Math Help - Is the limiting distribution unique?

1. Is the limiting distribution unique?

Is the limiting distribution of an infinite sequence of random variables unique if the convergence is
a) almost surely
b) in probability
c) in distribution

In other words, is it possible for a sequence of RVs to converge in distribution (resp. in probability, almost surely) to two different distributions?

2. Re: Is the limiting distribution unique?

Here's a possible proof that the limiting distribution is unique under all three stochastic convergence types, but is the proof valid? Your input would be most welcome.

Step #1: "The limiting distribution is unique if the convergence is in distribution"
Proof: According to Wikipedia
(a) convergence in distribution is metrizable,
(b) A metrizable space is Hausdorff,
(c) A Hausdorff space implies the uniqueness of limits of sequences.
The result follows.

Step #2: "The limiting distribution is unique if the convergence is in probability/almost surely"
Proof: According to Wikipedia,
(a) convergence in probability implies convergence in distribution,
(b) almost sure convergence implies convergence in probability
Hence the result.

3. Re: Is the limiting distribution unique?

The convergence in distribution, in probability and almost sure all have uniqueness of the limit.

The proof you thought is correct, but it is simpler to do it directly:

- a.s. limit is just a standard limit in R (or R^N) metric, so it's obviosly unique

- limit in probability: try to prove it's unique using the definition

- limit in distribution: again try to prove it's unique using the definition (Xn->X in distribution iff E[h(Xn)]->E[h(X)] for every h continuous bounded function)

4. Re: Is the limiting distribution unique?

It's always an almost sure uniqueness of course