View source for Recurrent events

''in a series of repeated trials with random results''

A series of events <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080160/r0801601.png" /> such that the occurrence of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080160/r0801602.png" /> is determined by the results of the first <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080160/r0801603.png" /> trials, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080160/r0801604.png" /> and under the condition that whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080160/r0801605.png" /> has occurred, the occurrence of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080160/r0801606.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080160/r0801607.png" />, is determined by the results of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080160/r0801608.png" />-st, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080160/r0801609.png" />-nd, etc., trial up to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080160/r08016010.png" />-th trial; furthermore, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080160/r08016011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080160/r08016012.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080160/r08016013.png" /> occur simultaneously, the results of the first <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080160/r08016014.png" /> and the last <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080160/r08016015.png" /> trials should be conditionally independent.

In more detail, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080160/r08016016.png" /> be the (finite or countable) collection of all results of the individual trials, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080160/r08016017.png" /> be the space of sequences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080160/r08016018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080160/r08016019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080160/r08016020.png" />, of the results in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080160/r08016021.png" /> trials, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080160/r08016022.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080160/r08016023.png" /> be the space of infinite sequences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080160/r08016024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080160/r08016025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080160/r08016026.png" /> of results, in which a certain probability distribution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080160/r08016027.png" /> is given. Let in each space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080160/r08016028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080160/r08016029.png" /> be chosen a subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080160/r08016030.png" /> such that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080160/r08016031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080160/r08016032.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080160/r08016033.png" />, the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080160/r08016034.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080160/r08016035.png" /> belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080160/r08016036.png" /> if and only if the sequence

<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/r/r080/r080160/r08016037.png" /></td> </tr></table>

If the last condition is fulfilled and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080160/r08016038.png" />, then

<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/r/r080/r080160/r08016039.png" /></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/r/r080/r080160/r08016040.png" /></td> </tr></table>

where for the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080160/r08016041.png" />, by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080160/r08016042.png" /> one denotes the sequence

<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/r/r080/r080160/r08016043.png" /></td> </tr></table>

The event

<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/r/r080/r080160/r08016044.png" /></td> </tr></table>

is called a recurrent event if it occurs after <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080160/r08016045.png" /> trials.

===Examples.===


1) In a sequence of independent coin tossing, the event consisting of the fact that in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080160/r08016046.png" /> trials, heads and tails will fall an equal number of times (such an event is only possible with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080160/r08016047.png" /> even) is recurrent.

2) In a [[Random walk|random walk]] on a one-dimensional lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080160/r08016048.png" /> starting at zero (with independent jumps at various steps into neighbouring points with probabilities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080160/r08016049.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080160/r08016050.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080160/r08016051.png" />), the event in which the moving point turns out to be at zero after the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080160/r08016052.png" />-th step, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080160/r08016053.png" /> is recurrent.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  W. Feller,   "An introduction to probability theory and its applications" , '''1''' , Wiley  (1968)</TD></TR></table>



====Comments====
Cf. [[Markov chain, recurrent|Markov chain, recurrent]]; [[Markov chain, class of positive states of a|Markov chain, class of positive states of a]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N.T.J. Bailey,   "The elements of stochastic processes" , Wiley  (1964)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  K.L. Chung,   "Markov chains with stationary transition probabilities" , Springer  (1960)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  I.I. [I.I. Gikhman] Gihman,   A.V. [A.V. Skorokhod] Skorohod,   "The theory of stochastic processes" , '''1''' , Springer  (1974)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  V. Spitzer,   "Principles of random walk" , v. Nostrand  (1964)</TD></TR></table>