Two-sided estimates for order statistics of log-concave random vectors

TL;DR

This paper derives precise two-sided bounds for the expectations of order statistics of log-concave random vectors, advancing understanding of their coordinate maxima and sums with universal constants.

Contribution

It provides the first sharp two-sided bounds for order statistics of log-concave vectors, applicable in unconditional and isotropic cases, with explicit bounds for sums of largest coordinates.

Findings

Exact bounds up to universal constants for order statistics in unconditional case.

Bounds valid for all k in unconditional case and for k ≤ n - c n^{5/6} in isotropic case.

Estimates for sums of the largest moduli of coordinates for specific classes of vectors.

Abstract

We establish two-sided bounds for expectations of order statistics ($k$-th maxima) of moduli of coordinates of centered log-concave random vectors with uncorrelated coordinates. Our bounds are exact up to multiplicative universal constants in the unconditional case for all $k$ and in the isotropic case for $k \leq n-cn^{5/6}$. We also derive two-sided estimates for expectations of sums of $k$ largest moduli of coordinates for some classes of random vectors.

Equations

$$\mathbb{E}\underset{|I|=k}{\max }\sum _{i\in I}|X_i|=\mathbb{E}\sum _{l=1}^{k}l\text{-}\underset{i}{\max }|X_i|.$$

$$t(k,X):=\inf \left\{t>0:\frac{1}{t}\sum _{i=1}^n\mathbb{E}|X_i|1_{\{|X_i|\ge t\}}\le k\right\}.$$

$$\mathbb{E}\underset{|I|=k}{\max }\sum _{i\in I}|X_i|\le 2kt(k,X).$$

$$\underset{|I|=k}{\max }\sum _{i\in I}|X_i|\le tk+\sum _{i=1}^n|X_i|1_{\{|X_i|\ge t\}}.\text{qed}$$

$$\mathbb{P}(|X_i|\ge s,|X_j|\ge t)\le \alpha \mathbb{P}(|X_i|\ge s)\mathbb{P}(|X_j|\ge t)\text{mbox}foralli\ne j\text{mbox}andalls,t>0.$$

$$c(\alpha )kt(k,X)\le \mathbb{E}\underset{|I|=k}{\max }\sum _{i\in I}|X_i|\le 2kt(k,X).$$

$$ckt(k,X)\le \mathbb{E}\underset{|I|=k}{\max }\sum _{i\in I}|X_i|\le 2kt(k,X).$$

$$\mathbb{E}\underset{|I|=k}{\max }\sum _{i\in I}|X_i|\sim \mathbb{E}\underset{|I|=k}{\max }\sum _{i\in I}|X_i^{\prime }|.$$

$$\mathbb{E}\Vert X\Vert \sim \mathbb{E}\Vert X^{\prime }\Vert ?$$

$$t^{*}(p,X):=\inf\biggl{\{}t>0\colon\ \sum_{i=1}^{n}{\mathbb{P}}(|X_{i}|\geq t)\leq p\biggr{\}}\quad\mbox{ for }0<p<n.$$

$${\mathbb{E}}k\text{-}\max_{i\leq n}|X_{i}|\geq\frac{1}{2}\mathrm{Med}\Bigl{(}k\text{-}\max_{i\leq n}|X_{i}|\Bigr{)}\geq ct^{*}\biggl{(}k-\frac{1}{2},X\biggr{)}.$$

$${\mathbb{E}}k\text{-}\max_{i\leq n}|X_{i}|\leq Ct^{*}\biggl{(}k-\frac{1}{2},X\biggr{)}.$$

$${\mathbb{P}}\biggl{(}k\text{-}\max_{i\leq n}|X_{i}|\geq Ct^{*}\biggl{(}k-\frac{1}{2},X\biggr{)}\biggr{)}\leq 1-c$$

$${\mathbb{E}}k\text{-}\max_{i\leq n}|X_{i}|\leq Ct^{*}\biggl{(}k-\frac{1}{2}k^{5/6},X\biggr{)}.$$

$$\mathbb{E}k\text{-}\underset{i\le n}{\max }|X_i|\sim t^{\ast }(k,X)\sim t(k,X)$$

$$c\frac{k}{n}\le \mathbb{E}k\text{-}\underset{i\le n}{\min }|X_i|=\mathbb{E}(n-k+1)\text{-}\underset{i\le n}{\max }|X_i|\le C\left(\frac{k}{n}+n^{-1/6}\right).$$

