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STOCHASTIC CONVERGENCE

EXAMPLES FOR STUDY & PRACTICE

1. wP1-convergence

a) 

b) 

2. the P-metric

a) Verify that ρ(X, Y ) = E|X−Y |^1 defines a metric on R for which 

b) Verify that δ(X, Y ) = E|X−Y |/(1+ |X−Y |) also defines a metric on R for which 

3. uniform integrability & L-convergence

a) 

b) Give any example for which 

c) Give any example for which 

d) If Xn U I does it follow that g(Xn) U I if g is continuous?

4. uniform integrability & the criterion of de la Vall´ee Poussin

a) 

b) 

5. L-completeness

a) 

b) Verify that L is complete in the L-metric.

6. L-convergence & sche↵´e’s theorem

a) For X, Xn ∈ L:

b) For X, Xn ∈ L:

c) For

7. uniform integrability & uniform absolute continuity