Disclaimer: As I am not very knowledgeable of the field to which this question pertains, I will introduce a temporary terminology to convey the idea of my question at the risk of conflicting with existing ones.

Let $(X_n)_{n\geq 0}$ be a sequence of random variables (not necessarily independent). Say such a sequence asymptotically concentrates (resp. $O$-concentrates) when

$$ X_n \stackrel{a.s.}{\sim} \mathbb{E}(X_n) \quad\text{(resp. } X_n \stackrel{a.s.}{\asymp} \mathbb{E}(X_n) \text{)}, $$ where the asymptotic is taken as $n\to +\infty$ and a.s. stands for almost surely (i.e. the probability of the associated limit is $1$)

An example of asymptotic concentration is $S_n := x_1 +\cdots + x_n$, where $x_n$ are independent boolean random variables, not necessarily identically distributed (Strong Law of Large Numbers: SLLN). Further instances of both the first and the second seem to be plenty in probabilistic combinatorics, though the one I am most familiar with is the Erdős-Tetali theorem (an example of $O$-concentration), where the $X_n$ are certain specific boolean polynomials (i.e. $f(x_1,\cdots,x_n)$ where $x_1,\cdots,x_n$ are independent boolean random variables).

As far as I gather, results on asymptotic mean concentration, as described, tend to be consequences of Hoeffding-type bounds and more concrete tail estimates and moment concentration inequalities (e.g. Janson's inequality, Kim-Vu inequality, and related estimates for boolean polynomials). In this sense, my question is:

Question: Is there a line of research directed at the "softer" problem of asymptotic concentration (and $O$-concentration) of families of certain types of r.v.s instead of the more concrete concentration inequalities? If so, where could I find more information about it?


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