§ Convergence in distribution is very weak
- consider . Also consider which will be identically distributed (by symmetry of and ).
- So we have that .
- But this tells us nothing about and ! so this type of "convergence of distribution" is very weak.
- Strongest notion of convergence (#2): Almost surely. iff . Consider a snowball left out in the sun. In a couple hours, It'll have a random shape, random volume, and so on. But the ball itself is a definite thing --- the . Almost sure says that for almost all of the balls, converges to .
- #2 notion of convergence: Convergence in probability. iff for all . This allows us to squeeze probability under the rug.
- Convergence in : iff . Eg. think of convergence in variance of a gaussian.
- Convergence in distrbution: (weakest): iff for all .
§ Characterization of convergence in distribution
- (2) For all continuous and bounded, we have .
- (2) we have . [characteristic function converges ].
§ Strength of different types of convergence
- Almost surely convergence implies convergence in probability. Also, the two limits (which are RVs) are almost surely equal.
- Convergence in implies convergence in probability and convergence in for all . Also, the limits (which are RVs) are almost surely equal.
- If converges in probability, it also converges in distribution (meaning the two sequences will have the same DISTRIBUTION, not same RV).
- All of almost surely, probabilistic convergence, convergence in distribution (not ) map properly by continuous fns. implies .
- almost surely implies P implies distribution convergence.
§ Slutsky's Theorem
- If and (That is, the sequence of is eventually deterministic),we then have that . In particular, we get that and .
- This is important, because in general, convergence in distribution says nothing about the RV! but in this special case, it's possible.