Regenerative process

<h2 id="history">History</h2>
Regenerative processes were first defined by <a href="/facts/Wally_Smith_(mathematician)/qC78CMNr">Walter L. Smith</a> in <a href="/facts/Proceedings_of_the_Royal_Society_A/qyn2m2ja">Proceedings of the Royal Society A</a> in 1955.<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a><a class="footnote-ref" id="fnref:3" href="#fn:3">3</a>

<h2 id="definition">Definition</h2>
A regenerative process is a <a href="/facts/Stochastic_process/Ng1n1dnB">stochastic process</a> with time points at which, from a probabilistic point of view, the process restarts itself.<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a> These time point may themselves be determined by the evolution of the process. That is to say, the process {X(t), t ≥ 0} is a regenerative process if there exist time points 0 ≤ T0 < T1 < T2 < ... such that the post-Tk process {X(Tk + t) : t ≥ 0}

<ul><li>has the same distribution as the post-T0 process {X(T0 + t) : t ≥ 0}</li>
<li>is independent of the pre-Tk process {X(t) : 0 ≤ t < Tk}</li></ul>
for k ≥ 1.<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a> Intuitively this means a regenerative process can be split into <a href="/facts/Independent_and_identically_distributed/othIRaWt">i.i.d.</a> cycles.<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a>
When T0 = 0, X(t) is called a nondelayed regenerative process. Else, the process is called a delayed regenerative process.<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a>

<h2 id="examples">Examples</h2>
<ul><li><a href="/facts/Renewal_process/1rDc2czB">Renewal processes</a> are regenerative processes, with T1 being the first renewal.<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a></li>
<li>Alternating <a href="/facts/Renewal_process/1rDc2czB">renewal processes</a>, where a system alternates between an 'on' state and an 'off' state.<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a></li>
<li>A recurrent <a href="/facts/Markov_chain/5oP5hntk">Markov chain</a> is a regenerative process, with T1 being the time of first recurrence.<a class="footnote-ref" id="fnref:10" href="#fn:10">10</a> This includes <a href="/facts/Harris_chain/gjKEG4Hc">Harris chains</a>.</li>
<li><a href="/facts/Reflected_Brownian_motion/Wb4Knnzv">Reflected Brownian motion</a> is a regenerative process (where one measures the time it takes particles to leave and come back).<a class="footnote-ref" id="fnref:11" href="#fn:11">11</a></li></ul>
<h2 id="properties">Properties</h2>
<ul><li>By the <a href="/facts/Renewal_reward_theorem/1rDc2czB">renewal reward theorem</a>, with probability 1,<a class="footnote-ref" id="fnref:12" href="#fn:12">12</a></li></ul>

lim
          
            t
            →
            ∞
          
        
        
          
            1
            t
          
        
        
          ∫
          
            0
          
          
            t
          
        
        X
        (
        s
        )
        d
        s
        =
        
          
            
              
                E
              
              [
              R
              ]
            
            
              
                E
              
              [
              τ
              ]
            
          
        
        .
      
    
    {\displaystyle \lim _{t\to \infty }{\frac {1}{t}}\int _{0}^{t}X(s)ds={\frac {\mathbb {E} [R]}{\mathbb {E} [\tau ]}}.}

where 
 
 
 
 τ
 
 
 {\displaystyle \tau }
 
 is the length of the first cycle and 
 
 
 
 R
 =
 
 ∫
 
 0
 
 
 τ
 
 
 X
 (
 s
 )
 d
 s
 
 
 {\displaystyle R=\int _{0}^{\tau }X(s)ds}
 
 is the value over the first cycle.
<ul><li>A <a href="/facts/Measurable_function/NgIHnb1b">measurable function</a> of a regenerative process is a regenerative process with the same regeneration time<a class="footnote-ref" id="fnref:13" href="#fn:13">13</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Ross, S. M. (2010). "Renewal Theory and Its Applications". Introduction to Probability Models. pp. 421–641. doi:10.1016/B978-0-12-375686-2.00003-0. ISBN 9780123756862. <a href="9780123756862" target="_blank">9780123756862</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Schellhaas, Helmut (1979). "Semi-Regenerative Processes with Unbounded Rewards". Mathematics of Operations Research. 4: 70–78. doi:10.1287/moor.4.1.70. JSTOR 3689240. <a href="/wiki/Mathematics_of_Operations_Research" target="_blank">/wiki/Mathematics_of_Operations_Research</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Smith, W. L. (1955). "Regenerative Stochastic Processes". Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 232 (1188): 6–31. Bibcode:1955RSPSA.232....6S. doi:10.1098/rspa.1955.0198. <a href="/wiki/Wally_Smith_(mathematician)" target="_blank">/wiki/Wally_Smith_(mathematician)</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Sheldon M. Ross (2007). Introduction to probability models. Academic Press. p. 442. ISBN 0-12-598062-0. <a href="0-12-598062-0" target="_blank">0-12-598062-0</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Haas, Peter J. (2002). "Regenerative Simulation". Stochastic Petri Nets. Springer Series in Operations Research and Financial Engineering. pp. 189–273. doi:10.1007/0-387-21552-2_6. ISBN 0-387-95445-7. <a href="0-387-95445-7" target="_blank">0-387-95445-7</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Asmussen, Søren (2003). "Regenerative Processes". Applied Probability and Queues. Stochastic Modelling and Applied Probability. Vol. 51. pp. 168–185. doi:10.1007/0-387-21525-5_6. ISBN 978-0-387-00211-8. <a href="978-0-387-00211-8" target="_blank">978-0-387-00211-8</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Haas, Peter J. (2002). "Regenerative Simulation". Stochastic Petri Nets. Springer Series in Operations Research and Financial Engineering. pp. 189–273. doi:10.1007/0-387-21552-2_6. ISBN 0-387-95445-7. <a href="0-387-95445-7" target="_blank">0-387-95445-7</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Sheldon M. Ross (2007). Introduction to probability models. Academic Press. p. 442. ISBN 0-12-598062-0. <a href="0-12-598062-0" target="_blank">0-12-598062-0</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Sheldon M. Ross (2007). Introduction to probability models. Academic Press. p. 442. ISBN 0-12-598062-0. <a href="0-12-598062-0" target="_blank">0-12-598062-0</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Sheldon M. Ross (2007). Introduction to probability models. Academic Press. p. 442. ISBN 0-12-598062-0. <a href="0-12-598062-0" target="_blank">0-12-598062-0</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">Asmussen, Søren (2003). "Regenerative Processes". Applied Probability and Queues. Stochastic Modelling and Applied Probability. Vol. 51. pp. 168–185. doi:10.1007/0-387-21525-5_6. ISBN 978-0-387-00211-8. <a href="978-0-387-00211-8" target="_blank">978-0-387-00211-8</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">Sigman, Karl (2009) Regenerative Processes, lecture notes <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
<li id="fn:13">Sigman, Karl (2009) Regenerative Processes, lecture notes <a href="#fnref:13" class="footnote-back-ref">↩</a></li>
</ol>

Regenerative process open-in-new

Regenerative process