Limiting vs Hitting Probability

Comparison: Limiting vs Hitting Probability

Both often use the notation $\pi$, but represent fundamentally different concepts:

Feature	Limiting Probability ${\pi }_j$	Hitting Probability ${\pi }_i(j)$
Question	”In the long run, what % of time is spent in state $j$?"	"Starting at state $i$, what is the chance I ever reach state $j$?”
System State	Describes steady state (long-run)	Describes the fate of a single path
Dependence	Independent of starting state	Heavily depends on starting state $i$
Matrix Math	Left-multiplication: $\pi =\pi \mathbf{P}$	Right-multiplication: $u=\mathbf{P}u$
Indices	Summing over columns: ${\pi }_j={\sum }_k{\pi }_k{P}_{kj}$	Summing over rows: ${\pi }_i={\sum }_k{P}_{ik}{\pi }_k$
Boundaries	No boundaries; requires $\sum {\pi }_j=1$	Requires boundaries (Target=1, Other Absorbing=0)

TIP

• Use Limiting Probability when asked for “Long-run proportion” or “Steady state”
• Use Hitting Probability (via First Step Analysis) when asked for “Probability of reaching/absorption” or “Chance of winning”

Related

• Limiting Probability
• Hitting Probability (Transient)
• First Step Analysis