In
probability theory
Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...
, a
stochastic process
In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables in a probability space, where the index of the family often has the interpretation of time. Sto ...
is said to be continuous in probability or stochastically continuous if its distributions converge whenever the values in the index set converge.
Definition
Let
be a
stochastic process
In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables in a probability space, where the index of the family often has the interpretation of time. Sto ...
in
.
The process
is continuous in probability when
converges in probability
In probability theory, there exist several different notions of convergence of sequences of random variables, including ''convergence in probability'', ''convergence in distribution'', and ''almost sure convergence''. The different notions of conve ...
to
whenever
converges to
.
Examples and Applications
Feller processes are continuous in probability at
. Continuity in probability is a sometimes used as one of the defining property for
Lévy process
In probability theory, a Lévy process, named after the French mathematician Paul Lévy, is a stochastic process with independent, stationary increments: it represents the motion of a point whose successive displacements are random, in which disp ...
.
Any process that is continuous in probability and has
independent increments has a
version
Version may refer to:
Computing
* Software version, a set of numbers that identify a unique evolution of a computer program
* VERSION (CONFIG.SYS directive), a configuration directive in FreeDOS
Music
* Cover version
* Dub version
* Remix
* ''V ...
that is
càdlàg
In mathematics, a càdlàg (), RCLL ("right continuous with left limits"), or corlol ("continuous on (the) right, limit on (the) left") function is a function defined on the real numbers (or a subset of them) that is everywhere right-continuous an ...
.
As a result, some authors immediately define Lévy process as being càdlàg and having independent increments.
References
[{{cite book , last1=Kallenberg , first1=Olav , author-link1=Olav Kallenberg , year=2002 , title=Foundations of Modern Probability, location= New York , publisher=Springer , edition=2nd, pages=290]
Stochastic processes