On Littlewood-Offord theory for arbitrary distributions
Let $X_1,\ldots,X_n$ be independent identically distributed random vectors in $\mathbb{R}^d$. We consider upper bounds on $\max_x \mathbb{P}(a_1X_1+\cdots+a_nX_n=x)$ under various restrictions on $X_i$ and the weights $a_i$. When $\mathbb{P}(X_i=\pm 1) = \frac {1} {2}$, this corresponds to the class...
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| Main Authors | , |
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| Format | Journal Article |
| Language | English |
| Published |
18.12.2019
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| Subjects | |
| Online Access | Get full text |
| DOI | 10.48550/arxiv.1912.08770 |
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| Summary: | Let $X_1,\ldots,X_n$ be independent identically distributed random vectors in
$\mathbb{R}^d$. We consider upper bounds on $\max_x
\mathbb{P}(a_1X_1+\cdots+a_nX_n=x)$ under various restrictions on $X_i$ and the
weights $a_i$. When $\mathbb{P}(X_i=\pm 1) = \frac {1} {2}$, this corresponds
to the classical Littlewood-Offord problem. We prove that in general for
identically distributed random vectors and even values of $n$ the optimal
choice for $(a_i)$ is $a_i=1$ for $i\leq \frac{n}{2}$ and $a_i=-1$ for $i >
\frac {n} 2$, regardless of the distribution of $X_1$. Applying these results
for Bernoulli random variables answers a recent question of Fox, Kwan and
Sauermann.
Finally, we provide sharp bounds for concentration probabilities of sums of
random vectors under the condition $\sup_{x}\mathbb{P}(X_i=x)\leq \alpha$,
where it turns out that the worst case scenario is provided by distributions on
an arithmetic progression that are in some sense as close to the uniform
distribution as possible. An important feature of this work is that unlike much
of the literature on the subject we use neither methods of harmonic analysis
nor those from extremal combinatorics. |
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| DOI: | 10.48550/arxiv.1912.08770 |