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This can be viewed as a toy version of this wonderful question. (I'm removing the TSP component, and I'll zoom in on the statistical part.)

Let $X_1,\ldots, X_n$ be iid, with uniform distribution in $[0,1]$. Let $\ell_n$ be the length of the shortest subinterval of $[0,1]$ that contains $n^{\alpha}$ of the points $X_j$, with $0<\alpha<1$. What can we say about the distribution of the normalized length $L_n=n^{1-\alpha}\ell_n$? For example, what are the asymptotics of $EL_n$?

As a warm-up of sorts, we can in fact also consider the shortest interval with $cn$ points and $L_n=\ell_n/c\,$; the answers to the linked question seem to suggest that $EL_n\to 1$ in this case.

(This sounds like it should be well studied, but I couldn't locate anything on the internet.)

This can be viewed as a toy version of this wonderful question. (I'm removing the TSP component, and I'll zoom in on the statistical part.)

Let $X_1,\ldots, X_n$ be iid, with uniform distribution in $[0,1]$. Let $\ell_n$ be the length of the shortest subinterval of $[0,1]$ that contains $n^{\alpha}$ of the points $X_j$, with $0<\alpha<1$. What can we say about the distribution of the normalized length $L_n=n^{1-\alpha}\ell_n$? For example, what are the asymptotics of $EL_n$?

As a warm-up of sorts, we can in fact also consider the shortest interval with $cn$ points and $L_n=\ell_n/c\,$; the answers to the linked question seem to suggest that $EL_n\to 1$ in this case.

(This sounds like it should be well studied, but I couldn't locate anything on the internet.)

This can be viewed as a toy version of this wonderful question. (I'm removing the TSP component, and I'll zoom in on the statistical part.)

Let $X_1,\ldots, X_n$ be iid, with uniform distribution in $[0,1]$. Let $\ell_n$ be the length of the shortest subinterval of $[0,1]$ that contains $n^{\alpha}$ of the points $X_j$, with $0<\alpha<1$. What can we say about the distribution of the normalized length $L_n=n^{1-\alpha}\ell_n$? For example, what are the asymptotics of $EL_n$?

We can in fact also consider the shortest interval with $cn$ points and $L_n=\ell_n/c\,$; the answers to the linked question seem to suggest that $EL_n\to 1$ in this case.

(This sounds like it should be well studied, but I couldn't locate anything on the internet.)

This can be viewed as a toy version of this wonderful question. (I'm removing the TSP component, and I'll zoom in on the statistical part.)

Let $X_1,\ldots, X_n$ be iid, with uniform distribution in $[0,1]$. Let $\ell_n$ be the length of the shortest subinterval of $[0,1]$ that contains $n^{\alpha}$ of the points $X_j$, with $0<\alpha<1$. What can we say about the distribution of the normalized length $L_n=n^{1-\alpha}\ell_n$? For example, what are the asymptotics of $EL_n$?

As a warm-up of sorts, we can in fact also consider the shortest interval with $cn$ points and $L_n=\ell_n/c\,$; the answers to the linked question seem to suggest that $EL_n\to 1$ in this case.

(This sounds like it should be well studied, but I couldn't locate anything on the internet.)

This can be viewed as a toy version of this wonderful question. (I'm removing the TSP component, and I'll zoom in on the statistical part.)

Let $X_1,\ldots, X_n$ be iid, with uniform distribution in $[0,1]$. Let $\ell_n$ be the length of the shortest subinterval of $[0,1]$ that contains $n^{\alpha}$ of the points $X_j$, with $0<\alpha<1$. What can we say about the distribution of the normalized length $L_n=n^{1-\alpha}\ell_n$? For example, what are the asymptotics of $EL_n$?

(This sounds like it should be well studied, but I couldn't locate anything on the internet.)

This can be viewed as a toy version of this wonderful question. (I'm removing the TSP component, and I'll zoom in on the statistical part.)

Let $X_1,\ldots, X_n$ be iid, with uniform distribution in $[0,1]$. Let $\ell_n$ be the length of the shortest subinterval of $[0,1]$ that contains $n^{\alpha}$ of the points $X_j$, with $0<\alpha<1$. What can we say about the distribution of the normalized length $L_n=n^{1-\alpha}\ell_n$? For example, what are the asymptotics of $EL_n$?

We can in fact also consider the shortest interval with $cn$ points and $L_n=\ell_n/c\,$; the answers to the linked question seem to suggest that $EL_n\to 1$ in this case.

(This sounds like it should be well studied, but I couldn't locate anything on the internet.)