$$\mathbb{E}k\text{-}\underset{i\le n}{\min }|X_i|=\mathbb{E}(n-k+1)\text{-}\underset{i\le n}{\max }|X_i|\sim \frac{k}{n}.$$

$$1-\mu (A+u{B}_2^n)\le e^{-u/\alpha }\text{mbox}forallu>0.$$

$${\mathrm{Var}}_{\mu }f\le \beta \int |\nabla f{|}^{2}d\mu \text{mbox}forallboundedsmoothfunctionsf:{\mathbb{R}}^n\mapsto \mathbb{R}$$

$${\mathbb{E}}\max_{|I|=k}\sum_{i\in I}|X_{i}|\geq\Bigl{(}8+64\frac{\alpha}{\sqrt{k}}\Bigr{)}^{-1}kt(k,X),$$

$$\max_{|I|=k}\sum_{i\in I}|z_{i}|=\int_{0}^{\infty}\min\biggl{\{}k,\sum_{i=1}^{n}{\mathbf{1}}_{\{|z_{i}|\geq s\}}\biggr{\}}ds.$$

$$\displaystyle\int_{0}^{\infty}\min\biggl{\{}k,\sum_{i=1}^{n}{\mathbf{1}}_{\{|z_{i}|\geq s\}}\biggr{\}}ds$$

$$=z_1+\dots +z_k=\underset{|I|=k}{\max }\sum _{i\in I}|z_i|.\text{qed}$$

$$N(s):=\sum _{i=1}^n{1}_{\{|X_i|\ge s\}}.$$

$$\mathbb{E}\underset{|I|=k}{\max }\sum _{i\in I}|X_i|={\int }_0^{\infty }\sum _{l=1}^k\mathbb{P}(N(s)\ge l)ds,$$

$$\mathbb{E}\sum _{i=1}^n|X_i|1_{\{|X_i|\ge t\}}=t\mathbb{E}N(t)+{\int }_t^{\infty }\sum _{l=1}^{\infty }\mathbb{P}(N(s)\ge l)ds.$$

$$\mathbb{E}\sum _{i=1}^n|X_i|1_{\{|X_i|\ge t\}}\le \mathbb{E}\underset{|I|=k}{\max }\sum _{i\in I}|X_i|+\sum _{l=k+1}^{\infty }\left(t\mathbb{P}(N(t)\ge l)+{\int }_t^{\infty }\mathbb{P}(N(s)\ge l)ds\right).$$

$${\int }_0^{\infty }\sum _{l=1}^k\mathbb{P}(N(s)\ge l)ds$$

$$=\mathbb{E}\underset{|I|=k}{\max }\sum _{i\in I}|X_i|,$$

$$t\mathbb{E}N(t)+{\int }_t^{\infty }\sum _{l=1}^{\infty }\mathbb{P}(N(s)\ge l)ds$$

$$=\mathbb{E}\sum _{i=1}^n\left(t{1}_{\{|X_i|\ge t\}}+{\int }_t^{\infty }{1}_{\{|X_i|\ge s\}}ds\right)$$

$$=\mathbb{E}\sum _{i=1}^n|X_i|1_{\{|X_i|\ge t\}}.$$

Full text

The research of RL was supported by the National Science Centre, Poland grant 2015/18/A/ST1/00553 and of MS by the National Science Centre, Poland grant 2015/19/N/ST1/02661

1. Introduction and main results

For a vector $x\in{\mathbb{R}}^{n}$ let $k{\text{-}}\max x_{i}$ (or $k{\text{-}}\min x_{i}$) denote its $k$-th maximum (respectively its * $k$-th minimum*), i.e. its $k$-th maximal (respectively $k$-th minimal) coordinate. For a random vector $X=(X_{1},\ldots,X_{n})$, $k{\text{-}}\min X_{i}$ is also called the $k$-th order statistic of $X$.

Let $X=(X_{1},\ldots,X_{n})$ be a random vector with finite first moment. In this note we try to estimate ${\mathbb{E}}k{\text{-}}\max_{i}|X_{i}|$ and

[TABLE]

Order statistics play an important role in various statistical applications and there is an extensive literature on this subject (cf. [2, 5] and references therein).

