Let $X_1,...,X_n$ be iid random variables. Consider $f:\mathbb{R}\times\mathbb{R}^{n-1}\to\mathbb{R}$ such that $f$ is symmetric in the last $n-1$ variables. Our goal is to show that $\sum_{i=1}^n f(X_i;X_{-i})$, where $X_{-i}=\{X_1,...,X_n\}\setminus\{X_i\}$, concentrates around its expected value. What approaches can be used for this purpose? Any references would be greatly appreciated.
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$\begingroup$ Maybe jackknife sampling (en.wikipedia.org/wiki/Jackknife_resampling) is what you look for? $\endgroup$– AlexCommented Feb 17 at 14:20
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1$\begingroup$ Do you really want to consider the sum (rather than, say, the average $(1/n)\sum\ldots$)? For example, if $f(x,y)=x$, then $\operatorname{Var}(\sum f)=n\operatorname{Var}(X_1)$, so the sum won't concentrate around anything. $\endgroup$– Christian RemlingCommented Feb 17 at 15:53
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$\begingroup$ @ChristianRemling: I am looking for a general technique to handle such a sum. Of course, there will be examples when the concentration doesn't hold. In my case, function f has a specific form, and I expect the concentration to hold. For me, the sum and the average are basically the same. $\endgroup$– legonCommented Feb 17 at 20:11
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$\begingroup$ @alex: Thanks, but jackknife corresponds to the case when $f(X_i;X_{-i})$ does not depend on $X_i$. In my setting, this is not the case. $\endgroup$– legonCommented Feb 17 at 20:21
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$\begingroup$ "In my case, function f has a specific form" ... I think it could be helpful to disclose what your $f$ is. Alternatively, you can describe what you think the relevant properties of $f$ are and ask whether the sum concentrates. Otherwise, the question seems way too open. $\endgroup$– Iosif PinelisCommented Feb 18 at 3:08
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