Stochastic equivalence

The equivalence relation between random variables that differ only on a set of probability zero. More precisely, two random variables and , defined on a common probability space , are called stochastically equivalent if . In most problems of probability theory one deals with classes of equivalent random variables, rather than with the random variables themselves.

Two stochastic processes and , , defined on a common probability space are called stochastically equivalent if for any stochastic equivalence holds between the corresponding random variables: . With regard to stochastic processes and with coinciding finite-dimensional distributions, the term "stochastic equivalence" is sometimes used in the broad sense.

Comments

The members of a stochastic equivalence class (of random variables or stochastic processes) are sometimes referred to as versions (of each other or of the equivalence class). A version of a random variable or stochastic process is also called a modification.

References

[a1]	J.L. Doob, "Classical potential theory and its probabilistic counterpart" , Springer (1984) pp. 390 MR0731258 Zbl 0549.31001
[a2]	I.I. [I.I. Gikhman] Gihman, A.V. [A.V. Skorokhod] Skorohod, "The theory of stochastic processes" , 1 , Springer (1974) pp. 43ff (Translated from Russian) MR0346882 Zbl 0291.60019
[a3]	C. Dellacherie, "Capacités et processus stochastiques" , Springer (1972) pp. 46 MR0448504 Zbl 0246.60032
[a4]	A.V. [A.V. Skorokhod] Skorohod, "Random processes with independent increments" , Kluwer (1991) pp. 9 (Translated from Russian) MR1155400
[a5]	R.Sh. Liptser, A.N. [A.N. Shiryaev] Shiryayev, "Theory of martingales" , Kluwer (1989) pp. 4 (Translated from Russian) MR1022664 Zbl 0728.60048