Let $X_1, X_2, ..., X_N$ be a sample from a distribution F, and N a positive integer-valued random variable. Thus, the sample size is unknown. Suppose we define the Empirical Distribution Function as per usual:

$$F_N^*(x) = \frac{1}{N} \sum_{i=1}^N I(X \leq x)$$

for $i=1, 2, ...$, and $I(X \leq x)=1$ if $X \leq x$ and $0$ otherwise.

Are there any well known results about this sort of setup? I know of the elegant result for finding $E(S_N)$, where $S_N = \sum_{i=1}^NX_i$, using $E(S_N)=E\{E(S_N | N)\}$, and using this one can show the above has the same expected value as the regular EDF. Then, if we assume $N=n+N'$ for a known n and another positive integer-valued random variable $N'$, I am inclined to believe one can apply the Weak or Strong Law of Large Numbers for $n \to \infty$ and for a fixed $x$ to obtain Consistency. I am also inclined to believe that uniform convergence under the Glivenko-Cantelli Theorem should also hold under such conditions, but I am unsure.

I am also interested in the multidimensional case of this problem, under the same setup as here. I was hoping someone could shed some light on any of the two.

Thanks in advance.



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