Convergence in Probability and Convergence with Probability 1

Definations:

For a random vector and a sequence of random vectors defined on a probability space .

1. Convergence in Probability or Convergence of probability measures, is kind of convergence in measure, in concept of the measure theory.
: for any fixed .

2. Convergence with Probability 1, is kind of pointwise convergence in real analysis, it is also called almost surely convergence in the measure theory.
: if or .

Convergence in Probability is weaker than Convergence with Probability 1, as stated below:

Theorem:

Proof: {: for each there exists an such that for all }

{{ there exists an such that for all }}

{{ for all }}

{{ }}

So

for all

for all

for all

for all , or for any fixed , that is .

if , let the set of unconvergence points as then .
if , let the set of unconvergence points as then we donot have .

Convergence in Probability do not require but release the condition to , where is a set which will go smaller as .

From the proof above we can see that and , that is to say is bigger than

References:
A Course in Large Sample Theory(Lecture notes), Xianyi Wu
A Course in Large Sample Theory, Thomas S. Ferguson