distribution
x = x +u
//addition of step random walk
t=t+1 //number of time
end //end loop for check condition of random walk
if x>1 then //condition for check right exit probability
R=R+1;
//count for right exit probability
elseif x<0 //condition for check left exit probability
L=L+1; //count for left exit probability
else
//condition for check survival probability
S=S+1
//count for survival
end
//end for check condition exit probability
x=0.9//reset starting point
m=m+t //count for time
t=0; //reset time
end //end loop for number experiment
m=m/n //average time
R=R/n //right exit probability
L=L/n //left exit probability
S=S/n //survival probability
disp(R) //show right exit probability
disp(L) //show left exit probability
disp(S) //show survival probability
disp(m) //show average time
Acknowledgment. The authors are very grateful to the referees for their careful reading of
the manuscript and their useful comments.
References
[1] E. Kosygina and J, Peterson, 15 December 2016. Excited random walks with Markovian
cookie stacks, arXiv:1504.06280. (2015).
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[2] M. D. Volovik and S. Redner, 23 September 2016. First-passage properties of bursty random
walks, arXiv:1005.1579. (2010).
[3] K. Lakatos-Lindenberg and K. E. Shuler, Random Walks with Non nearest Neighbor Transitions. I. Analytic 1D Theory for Next Nearest Neighbor and Exponentially Distributed
Steps, J. Math. Phys. 12, 633 (1971).
[4] T. Antal and S. Redner, Escape of a Uniform Random Walk from an Interval, J. Stat.
Phys. 123, 1129 (2006).
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