Mathematics Prelims

December 22, 2008

Convergence in Probability/Measure

Filed under: Measure Theory,Probability Theory — cjohnson @ 11:05 pm

If X_n : \Omega \to \mathbb{R} is a random variable on the probability space (\Omega, \mathcal{F}, P), then one way to define the convergence of a sequence of random variables (X_n)_{n \in \mathbb{N}} is to simply define the limit pointwise, as you would do with any old function.  That is, if there exists a random variable X such that for every \omega \in \Omega we have

\displaystyle X_n(\omega) \to X(\omega)

Similarly, we say that X_n \to X almost everywhere (almost surely) if the measure (probability) of the set of points where X_n(\omega) \not\to X(\omega) is zero.

Another way to define convergence of measurable functions in general is to talk about convergence in measure.  This is a notion of convergence where the set of points where the functions in the sequence do not get arbitrarily close to the corresponding point in the limit function has measure zero.  That is if for each \delta > 0,

\displaystyle P(\{\omega \in \Omega : |X_n(\omega) - X(\omega)| > \delta\}) \to 0

(Here we’re using random variables and a probability space, but the same thing is used for measurable functions of general measure spaces.)  To say that X_n converges to X in probability (denoted X_n \xrightarrow{P} X), is to say that for any \epsilon > 0, there exists an N \in \mathbb{N} such that for every n > N, the set of points where X_n isn’t “close” to X (here “close” means within \delta distance of X) is less than \epsilon.

As noted in Cohn, convergence in measure does not imply convergence pointwise (not even almost everywhere!), or vice versa.  For instance, the set of functions f_n = 1_{[n, \infty)} pointwise converges to the zero function, but does not converge in measure.   However, in the case of a finite measure (e.g., a probability) we have the following proposition.

Proposition If (\Omega, \mathcal{F}, P) is a measure space with P(\Omega) finite, then if (X_n)_{n \in \mathbb{N}} is a sequence of measurable functions with X_n \to X a.e., then X_n \xrightarrow{P} X.

Proof: Let \delta > 0 given.  Define A_n = \{ \omega \in \Omega : |X_n(\omega) - X(\omega)| > \delta \} and B_n = \bigcup_{k=n}^\infty A_k.  Clearly B_n forms a decreasing sequence whose intersection is a subset of the points \omega \in \Omega for which X_n(\omega) \not\to \Omega, by assumption this set has measure zero, P(\bigcap B_n) = 0, which gives that the functions converge in measure to X.

(These examples and proofs are from Donald L. Cohn’s “Measure Theory.”)

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