Recurrent events

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)