We put special emphasis on the case of log-concave vectors, i.e. random vectors $X$ satisfying the property ${\mathbb{P}}(X\in\lambda K+(1-\lambda)L)\geq{\mathbb{P}}(X\in K)^{\lambda}{\mathbb{P}}(X\in L)^{1-\lambda}$ for any $\lambda\in[0,1]$ and any nonempty compact sets $K$ and $L$. By the result of Borell [3] a vector $X$ with full dimensional support is log-concave if and only if it has a log-concave density, i.e. the density of a form $e^{-h(x)}$ where $h$ is convex with values in $(-\infty,\infty]$. A typical example of a log-concave vector is a vector uniformly distributed over a convex body. In recent years the study of log-concave vectors attracted attention of many researchers, cf. monographs [1, 4].

To bound the sum of $k$ largest coordinates of $X$ we define

[TABLE]

and start with an easy upper bound.

Proposition 1**.**

For any random vector $X$ with finite first moment we have

[TABLE]

Proof.

For any $t>0$ we have

[TABLE]

It turns out that this bound may be reversed for vectors with independent coordinates or, more generally, vectors satisfying the following condition

[TABLE]

If $\alpha=1$ this means that moduli of coordinates of $X$ are negatively correlated.

Theorem 2**.**

Suppose that a random vector $X$ satisfies condition (3) with some $\alpha\geq 1$. Then there exists a constant $c(\alpha)>0$ which depends only on $\alpha$ such that for any $1\leq k\leq n$,

[TABLE]

We may take $c(\alpha)=(36(5+4\alpha)(1+2\alpha))^{-1}$.

In the case of i.i.d. coordinates two-sided bounds for ${\mathbb{E}}\max_{|I|=k}\sum_{i\in I}|a_{i}X_{i}|$ in terms of an Orlicz norm (related to the distribution of $X_{i}$) of a vector $(a_{i})_{i\leq n}$ where known before, see [7].

Log-concave vectors with diagonal covariance matrices behave in many aspects like vectors with independent coordinates. This is true also in our case.

Theorem 3**.**

Let $X$ be a log-concave random vector with uncorrelated coordinates (i.e. ${\operatorname{Cov}}(X_{i},X_{j})=0$ for $i\neq j$). Then for any $1\leq k\leq n$,

[TABLE]

In the above statement and in the sequel $c$ and $C$ denote positive universal constants.

The next two examples show that the lower bound cannot hold if $n\gg k$ and only marginal distributions of $X_{i}$ are log-concave or the coordinates of $X$ are highly correlated.

Example 1. Let $X=(\varepsilon_{1}g,\varepsilon_{2}g,\ldots,\varepsilon_{n}g)$, where $\varepsilon_{1},\ldots,\varepsilon_{n},g$ are independent, ${\mathbb{P}}(\varepsilon_{i}=\pm 1)=1/2$ and $g$ has the normal ${\mathcal{N}}(0,1)$ distribution. Then ${\operatorname{Cov}}X={\operatorname{Id}}$ and it is not hard to check that ${\mathbb{E}}\max_{|I|=k}\sum_{i\in I}|X_{i}|=k\sqrt{2/\pi}$ and $t(k,X)\sim\ln^{1/2}(n/k)$ if $k\leq n/2$.

Example 2. Let $X=(g,\ldots,g)$, where $g\sim{\mathcal{N}}(0,1)$. Then, as in the previous example, ${\mathbb{E}}\max_{|I|=k}\sum_{i\in I}|X_{i}|=k\sqrt{2/\pi}$ and $t(k,X)\sim\ln^{1/2}(n/k)$.

Question 1. Let $X^{\prime}=(X^{\prime}_{1},X^{\prime}_{2},\ldots,X^{\prime}_{n})$ be a decoupled version of $X$, i.e. $X^{\prime}_{i}$ are independent and $X^{\prime}_{i}$ has the same distribution as $X_{i}$. Due to Theorem 2 (applied to $X^{\prime}$), the assertion of Theorem 3 may be stated equivalently as

[TABLE]

Is the more general fact true that for any symmetric norm and any log-concave vector $X$ with uncorrelated coordinates

[TABLE]

Maybe such an estimate holds at least in the case of unconditional log-concave vectors?

We turn our attention to bounding $k$-maxima of $|X_{i}|$. This was investigated in [8] (under some strong assumptions on the function $t\mapsto{\mathbb{P}}(|X_{i}|\geq t)$) and in the weighted i.i.d. setting in [7, 9, 15]. We will give different bounds valid for log-concave vectors, in which we do not have to assume independence, nor any special conditions on the growth of the distribution function of the coordinates of $X$. To this end we need to define another quantity:

[TABLE]

Theorem 4**.**

Let $X$ be a mean zero log-concave $n$-dimensional random vector with uncorrelated coordinates and $1\leq k\leq n$. Then

[TABLE]

Moreover, if $X$ is additionally unconditional then

[TABLE]

The next theorem provides an upper bound in the general log-concave case.

