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Independent increments

Let ${\displaystyle (X_{t})_{t\in T}}$ be a stochastic process. In most cases, ${\displaystyle T=\mathbb {N} }$or${\displaystyle T=\mathbb {R} ^{+}}$. Then the stochastic process has independent increments if and only if for every ${\displaystyle m\in \mathbb {N} }$ and any choice ${\displaystyle t_{0},t_{1},t_{2},\dots ,t_{m-1},t_{m}\in T}$ with

${\displaystyle t_{0}<t_{1}<t_{2}<\dots <t_{m}}$

the random variables

${\displaystyle (X_{t_{1}}-X_{t_{0}}),(X_{t_{2}}-X_{t_{1}}),\dots ,(X_{t_{m}}-X_{t_{m-1}})}$

are stochastically independent.^[2]

Definition for random measures[edit]

Arandom measure ${\displaystyle \xi }$ has got independent increments if and only if the random variables ${\displaystyle \xi (B_{1}),\xi (B_{2}),\dots ,\xi (B_{m})}$ are stochastically independent for every selection of pairwise disjoint measurable sets ${\displaystyle B_{1},B_{2},\dots ,B_{m}}$ and every ${\displaystyle m\in \mathbb {N} }$. ^[3]

Independent S-increments[edit]

Let ${\displaystyle \xi }$ be a random measure on ${\displaystyle S\times T}$ and define for every bounded measurable set ${\displaystyle B}$ the random measure ${\displaystyle \xi _{B}}$on${\displaystyle T}$as

${\displaystyle \xi _{B}(\cdot ):=\xi (B\times \cdot )}$

Then ${\displaystyle \xi }$ is called a random measure with independent S-increments, if for all bounded sets ${\displaystyle B_{1},B_{2},\dots ,B_{n}}$ and all ${\displaystyle n\in \mathbb {N} }$ the random measures ${\displaystyle \xi _{B_{1}},\xi _{B_{2}},\dots ,\xi _{B_{n}}}$ are independent.^[4]

Application[edit]

Independent increments are a basic property of many stochastic processes and are often incorporated in their definition. The notion of independent increments and independent S-increments of random measures plays an important role in the characterization of Poisson point process and infinite divisibility

References[edit]