# Tagged: Random Walks

## Simple Random Walk: Unrestricted Random Walk

A simple random (or unrestricted random walk) walk on a line or in one dimension occurs with probability $p$ when walker step forward (+1) and/or has probability $q=1-p$ if walker steps back ($-1$). For ith step, the modified Bernoulli random variable $W_i$ (takes the value $+1$ or $-1$ instead of {0,1}) is observed and the position of the walk at the nth step can be found by
\begin{align}
X_n&=X_0+W_1+W_2+\cdots+W_n\nonumber\\
&=X_0+\sum_{i=1}^nW_i\nonumber\\
&=X_{n-1}+W_n
\end{align}
In the gambler’s ruin problems X0=k, but here we assume (without loss of generality) that walks start from the origin so that Xo=0.

Several derived results for random walks are restricted by boundaries. We consider here random walks without boundaries called unrestricted random walks. We are interested in

1. The position of the walk after a number of steps and
2. The probability of a return to the origin, the start of the walker.

From equation (1) the position of the walker at step n simply depends on the position at (n-1)th step, because the simple random walk possesses the Markov property (the current state of the walk depends on its immediate previous state, not on the history of the walks up to the present state)

Furthermore Xn=Xn-1 ±1

and the transition probabilities from one position to another is
P(Xn=j|Xn-1=j-1)=p, and P(Xn=j|Xn-1=j+1)=q is independent of the number of plays in the game or steps is represented by n.

Mean and Variance of Xn can be calculated as:
\begin{align*}
E(X_n)&=E\left(X_0+\sum_{i=1}^n W_i\right)\\
&=E\left(\sum_{i=1}^n W_i\right)=nW_n\\
V(X_n)&=V\left(\sum_{i=1}^n W_i\right)=nV(W)
\end{align*}
Since $W_i$ are independent and identically distributed (iid) random variables and where $W$ is the common or typical Bernoulli random variable in the sequence {Wi}. Thus
\begin{align*}
E(W)&=1.p+(-1)q=p-q\\
V(W)&=E(W^2)-[E(W)]^2\\
&=1^2p+(-1)^2q-(p-q)^2\\
&=p+q-(p^2+q^2-2pq)\\
&=1-p^2-q^2+2pq\\
&=1-p^2-(1-p)^2+2pq\\
&=1-p^2-(1+p^2-2p)+2pq\\
&=1-p^2-1-p^2+2p+2pq\\
&=-2p^2+2p+2pq\\
&=2p(1-p)+2pq=4pq
\end{align*}
So the probability distribution of the position of the random walk at stage n has to mean E(Xn)=n(p-q) and variance V(Xn)=4npq.

For the symmetric random walk (where p=½) after n steps, the expected position is the origin, and it yields the maximum value of V(Xn)=4npq=4np(1-p).

If p>½ then drift is expected away from the origin in a positive direction and if p<½ it would be expected that the drift would be in the negative direction.

Since V(Xn) is proportional to $n$, it grows with increasing n, and we would be increasingly uncertain about the position of the walker as n increases.
i.e.
\begin{align*}
\frac{\partial V(X_n)}{\partial p}&=\frac{\partial}{\partial p} {4npq}\\
&=\frac{\partial}{\partial p} \{4np-4np^2 \}=4n-8np \quad \Rightarrow p=\frac{1}{2}
\end{align*}
Just knowing the mean and standard deviation of a random variable does not enable us to identify its probability distribution. But for large n, we can apply the CLT.
$Z_n=\frac{X_n-n(p-q)}{\sqrt{4npq}}\thickapprox N(0,1)$
Applying continuity correction, approximate probabilities may be obtained for the position of the walk.

Example : Consider unrestricted random walk with n=100, p=0.6 then
\begin{align*}
E(X_n)&=E(X_{100})=nE(W)=n(p-q)\\
&=100(0.6-0.4)=20\\
V(X_n)&=4npq=4\times 100\times 0.6 \times 0.4=96
\end{align*}
The position of walk at the 100th step between 15 and 25 pace/step from the origin is
$P(15\leq X_{100}\leq30)\thickapprox P(14.5<X_{100}<25.5)$
$-\frac{5.5}{\sqrt{96}}<Z_{100}=\frac{X_{100}-20}{\sqrt{96}}<\frac{5.5}{96}$
hence
$P(-0.5613<Z_{100}<0.5613)=\phi(0.5613)-\phi(-0.5613)=0.43$
where Φ(Z) is the standard normal distribution function.

## Random Walks Model: A Mathematical Formalization of Path

A random walk (first introduced by Karl Pearson in 1905) is a mathematical formalization of a path consisting series of random steps. Some examples may include,

1. The path traced by a molecule as it travels in a liquid or gas,
2. The search path of a foraging animal,
3. The price of fluctuating stock, and (iv) the financial status of a gambler.
All these random steps in the example can be modeled as random walks, although they may not be truly random in reality.

Suppose there are $a+1$ positions are marked out on a straight line and numbered 0,1,2,…, a. A person starts at $k$ where $0<k<a$. The walk proceeds in such a way that, at each step, there is probability p that the walker goes forward one step to $k+1$ and a probability $q=1-p$ that the walker goes back one step to $k-1$. The walk continues until either $0$ or $a$ is reached and then ends. In a random walk, the position of a walker after having moved $n$ times is known as the state of the walk after $n$ steps or after covering $n$ stages. Thus the walk described above starts at stage $k$ at step $0$ and moves to either stage $k-1$ or stage $k+1$ after 1 step and so on.

If the walk is bounded, then the ends of the walk are known as barriers and they may have various properties. In this case, the barriers are said to be absorbing implying that the walk must end once a barrier is reached since there is no escape.

A useful diagrammatic way of representing random walk is by a transition or process diagram. In a transition diagram, the possible states of the walker can be represented by points on a line. If a transition between two points can occur in one step then those points are joined by a curve or edge as shown with an arrow indicating the direction of the walk and a weighting denoting the probability of the step occurring. Transition diagram is also known as a direct graph.

For small Markov processes the simplest way to represent the process is often in terms of its state transition diagram. In-state transition diagram each state (outcome) represents the process as a node in a graph. The arcs in the graph represent possible transitions between states of the process. The arcs are labeled by the transition rates between the states.

Example:  Suppose a meteorologist notices that the weather on a given day seems to depend on the weather conditions of the previous day. He/ She observes that if it is raining one day, then the next day is sunny 60% of the time and rainy 40% of the time; on the other hand, if it is sunny, the next day is sunny with probability 30% and rainy with probability 70%. Note that there are two outcomes (i) sunny (ii) rainy in this Markov process. The transition probability between sunny and rainy is 70%, between sunny and sunny is 30%, between rainy and sunny is 60% and between rainy and rainy is 40%. The simple weather forecasting Markov Process in transition diagram is

Random walks are widely being used in many fields such as Ecology, Economics, Psychology, Computer Science, Physics, Chemistry, and Biology, etc. Random walks explain the observed behavior of processes in all these fields, serving as a fundamental model for the recorded stochastic activity.