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''in a series of repeated trials with random results''
 
''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.
+
A series of events $  A _ {1} , A _ {2} \dots $
 +
such that the occurrence of $  A _ {n} $
 +
is determined by the results of the first $  n $
 +
trials, $  n = 1, 2 \dots $
 +
and under the condition that whenever $  A _ {n} $
 +
has occurred, the occurrence of $  A _ {m} $,  
 +
$  m > n $,  
 +
is determined by the results of the $  ( n+ 1) $-
 +
st, $  ( n+ 2) $-
 +
nd, etc., trial up to the $  m $-
 +
th trial; furthermore, when $  A _ {n} $
 +
and $  A _ {m} $
 +
$  ( m > n) $
 +
occur simultaneously, the results of the first $  n $
 +
and the last $  ( m- n) $
 +
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
+
In more detail, let $  X $
 +
be the (finite or countable) collection of all results of the individual trials, let $  X ^ {[ 1,n] } $
 +
be the space of sequences $  ( x _ {1} \dots x _ {n} ) $,  
 +
$  x _ {i} \in X $,  
 +
$  i = 1 \dots n $,  
 +
of the results in $  n $
 +
trials, $  n = 1, 2 \dots $
 +
and let $  X ^ {[ 1, \infty ] } $
 +
be the space of infinite sequences $  ( x _ {1} , x _ {2} , . . . ) $,  
 +
$  x _ {i} \in X $,
 +
$  i = 1, 2 \dots $
 +
of results, in which a certain probability distribution $  P $
 +
is given. Let in each space $  X ^ {[ 1,n] } $,
 +
$  n = 1, 2 \dots $
 +
be chosen a subspace $  \epsilon _ {n} \subseteq X ^ {[ 1,n] } $
 +
such that for any $  n $
 +
and $  m $,
 +
$  1 \leq  n < m < \infty $,  
 +
the sequence $  \overline{x} = ( \overline{x} _ {1} \dots \overline{x} _ {m} ) \in X ^ {[ 1,m] } $
 +
for which $  \overline{x} \mid  _ {1}  ^ {n} \equiv ( \overline{x} _ {1} \dots \overline{x} _ {n} ) \in \epsilon _ {n} $
 +
belongs to $  \epsilon _ {m} $
 +
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>
+
$$
 +
\overline{x} \mid  _ {n+1}  ^ {m}  \equiv  ( \overline{x} _ {n+1} \dots \overline{x} _ {m} ) \
 +
\in  \epsilon _ {m-n} .
 +
$$
  
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
+
If the last condition is fulfilled and if $  \overline{x} \in \epsilon _ {m} $,  
 +
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>
+
$$
 +
P \{ {x \in X ^ {[ 1, \infty ] } } : {x \mid  _ {1}  ^ {m} = \overline{x} } \} =
 +
$$
  
<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>
+
$$
 +
= \
 +
P \{ x \in X ^ {[ 1, \infty ] } : x \mid  _ {1}  ^ {n} =
 +
\overline{x} \mid  _ {1}  ^ {n} \} P \{ x \in X ^ {[ 1, \infty ] }
 +
: x | _ {n+1}  ^ {m} = \overline{x} | _ {n+1}  ^ {m} \} ,
 +
$$
  
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
+
where for the sequence $  x = ( x _ {1} , x _ {2} ,\dots ) \in X ^ {[ 1, \infty ] } $,  
 +
by $  x \mid  _ {i}  ^ {j} $
 +
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>
+
$$
 +
x \mid  _ {i}  ^ {j}  = ( x _ {i} , x _ {i+1} \dots x _ {j} ),\ \
 +
i \leq  j,\ \
 +
( i, j) = 1, 2 , .  . . .
 +
$$
  
 
The event
 
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>
+
$$
 +
A _ {n}  = \
 +
\{ {x \in X ^ {[ 1, \infty ] } } : {x \mid  _ {1}  ^ {n} \in \epsilon _ {n} } \}
 +
$$
  
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.
+
is called a recurrent event if it occurs after $  n $
 +
trials.
  
===Examples.===
+
====Examples====
  
 +
1) In a sequence of independent coin tossing, the event consisting of the fact that in  $  n $
 +
trials, heads and tails will fall an equal number of times (such an event is only possible with  $  n $
 +
even) is recurrent.
  
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 $  Z  ^ {1} $
 
+
starting at zero (with independent jumps at various steps into neighbouring points with probabilities $  p $
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.
+
and $  q $,  
 +
$  p+ q = 1 $),  
 +
the event in which the moving point turns out to be at zero after the $  n $-
 +
th step, $  n = 2, 4 \dots $
 +
is recurrent.
  
