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
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If the last condition is fulfilled and if , then
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where for the sequence , by
one denotes the sequence
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The event
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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) |
Recurrent events. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Recurrent_events&oldid=25531