Finite dimensional distributions of a stochastic process #
For a stochastic process X : T → Ω → 𝓧
and a finite measure P
on Ω
, the law of the process is
P.map (fun ω ↦ (X · ω))
, and its finite dimensional distributions are
P.map (fun ω ↦ I.restrict (X · ω))
for I : Finset T
.
We show that two stochastic processes have the same laws if and only if they have the same finite dimensional distributions.
Main statements #
map_eq_iff_forall_finset_map_restrict_eq
: two processes have the same law if and only if their finite dimensional distributions are equal.identDistrib_iff_forall_finset_identDistrib
: same statement, but stated in terms ofIdentDistrib
.map_restrict_eq_of_forall_ae_eq
: if two processes are modifications of each other, then their finite dimensional distributions are equal.map_eq_of_forall_ae_eq
: if two processes are modifications of each other, then they have the same law.
The finite dimensional distributions of a stochastic process are a projective measure family.
The projective limit of the finite dimensional distributions of a stochastic process is the law of the process.
Two stochastic processes have same law iff they have the same finite dimensional distributions.
Two stochastic processes are identically distributed iff they have the same finite dimensional distributions.
If two processes are modifications of each other, then they have the same finite dimensional distributions.
If two processes are modifications of each other, then they have the same distribution.