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Independent increments
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<h2 id="definition-for-stochastic-processes">Definition for stochastic processes</h2> <p>Let ( X t ) t ∈ T {\displaystyle (X_{t})_{t\in T}} be a <a href="/facts/Stochastic_process/Ng1n1dnB">stochastic process</a>. In most cases, T = N {\displaystyle T=\mathbb {N} } or T = R + {\displaystyle T=\mathbb {R} ^{+}} . Then the stochastic process has independent increments if and only if for every m ∈ N {\displaystyle m\in \mathbb {N} } and any choice t 0 , t 1 , t 2 , … , t m − 1 , t m ∈ T {\displaystyle t_{0},t_{1},t_{2},\dots ,t_{m-1},t_{m}\in T} with </p> t 0 < t 1 < t 2 < ⋯ < t m {\displaystyle t_{0}<t_{1}<t_{2}<\dots <t_{m}} <p>the <a href="/facts/Random_variables/TwTBXnLT">random variables</a> </p> ( X t 1 − X t 0 ) , ( X t 2 − X t 1 ) , … , ( X t m − X t m − 1 ) {\displaystyle (X_{t_{1}}-X_{t_{0}}),(X_{t_{2}}-X_{t_{1}}),\dots ,(X_{t_{m}}-X_{t_{m-1}})} <p>are <a href="/facts/Independence_(probability_theory)/NUzQtnUL">stochastically independent</a>.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> </p> <h2 id="definition-for-random-measures">Definition for random measures</h2> <p>A <a href="/facts/Random_measure/kBFkYw1G">random measure</a> ξ {\displaystyle \xi } has got independent increments if and only if the random variables ξ ( B 1 ) , ξ ( B 2 ) , … , ξ ( B m ) {\displaystyle \xi (B_{1}),\xi (B_{2}),\dots ,\xi (B_{m})} are <a href="/facts/Independence_(probability_theory)/NUzQtnUL">stochastically independent</a> for every selection of <a href="/facts/Pairwise_disjoint/nLr8XRVd">pairwise disjoint</a> measurable sets B 1 , B 2 , … , B m {\displaystyle B_{1},B_{2},\dots ,B_{m}} and every m ∈ N {\displaystyle m\in \mathbb {N} } . <a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> </p> <h2 id="independent-s-increments">Independent S-increments</h2> <p>Let ξ {\displaystyle \xi } be a random measure on S × T {\displaystyle S\times T} and define for every bounded measurable set B {\displaystyle B} the random measure ξ B {\displaystyle \xi _{B}} on T {\displaystyle T} as </p> ξ B ( ⋅ ) := ξ ( B × ⋅ ) {\displaystyle \xi _{B}(\cdot ):=\xi (B\times \cdot )} <p>Then ξ {\displaystyle \xi } is called a random measure with independent S-increments, if for all bounded sets B 1 , B 2 , … , B n {\displaystyle B_{1},B_{2},\dots ,B_{n}} and all n ∈ N {\displaystyle n\in \mathbb {N} } the random measures ξ B 1 , ξ B 2 , … , ξ B n {\displaystyle \xi _{B_{1}},\xi _{B_{2}},\dots ,\xi _{B_{n}}} are independent.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> </p> <h2 id="application">Application</h2> <p>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 <a href="/facts/Poisson_point_process/8uphjAUc">Poisson point process</a> and <a href="/facts/Infinite_divisibility/5tp0mYjN">infinite divisibility</a>. </p> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Sato, Ken-Ito (1999). Lévy processes and infinitely divisible distributions. Cambridge University Press. pp. 31–68. ISBN 9780521553025. <a href="9780521553025" target="_blank">9780521553025</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Klenke, Achim (2008). Probability Theory. Berlin: Springer. p. 190. doi:10.1007/978-1-84800-048-3. ISBN 978-1-84800-047-6. <a href="978-1-84800-047-6" target="_blank">978-1-84800-047-6</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Klenke, Achim (2008). Probability Theory. Berlin: Springer. p. 527. doi:10.1007/978-1-84800-048-3. ISBN 978-1-84800-047-6. <a href="978-1-84800-047-6" target="_blank">978-1-84800-047-6</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Kallenberg, Olav (2017). Random Measures, Theory and Applications. Switzerland: Springer. p. 87. doi:10.1007/978-3-319-41598-7. ISBN 978-3-319-41596-3. <a href="978-3-319-41596-3" target="_blank">978-3-319-41596-3</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> </ol>