Given a sequence of positive numbers $X = (x_1, x_2, \dots)$, define the $k$th elementary symmetric mean of the first $n$ entries to be $$ S(X, n, k) := \frac{\displaystyle\sum_{1 \leq i_1 < \cdots < i_k \leq n} x_{i_1}x_{i_2}\cdots x_{i_k}}{\dbinom{n}{k}} $$

Note that $S(X,n,1)$ is the arithmetic mean and $S(X,n,n)^{1/n}$ is the geometric mean of the first $n$ entries of $X$.

Now we are going to impose the following conditions on our sequence $X$:

$$ \lim_{n \to \infty} S(X,n,n)^{1/n} = G < A = \lim_{n \to \infty} S(X, n, 1) < \infty$$

For any fixed $k$, I believe it's easy to show that these conditions imply

$$\lim_{n \to \infty} S(X,n,n-k)^{1/n-k} = G$$ and $$\lim_{n \to \infty} S(X,n,k)^{1/k} = A.$$

However, I'm interested in what happens when $k$ is allowed to depend on $n$. Specifically, is the existence of the AM and GM enough to imply the existence of

$$\lim_{n \to \infty} S(X,n,cn)^{1/cn} =: F(c)$$

where $c$ is any proportion in $(0,1)$? Maclaurin's inequalities tell us that the liminf/limsup lie inside [G,A], but do we also know these intermediate means don't oscillate strangely inside the interval?