Difference between revisions of "Recurrent events"
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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 | + | 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 | + | 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 | + | 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 | + | 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 | 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 | + | 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. | ||
− | + | 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 | + | 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==== |
Revision as of 08:10, 6 June 2020
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) |
Recurrent events. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Recurrent_events&oldid=48453