Recurrent events

in a series of repeated trials with random results

A series of events such that the occurrence of is determined by the results of the first trials, and under the condition that whenever has occurred, the occurrence of , , is determined by the results of the -st, -nd, etc., trial up to the -th trial; furthermore, when and occur simultaneously, the results of the first and the last trials should be conditionally independent.

In more detail, let be the (finite or countable) collection of all results of the individual trials, let be the space of sequences , , , of the results in trials, and let be the space of infinite sequences , , of results, in which a certain probability distribution is given. Let in each space , be chosen a subspace such that for any and , , the sequence for which belongs to if and only if the sequence

If the last condition is fulfilled and if , then

where for the sequence , by one denotes the sequence

The event

is called a recurrent event if it occurs after trials.

Examples.

1) In a sequence of independent coin tossing, the event consisting of the fact that in trials, heads and tails will fall an equal number of times (such an event is only possible with even) is recurrent.

2) In a random walk on a one-dimensional lattice starting at zero (with independent jumps at various steps into neighbouring points with probabilities and , ), the event in which the moving point turns out to be at zero after the -th step, 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)