Theorem 5**.**

Let $X$ be a mean zero log-concave $n$-dimensional random vector with uncorrelated coordinates and $1\leq k\leq n$. Then

[TABLE]

and

[TABLE]

In the isotropic case (i.e. ${\mathbb{E}}X_{i}=0,{\operatorname{Cov}}X={\operatorname{Id}}$) one may show that $t^{*}(k/2,X)\sim t^{*}(k,X)\sim t(k,X)$ for $k\leq n/2$ and $t^{*}(p,X)\sim\frac{n-p}{n}$ for $p\geq n/4$ (see Lemma 24 below). In particular $t^{*}(n-k+1-(n-k+1)^{5/6}/2,X)\sim k/n-n^{-1/6}$ for $k\leq n/2$. This together with the two previous theorems implies the following corollary.

Corollary 6**.**

Let $X$ be an isotropic log-concave $n$-dimensional random vector and $1\leq k\leq n/2$. Then

[TABLE]

and

[TABLE]

If $X$ is additionally unconditional then

[TABLE]

Question 2. Does the second part of Theorem 4 hold without the unconditionality assumptions? In particular, is it true that ${\mathbb{E}}k\text{-}\min_{i\leq n}|X_{i}|\sim k/n$ for $1\leq k\leq n/2$?

Notation. Throughout this paper by letters $C,c$ we denote universal positive constants and by $C(\alpha),c(\alpha)$ constants depending only on the parameter $\alpha$. The values of constants $C,c,C(\alpha),c(\alpha)$ may differ at each occurrence. If we need to fix a value of constant, we use letters $C_{0},C_{1},\ldots$ or $c_{0},c_{1},\ldots$. We write $f\sim g$ if $cf\leq g\leq Cg$. For a random variable $Z$ we denote $\|Z\|_{p}=({\mathbb{E}}|Z|^{p})^{1/p}$. Recall that a random vector $X$ is called isotropic, if ${\mathbb{E}}X=0$ and ${\operatorname{Cov}}X={\operatorname{Id}}$.

This note is organised as follows. In Section 2 we provide a lower bound for the sum of $k$ largest coordinates, which involves the Poincaré constant of a vector. In Section 3 we use this result to obtain Theorem 3. In Section 4 we prove Theorem 2 and provide its application to comparison of weak and strong moments. In Section 5 we prove the first part of Theorem 4 and in Section 6 we prove the second part of Theorem 4, Theorem 5, and Lemma 24.

2. Exponential concentration

A probability measure $\mu$ on ${\mathbb{R}}^{n}$ satisfies exponential concentration with constant $\alpha>0$ if for any Borel set $A$ with $\mu(A)\geq 1/2$,

[TABLE]

We say that a random $n$-dimensional vector satisfies exponential concentration if its distribution has such a property.

It is well known that exponential concentration is implied by the Poincaré inequality

[TABLE]

and $\alpha\leq 3\sqrt{\beta}$ (cf. [12, Corollary 3.2]).

Obviously, the constant in the exponential concentration is not linearly invariant. Typically one assumes that the vector is isotropic. For our purposes a more natural normalization will be that all coordinates have $L_{1}$-norm equal to $1$.

The next proposition states that bound (2) may be reversed under the assumption that $X$ satisfies the exponential concentration.

Proposition 7**.**

Assume that $Y=(Y_{1},\ldots,Y_{n})$ satisfies the exponential concentration with constant $\alpha>0$ and ${\mathbb{E}}|Y_{i}|\geq 1$ for all $i$. Then for any sequence $a=(a_{i})_{i=1}^{n}$ of real numbers and $X_{i}:=a_{i}Y_{i}$ we have

[TABLE]

where $t(k,X)$ is given by (1).

We begin the proof with a few simple observations.

Lemma 8**.**

For any real numbers $z_{1},\ldots,z_{n}$ and $1\leq k\leq n$ we have

[TABLE]

Proof.

