convergence in probability
Let be a sequence of random variables defined on a probability space taking values in a separable metric space , where is the metric. Then we say the sequence converges in probability or converges in measure to a random variable if for every ,
We denote convergence in probability of to by
Unlike ordinary convergence, and only implies that almost surely.
The need for separability on is to ensure that the metric, , is a random variable, for all random variables and .
Convergence almost surely implies convergence in probability but not conversely.
|Title||convergence in probability|
|Date of creation||2013-03-22 15:01:05|
|Last modified on||2013-03-22 15:01:05|
|Last modified by||CWoo (3771)|
|Synonym||converge in probability|
|Synonym||converges in measure|
|Synonym||converge in measure|
|Synonym||convergence in measure|
|Defines||converges in probability|