Simple Random Walk (Unrestricted Random Walk)

A simple random walk (or unrestricted random walk) on a line or in one dimension occurs with probability $p$ when the walker steps forward (+1) and/or has probability $q=1-p$ if the 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 $X_0=0$.

Simple Random Walk

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, $X_n=X_{n-1}\pm 1$

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

The mean and Variance of $X_n$ 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$\{W_i\}$. 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 $E(X_n)=n(p-q)$ and $V(X_n)=4npq$ and variance.

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

If $p>\frac{1}{2}$ then drift is expected away from the origin in a positive direction and if $p<\frac{1}{2}$ it would be expected that the drift would be in the negative direction.

Since $V(X_n)$ 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 the walk at the 100th step between 15 and 25 pace/steps 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 $\phi(Z)$ is the standard normal distribution function.

Read more about Simple Random Walk: Random Walks Model

References

1. https://www.encyclopediaofmath.org/index.php/Random_walk
2. https://mathworld.wolfram.com/RandomWalk1-Dimensional.html