Counting process

Dependencies:

  1. Random variable
  2. Independence of random variables (incomplete)

Let $N$ be a function from $\mathbb{R}_{\ge 0}$ to a random variable. (In other words, for any $t \ge 0$, $N(t)$ is a random variable.) Then the set $\{N(t): t \ge 0\}$ is called a counting process if all of the following hold:

  1. $N(0) = 0$.
  2. $N(t) \in \mathbb{Z}$.
  3. $s < t \implies N(s) \le N(t)$ (i.e., the support of $N$ only contains monotonic functions).

Intuitively, $N(t) - N(s)$ is said to be the number of events occuring in the time interval $(s, t]$.

The counting process $\{N(t): t \ge 0\}$ is said to have independent increments iff the number of events in two disjoint intervals are independent. Formally, for any $s_1 < t_1$ and $s_2 < t_2$, we have \[ (s_1, t_1] \cap (s_2, t_2] = \emptyset \implies N(t_1) - N(s_1) \textrm{ is independent of } N(t_2) - N(s_2). \]

The counting process $\{N(t): t \ge 0\}$ is said to have stationary increments iff the distribution of the number of events in an interval only depends on the length of the interval, i.e., for any $s < t$, the distribution of $N(t) - N(s)$ is identical to the distribution of $N(t-s)$.

Dependency for:

  1. Poisson process

Info:

Transitive dependencies:

  1. /analysis/topological-space
  2. /sets-and-relations/countable-set
  3. /sets-and-relations/de-morgan-laws
  4. σ-algebra
  5. Generated σ-algebra
  6. Borel algebra
  7. Measurable function
  8. Generators of the real Borel algebra (incomplete)
  9. Measure
  10. σ-algebra is closed under countable intersections
  11. Probability
  12. Conditional probability (incomplete)
  13. Independence of events
  14. Independence of composite events
  15. Random variable
  16. Independence of random variables (incomplete)