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 $N$?

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    $\begingroup$ If you miss even a single realization you may have an error as large as $1/\epsilon^3$ with probabiity $\epsilon/L$ that vanishes only linearly in $\epsilon$, so this will never provide a good approximation. $\endgroup$ – Carlo Beenakker Jan 25 at 15:14
  • $\begingroup$ But, this occurs with a very small probability. By choosing $|\mathcal{I}|$ largest value of $d_n^{-3}$, it seems that by properly choosing $|\mathcal{I}|$ we have a good approximation. $\endgroup$ – Math_Y Jan 25 at 15:21
  • $\begingroup$ But the probability that all of $N$ points be in $(0,\epsilon)$ is $(\epsilon/L)^N$ which is so small. $\endgroup$ – Math_Y Jan 25 at 15:39
  • $\begingroup$ For example, assume that $N$ points are $L/N,2L/N,\cdots,L$. Then, the question is that is it possible to truncate the series $\sum_{i=1}^{N}i^-3$ to have a good approximation? $\endgroup$ – Math_Y Jan 25 at 15:43
  • $\begingroup$ It is computationally efficient for me to choose only a small fraction of them. $\endgroup$ – Math_Y Jan 25 at 15:44

(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 $\delta = \tfrac{3}{2}$:

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 $1 - \beta$.

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:

Bentkus, V., Juozulynas, A. & Paulauskas, V. Lévy–LePage Series Representation of Stable Vectors: Convergence in Variation. Journal of Theoretical Probability 14, 949–978 (2001)

One may also search in the standard reference for simulation of stable random variables:

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

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If we first choose $d_n=\frac{L}{nN}$ and include the $M$ smallest $d_n$'s in the restricted sum, then the relative error is $$E=\frac{\sum_{n=M+1}^N 1/n^3}{\sum_{n=1}^N 1/n^3}=\frac{\psi ^{(2)}(M+1)}{\psi ^{(2)}(1)}+{\cal O}(N^{-2}).$$ This ratio of polygamma functions amounts to less than a 1% error for $M=6$, independent of $N$.

For a statistical test I compared $N=50$ and $N=500$ at a fixed $M=10$: shown below are the two histograms of the cumulative distribution of the error $E$, obtained from $10^3$ realizations of the set of random variables $d_1,d_2,\ldots d_N$. As you can see, the two histograms ($N=50$ on the left, $N=500$ on the right) are nearly the same, with $E<1\%$ happening with probability 0.9, so you can keep $M$ fixed as you scale up $N$.

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  • $\begingroup$ Thank you so much. I only said $nL/N$ as an example. Maybe, it is better to write the problem in probabilistic way. I mean that $\mathbb{P}(|\alpha-\bar{\alpha}|\leq\beta)$. $\endgroup$ – Math_Y Jan 25 at 16:22

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