I have the following problem (any help is really appreciated).
Given a sample of size $N$, I estimate a quantity (it does not matter what, but for the records it is a scalar, fixed coefficient), and I get an estimation error, which we may call $X_{N}$.
Defining $P^{\ast}$ as the probability conditional on the sample, I can show that
$P^{\ast}[|X_{N}|>\epsilon]=0$
almost surely, for all $\epsilon>0$. In other words, I can show that my estimator is a.s. consistent conditional on the sample.
Does this imply anything as far as unconditional convergence is concerned, i.e. can I say anything on $P[|X_{N}|>\epsilon]$, where $P$ denotes the probability "unconditional" on the sample? Under which conditions can I say that $P[|X_{N}|>\epsilon]=0$ as $N$ grows large?