Random Walk Model Example

Example: Random Walk Model

A Markov chain whose state space is given by the integers $i=0,\pm 1,\pm 2,\dots$ is said to be a random walk if, for some number $0<p<1$:

$P_{i,i+1}=p=1-P_{i,i-1},i=0,\pm 1,\dots$

Interpretation

Think of it as a model for an individual walking on a straight line who at each point in time either takes one step to the right with probability $p$ or one step to the left with probability $1-p$.

Transition Probabilities

• $P_{i,i+1}=p$ — probability of moving right
• $P_{i,i-1}=1-p$ — probability of moving left

Related

• Random Walk Markov Chain
• Gambling Model