A problem in convergence and limit

Question

For any non-negative integer $n$ and some finite $r$, we introduce the notation $n_k$ which indicates the number of $\{X_1, X_2, \cdots , X_n\}$ belonging to the $k$-th distribution type, for $k=1, 2, \cdots, r$. Here, it is assumed that $X_i$'s are independent, and that $E[X_i|S_n]$ converges to $E[X_i]$, for $i=1, 2, \cdots, n$. Then, the following condition holds true.

There are at most $r$ different elements in the set of all distributions of the sequence $X_1, X_2, \cdots,$. Moreover, we have that

\begin{equation} \lim_{n\rightarrow \infty} \min_{1\leq k\leq r} \frac{n_k}{ln \sigma[S_n]} = \infty, \end{equation} and for $i$ given, there exists an $n\geq i$ such that

\begin{equation} \sup_x \bar{f}_{S_{n/i}}(x) < \infty \end{equation} where $S_n = \sum_{i=1}^{n}X_i$ and $S_{n/i} = S_n - X_i$, for $i=1, 2, \cdots , n$, and $\bar{f}_{S_{n/i}}(x)$ is the probability density function of the normalized version of $S_{n/i}$, i.e., $\frac{S_{n/i} - E[S_{n/i}]}{\sigma[S_{n/i}]}$ with expectation and standard deviation $E[S_{n/i}]$ and $\sigma[S_{n/i}]$, respectively.

My question is: Why $r$ should be a finite number, what if r is infinite (the number of different distributions goes to infinity)? Can we provide another condition that satisfies the above condition?

Answer 1

$r$ need not be finite. Assuming your result is correct, it is sufficient that the condition on the $n_k$ holds for all $r$.

Proof by taking subsequences. Let $i$ and $r<\infty$ be given. Fix a set of $r$ distribution types including the type of $X_i$, and take the subsequence consisting only of those types. Write $\tilde S_n$ for the subsequence partial sums. Your condition on $n_k$ still applies to the subsequence (the numerator is the same and the denominator becomes smaller).

Your result thus says $E[X_i|\tilde S_n]\to E[X_i]$. But the full sequence partial sums $S_n$ cannot contain any more information about $X_i$ than $\tilde S_n$ do, because they are a function of $\tilde S_n$ and variables independent of the subsequence.

It does seem necessary that each type appears infinitely often, and there's still a gap between that and your condition, potentially allowing for more refinement.