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Difference between revisions of "Recurrent events"

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and  $  m $,  
 
and  $  m $,  
 
$  1 \leq  n < m < \infty $,  
 
$  1 \leq  n < m < \infty $,  
the sequence  $  \overline{x}\; = ( \overline{x}\; _ {1} \dots \overline{x}\; _ {m} ) \in X ^ {[ 1,m] } $
+
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} $
+
for which  $  \overline{x} \mid  _ {1}  ^ {n} \equiv ( \overline{x} _ {1} \dots \overline{x} _ {n} ) \in \epsilon _ {n} $
 
belongs to  $  \epsilon _ {m} $
 
belongs to  $  \epsilon _ {m} $
 
if and only if the sequence
 
if and only if the sequence
  
 
$$  
 
$$  
\overline{x}\; \mid  _ {n+} ^ {m}  \equiv  ( \overline{x}\; _ {n+} 1 \dots \overline{x}\; _ {m} ) \  
+
\overline{x} \mid  _ {n+1}   ^ {m}  \equiv  ( \overline{x} _ {n+1} \dots \overline{x} _ {m} ) \  
\in  \epsilon _ {m-} n .
+
\in  \epsilon _ {m-n} .
 
$$
 
$$
  
If the last condition is fulfilled and if  $  \overline{x}\; \in \epsilon _ {m} $,  
+
If the last condition is fulfilled and if  $  \overline{x} \in \epsilon _ {m} $,  
 
then
 
then
  
 
$$  
 
$$  
P \{ {x \in X ^ {[ 1, \infty ] } } : {x \mid  _ {1}  ^ {m} = \overline{x}\; } \} =
+
P \{ {x \in X ^ {[ 1, \infty ] } } : {x \mid  _ {1}  ^ {m} = \overline{x} } \} =
 
$$
 
$$
  
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= \  
 
= \  
 
P \{ x \in X ^ {[ 1, \infty ] } :  x \mid  _ {1}  ^ {n} =
 
P \{ x \in X ^ {[ 1, \infty ] } :  x \mid  _ {1}  ^ {n} =
\overline{x}\; \mid  _ {1}  ^ {n} \} P \{ x \in X ^ {[ 1, \infty ] }
+
\overline{x} \mid  _ {1}  ^ {n} \} P \{ x \in X ^ {[ 1, \infty ] }
:  x | _ {n+} 1 ^ {m} = \overline{x}\; | _ {n+} 1 ^ {m} \} ,
+
:  x | _ {n+1}  ^ {m} = \overline{x} | _ {n+1}  ^ {m} \} ,
 
$$
 
$$
  
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$$  
 
$$  
x \mid  _ {i}  ^ {j}  =  ( x _ {i} , x _ {i+} 1 \dots x _ {j} ),\ \  
+
x \mid  _ {i}  ^ {j}  =  ( x _ {i} , x _ {i+1} \dots x _ {j} ),\ \  
 
i \leq  j,\ \  
 
i \leq  j,\ \  
 
( i, j) = 1, 2 , .  .  . .
 
( i, j) = 1, 2 , .  .  . .

Revision as of 17:59, 2 January 2021


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=51175
This article was adapted from an original article by T.Yu. Popova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article