View source for Poisson process

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[[Category:Stochastic processes]]

A [[Stochastic process|stochastic process]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073340/p0733401.png" /> with independent increments <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073340/p0733402.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073340/p0733403.png" />, having a [[Poisson distribution|Poisson distribution]]. In the homogeneous Poisson process

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073340/p0733404.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073340/p0733405.png" /></td> </tr></table>

for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073340/p0733406.png" />. The coefficient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073340/p0733407.png" /> is called the intensity of the Poisson process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073340/p0733408.png" />. The trajectories of the Poisson process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073340/p0733409.png" /> are step-functions with jumps of height 1. The jump points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073340/p07334010.png" /> form an [[Elementary flow|elementary flow]] describing the demand flow in many queueing systems. The distributions of the random variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073340/p07334011.png" /> are independent for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073340/p07334012.png" /> and have exponential density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073340/p07334013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073340/p07334014.png" />.

One of the properties of a Poisson process is that the conditional distribution of the jump points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073340/p07334015.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073340/p07334016.png" /> is the same as the distribution of the [[Variational series|variational series]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073340/p07334017.png" /> independent samples with uniform distribution on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073340/p07334018.png" />. On the other hand, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073340/p07334019.png" /> is the variational series described above, then as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073340/p07334020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073340/p07334021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073340/p07334022.png" /> one obtains in the limit the distribution of the jumps of the Poisson process.

In an inhomogeneous process the intensity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073340/p07334023.png" /> depends on the time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073340/p07334024.png" /> and the distribution of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073340/p07334025.png" /> is defined by the formula

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073340/p07334026.png" /></td> </tr></table>

Under certain conditions a Poisson process can be shown to be the limit of the sum of a number of independent "sparse" flows of fairly general form as this number increases to infinity. For certain paradoxes which have been obtained in connection with Poisson processes see [[#References|[3]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.A. Borovkov, "Wahrscheinlichkeitstheorie" , Birkhäuser (1976) (Translated from Russian) {{MR|0410818}} {{ZBL|}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> I.I. Gikhman, A.V. Skorokhod, M.I. Yadrenko, "Probability theory and mathematical statistics" , Kiev (1979) (In Russian) {{MR|2026607}} {{ZBL|0673.60001}} </TD></TR>
<TR><TD valign="top">[3]</TD> <TD valign="top"> W. Feller, [[Feller, "An introduction to probability theory and its  applications"|"An introduction to probability theory and its  applications"]], '''2''' , Wiley (1971) pp. Chapt. 1 </TD></TR></table>

====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J.W. Cohen, "The single server queue" , North-Holland (1982) {{MR|0668697}} {{ZBL|0481.60003}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> G.G. Székely, "Paradoxes in probability theory and mathematical statistics" , Reidel (1986) {{MR|0880020}} {{ZBL|0605.60002}} </TD></TR></table>