Poisson (point) process is the simplest and most fundamental point process, which arose independently in multiple fields around 1909.
An arrival process is a sequence of increasing positive r.v.’s, \( ( S_1, S_2, \cdots ) \). And the r.v.’s \( S_1, S_2, \cdots \) are called arrival epochs.
A renewal process is an arrival process for which the sequence of inter-arrival times is a sequence of IID r.v.’s.
A counting process is a continuous time stochastic process where the random variable at any given epoch represents the number of arrivals up to and including that epoch. \( \{ N(t); t > 0 \} \)
A counting process has stationary increments if for every \( t' > t > 0 \), \( N(t') - N(t) \) has the same distribution function as \( N(t' - t) \).
A counting process has independent increments if for every integer \( k>0 \) and every k-tuple of times \( 0 < t_1 < t_2 < \cdots < t_k \), the k-tuple of rv’s \( N(t_1), \tilde{N}(t_1,t_2), \cdots ,\tilde{N}(t_{k-1},t_k) \) are statistically independent.
Two random processes are independent if any finite subsets of r.v.'s in these processes are independent from each other.
Although proof of equivalence is omitted, the following definitions of Poisson process are equivalent.
A Poisson process is a renewal process with exponentially distributed inter-arrival intervals. The arrival rate of a Poisson process is \( \lambda \), if the inter-arrival times \( X \sim \text{Exp}(\lambda) \).
A Poisson process is a counting process that has independent and stationary increments, with Poisson distributed number of arrivals: \( N(t) \sim \text{Poisson}(\lambda t), \forall t>0 \).
A Poisson process is a counting process that has stationary and independent increments, and satisfies
Denoting Poisson process as an arrival process: \(\mathbf{X} = (X_{(1)}, X_{(2)}, \cdots)\)
Complete independence: Subprocesses of a Poisson process on disjoint domains are mutually independent random processes.
\[ \forall B_1, B_2 \in [0,+\infty), B_1 \cap B_2 = \emptyset : \mathbf{X} \cap B_1 \perp \mathbf{X} \cap B_1 \]
The next occurrence epoch in a Poisson process, regardless of occurrence at and before the current epoch, is exponentially distributed with occurrence rate λ. So is any inter-arrival time.
\[ X_{(1)}, X_{(n+1)} - X_{(n)} \sim \text{Exp}(\lambda) \]
The occurrence epoch of a certain order in a Poisson process, regardless of occurrence at and before the current epoch, is Gamma distributed with parameters being occurrence rate λ and ordinal number k. So is the time between a certain number of arrivals.
\[ X_{(k)}, X_{(n+k)} - X_{(n)} \sim \Gamma(\lambda,k) \]
The number of occurrences during any time interval in a Poisson process is Poisson distributed with expected occurrence λt.
\[ K_t \equiv |\mathbf{X} \cap (t_0, t_0 +t]| \sim \text{Poisson}(\lambda t) \]
Conditional on the number of occurrences in a certain time interval, the distribution of occurrences in a homogeneous Poisson process is uniform over the time interval.
\[ \mathbf{X} | K_t \sim U(0,t)^{K_t} \]
Inspection paradox: For any renewal process, given the renewal time is larger than certain value, the total renewal interval is stochastically larger than an unconditioned renewal interval. \[\forall t > 0, x > 0, P(S_{X_t + 1} > x) \ge P(S_1 > x)\], or equivalently, \[\forall t > 0, x > 0, P(S > x | S > t) \ge P(S > x)\]. In particular, for Poisson processes $P(S > x | S > t) = P(S > x - t) > P(S > x)$. This means if a public transit system is not perfectly on schedule, the population average waiting time will be larger than half of the scheduled inter-arrival time.
Independent thinning results in another Poisson process.
If a Poisson process is split by a Bernoulli process, the resulting processes are independent Poisson processes, with rates \( \lambda_1 = p \lambda \) and \( \lambda_2 = (1-p) \lambda \) respectively. Conditional on the original process, however, the two new processes are complement to each other.
A p(x)-thinning operation applied to a Poisson process with intensity measure Λ gives a Poisson process of removed points with intensity measure \(\Lambda_p\): for any bounded Borel set B,
\[ \Lambda_p(B)= \int_B p(x) \Lambda(dx) \]
The sum process of two independent Poisson processes of rates \( \lambda_1 \) and \( \lambda_2 \) is a Poisson process of rate \( \lambda_1 + \lambda_2 \).
Superposition theorem: The superposition of independent Poisson processes with intensity measures \( \Lambda_1,\Lambda_2,\dots \) is a Poisson process with intensity measure
\[ \Lambda = \sum \limits_{i=1}^{\infty} \Lambda_i \]
The clustering operation is replacing each point of some point process by another (possibly different) point process.
Displacement theorem: A Poisson process on \( \mathbb{R}^d \) with intensity measure \( \lambda(x) \), after independent random transformation, or displacement, with probability density \(\rho(x,\cdot)\), is a Poisson process with intensity measure:
\[ \lambda'(y) = \int_{\mathbb{R}^d} \lambda(x) \rho(x,y) dx \]
Mapping theorem: A Poisson process on \( \mathbb{R}^d \) with intensity measure \( \lambda(x) \), after transformation \(f: \mathbb{R}^d \rightarrow \mathbb{R}^{d'} \), is a Poisson point process with intensity measure: for any bounded Borel set B on \(\mathbb{R}^{d'}\),
\[ \Lambda'(B)=\Lambda(f^{-1}(B)) \]
A non-homogeneous Poisson process with time varying arrival rate \( \lambda(t) \) is a counting process that has independent increments, and satisfies
A non-homogeneous Poisson process could be viewed as a homogeneous Poisson process on a nonlinear time scale.