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)