 
====References====
 
====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>
+
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  W. Feller, [[Feller, "An introduction to probability theory and its applications"|"An introduction to probability theory and its  applications"]], '''1''' , Wiley  (1968)</TD></TR></table>
 
 
 
 
  
 
====Comments====
 
====Comments====

Latest revision as of 20:21, 25 March 2024


in a series of repeated trials with random results

A series of events $ A _ {1} , A _ {2} \dots $ such that the occurrence of $ A _ {n} $ is determined by the results of the first $ n $ trials, $ n = 1, 2 \dots $ and under the condition that whenever $ A _ {n} $ has occurred, the occurrence of $ A _ {m} $, $ m > n $, is determined by the results of the $ ( n+ 1) $- st, $ ( n+ 2) $- nd, etc., trial up to the $ m $- th trial; furthermore, when $ A _ {n} $ and $ A _ {m} $ $ ( m > n) $ occur simultaneously, the results of the first $ n $ and the last $ ( m- n) $ trials should be conditionally independent.

In more detail, let $ X $ be the (finite or countable) collection of all results of the individual trials, let $ X ^ {[ 1,n] } $ be the space of sequences $ ( x _ {1} \dots x _ {n} ) $, $ x _ {i} \in X $, $ i = 1 \dots n $, of the results in $ n $ trials, $ n = 1, 2 \dots $ and let $ X ^ {[ 1, \infty ] } $ be the space of infinite sequences $ ( x _ {1} , x _ {2} , . . . ) $, $ x _ {i} \in X $, $ i = 1, 2 \dots $ of results, in which a certain probability distribution $ P $ is given. Let in each space $ X ^ {[ 1,n] } $, $ n = 1, 2 \dots $ be chosen a subspace $ \epsilon _ {n} \subseteq X ^ {[ 1,n] } $ such that for any $ n $ and $ m $, $ 1 \leq n < m < \infty $, the sequence $ \overline{x} = ( \overline{x} _ {1} \dots \overline{x} _ {m} ) \in X ^ {[ 1,m] } $ for which $ \overline{x} \mid _ {1} ^ {n} \equiv ( \overline{x} _ {1} \dots \overline{x} _ {n} ) \in \epsilon _ {n} $ belongs to $ \epsilon _ {m} $ if and only if the sequence

$$ \overline{x} \mid _ {n+1} ^ {m} \equiv ( \overline{x} _ {n+1} \dots \overline{x} _ {m} ) \ \in \epsilon _ {m-n} . $$

If the last condition is fulfilled and if $ \overline{x} \in \epsilon _ {m} $, then

$$ P \{ {x \in X ^ {[ 1, \infty ] } } : {x \mid _ {1} ^ {m} = \overline{x} } \} = $$

$$ = \ P \{ x \in X ^ {[ 1, \infty ] } : x \mid _ {1} ^ {n} = \overline{x} \mid _ {1} ^ {n} \} P \{ x \in X ^ {[ 1, \infty ] } : x | _ {n+1} ^ {m} = \overline{x} | _ {n+1} ^ {m} \} , $$

where for the sequence $ x = ( x _ {1} , x _ {2} ,\dots ) \in X ^ {[ 1, \infty ] } $, by $ x \mid _ {i} ^ {j} $ one denotes the sequence

$$ x \mid _ {i} ^ {j} = ( x _ {i} , x _ {i+1} \dots x _ {j} ),\ \ i \leq j,\ \ ( i, j) = 1, 2 , . . . . $$

The event

$$ A _ {n} = \ \{ {x \in X ^ {[ 1, \infty ] } } : {x \mid _ {1} ^ {n} \in \epsilon _ {n} } \} $$

is called a recurrent event if it occurs after $ n $ trials.

Examples

1) In a sequence of independent coin tossing, the event consisting of the fact that in $ n $ trials, heads and tails will fall an equal number of times (such an event is only possible with $ n $ even) is recurrent.

2) In a random walk on a one-dimensional lattice $ Z ^ {1} $ starting at zero (with independent jumps at various steps into neighbouring points with probabilities $ p $ and $ q $, $ p+ q = 1 $), the event in which the moving point turns out to be at zero after the $ n $- th step, $ n = 2, 4 \dots $ is recurrent.

References

[1] W. Feller, "An introduction to probability theory and its applications", 1 , Wiley (1968)

Comments

Cf. Markov chain, recurrent; Markov chain, class of positive states of a.

References

[a1] N.T.J. Bailey, "The elements of stochastic processes" , Wiley (1964)
[a2] K.L. Chung, "Markov chains with stationary transition probabilities" , Springer (1960)
[a3] I.I. [I.I. Gikhman] Gihman, A.V. [A.V. Skorokhod] Skorohod, "The theory of stochastic processes" , 1 , Springer (1974) (Translated from Russian)
[a4] V. Spitzer, "Principles of random walk" , v. Nostrand (1964)
How to Cite This Entry:
Recurrent events. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Recurrent_events&oldid=11616
This article was adapted from an original article by T.Yu. Popova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article