Let $d\_n, n\in\{1,2,\cdots,N\}$ be $N$ realizations drawn independent and identically from uniform distribution on $(0,L)$ where $L=\gamma\sqrt{N}$ with constant $\gamma$. Suppose that we need to approximate the sum
$$\alpha=\sum\_{n=1}^{N}d\_n^{-3},$$
with the restricted sum
$$\hat{\alpha}=\sum\_{n\in\mathcal I}d\_n^{-3},$$
where $\mathcal I\subset N$. We define the set $\mathcal I$ as the largest $|\mathcal I|$ element of random variables $d\_1^{-3},d\_2^{-3},\cdots,d\_N^{-3}$. Now, the question is to find the order of size of subset $\mathcal{I}$ which produce a good approximation ($\hat{\alpha}\simeq\alpha$), i.e.,
$$\mathbb{P}[|\alpha-\hat{\alpha}|\leq\epsilon]\geq 1 -\beta.$$
Is $|\mathcal I|$ in order of sub-linear respect to $N$? 



Limited sum for whole sum approximation
*(Edited after noticing an error which completely changes the answer.)*


*This is not a complete solution; however, it strongly suggests that the answer is **positive**.*


Let $U$ be a random variable with uniform distribution on $[0, 1]$. The random variable $U^{-3}$ has tail $\mathbb{P}[U^{-3} > x] = x^{-1/3}$, and hence it is in the domain of attraction of a stable law with index $\alpha = \tfrac{1}{3}$.


Observe that $d\_n = L U\_n = \gamma \sqrt{N} U\_n$ for an i.i.d. sequence $U\_n$ with uniform distribution on $[0, 1]$. By the invariance principle, the processes
$$ X^N\_t = \frac{1}{N^3} \sum\_{n = 1}^{\lfloor N t \rfloor} U\_n^{-3} $$
converge (in the appropriate Skorokhod topology) to the increasing $\tfrac{1}{3}$-stable Lévy process (i.e. the $\tfrac{1}{3}$-stable subordinator) $X\_t$. Clearly,
$$ \sum\_{n = 1}^{\lfloor N t \rfloor} d\_n^{-3} = \frac{N^{3/2} X^N\_t}{\gamma^3} \, . $$
Denote $J = |\mathcal{I}|$, and let $\hat\alpha\_N$ be the sum of $J$ largest variables among $d\_n^{-3}$, $n = 1, 2, \ldots, N$. Then
$$ \hat\alpha\_N \approx \frac{N^{3/2}}{\gamma} \times (\text{sum of $J$ largest jumps of $X\_t$, $t \in [0, 1]$}) . $$
(I am not rigorous here. However, I am rather convinced one can turn this into a completely argument.)


The question thus turns into the following problem, with $\alpha = \tfrac{1}{3}$ and $\delta = \tfrac{3}{2}$:



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> How many largest jumps of the $\alpha$-stable subordinator $X\_t$, $t \in [0, 1]$, one has to add in order to get an approximation which is within $\pm\gamma N^{-\delta} \times \epsilon$ from $X\_1$ with probability $1 - \beta$.
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This has been studied a lot: it is the question how fast does the *Ferguson–Klass–LePage series* of a stable subordinator converge. In particular, since the $n$-th largest jump of $X\_t$ is comparable with $n^{-1/\alpha}$, the error should be of the order $J^{1 - 1/\alpha}$ when $J$ largest jumps are taken into account. This suggests that $J \approx N^{\delta / (1/\alpha - 1)}$ should do the job. In our case, this gives $J \approx N^{(3/2) / (3 - 1)} = N^{3/4}$, and thus it suggests that it is sufficient to take roughly $|\mathcal{I}| = N^{3/4}$ largest values of $d\_n^{-3}$.


I bet one can find a reference for what is written above (and in fact at least some authors do work with Pareto-distributed jumps $U\_n^{-1/\alpha}$ rather than the ordered jumps of $X\_t$). I am not an expert in this area, though, and I failed to find a reference in a (very) quick Internet search. The closest one that I encountered is:



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> [Bentkus, V., Juozulynas, A. & Paulauskas, V. *Lévy–LePage Series Representation of Stable Vectors: Convergence in Variation*. Journal of Theoretical Probability 14, 949–978 (2001)](https://doi.org/10.1023/A:1017520702943)
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One may also search in the standard reference for simulation of stable random variables:



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> Janicki, A., and Weron, A. (1994). *Simulation and Chaotic Behaviour of $\alpha$-stable Stochastic Processes*. Marcel Dekker, New York
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