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''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|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====
<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====
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>