Without loss of generality we may assume that $z_{1}\geq z_{2}\geq\ldots\geq z_{n}\geq 0$. Then

[TABLE]

Fix a sequence $(X_{i})_{i\leq n}$ and define for $s\geq 0$,

[TABLE]

Corollary 9**.**

For any $k=1,\ldots,n$,

[TABLE]

and for any $t>0$,

[TABLE]

In particular

[TABLE]

Proof.

We have

[TABLE]

where the last equality follows by Lemma 8.

Moreover,

[TABLE]

The last part of the assertion easily follows, since

[TABLE]

Proof of Proposition 7.

To shorten the notation put $t_{k}:=t(k,X)$. Without loss of generality we may assume that $a_{1}\geq a_{2}\geq\ldots\geq a_{n}\geq 0$ and $a_{\lceil k/4\rceil}=1$. Observe first that

[TABLE]

so we may assume that $t_{k}\geq 16\alpha/\sqrt{k}$.

Let $\mu$ be the law of $Y$ and

[TABLE]

We have

[TABLE]

so we may assume that $\mu(A)\geq 1/2$.

Observe that if $y\in A$ and $\sum_{i=1}^{n}{\mathbf{1}}_{\{|a_{i}z_{i}|\geq s\}}\geq l>k$ for some $s\geq t_{k}$ then

[TABLE]

Thus we have

[TABLE]

Therefore

[TABLE]

and

[TABLE]

where to get the next-to-last inequality we used the fact that $t_{k}\geq 16\alpha/\sqrt{k}$.

Hence Corollary 9 and the definition of $t_{k}$ yields

[TABLE]

so ${\mathbb{E}}\max_{|I|=k}\sum_{i\in I}|X_{i}|\geq\frac{1}{2}kt_{k}$. ∎

We finish this section with a simple fact that will be used in the sequel.

Lemma 10**.**

Suppose that a measure $\mu$ satisfies exponential concentration with constant $\alpha$. Then for any $c\in(0,1)$ and any Borel set $A$ with $\mu(A)>c$ we have

[TABLE]

Proof.

Let $D:={\mathbb{R}}^{n}\setminus(A+rB_{2}^{n})$. Observe that $D+rB_{2}^{n}$ has an empty intersection with $A$ so if $\mu(D)\geq 1/2$ then

[TABLE]

and $r<\alpha\ln(1/c)$. Hence $\mu(A+\alpha\ln(1/c)B_{2}^{n})\geq 1/2$, therefore for $s\geq 0$,

[TABLE]

and the assertion easily follows. ∎

3. Sums of largest coordinates of log-concave vectors

We will usethe regular growth of moments of norms of log-concave vectors multiple times. By [4, Theorem 2.4.6], if $f:{\mathbb{R}}^{n}\to{\mathbb{R}}$ is a seminorm, and $X$ is log-concave, then

[TABLE]

where $C_{1}$ is a universal constant.

We will also apply a few times the functional version of the Grünbaum inequality (see [14, Lemma 5.4]) which states that

[TABLE]

Let us start with a few technical lemmas. The first one will be used to reduce the proof of Theorem 3 to the symmetric case.

Lemma 11**.**

Let $X$ be a log-concave $n$-dimensional vector and $X^{\prime}$ be an independent copy of $X$. Then for any $1\leq k\leq n$,

[TABLE]

and

[TABLE]

Proof.

The first estimate follows by the easy bound

[TABLE]

To get the second bound we may and will assume that ${\mathbb{E}}|X_{1}|\geq{\mathbb{E}}|X_{2}|\geq\ldots\geq{\mathbb{E}}|X_{n}|$. Let us define $Y:=X-{\mathbb{E}}X$, $Y^{\prime}:=X^{\prime}-{\mathbb{E}}X$ and $M:=\frac{1}{k}\sum_{i=1}^{k}{\mathbb{E}}|X_{i}|\geq\max_{i\geq k}{\mathbb{E}}|X_{i}|$. Obviously

[TABLE]

We have ${\mathbb{E}}Y_{i}=0$, thus ${\mathbb{P}}(Y_{i}\leq 0)\geq 1/e$ by (8). Hence

[TABLE]

for $t\geq 0$. In the same way we show that

[TABLE]

Therefore

[TABLE]

We have

[TABLE]

Together with (10) we get

[TABLE]

and (9) easily follows. ∎

Lemma 12**.**

Suppose that $V$ is a real symmetric log-concave random variable. Then for any $t>0$ and $\lambda\in(0,1]$,

[TABLE]

Moreover, if ${\mathbb{P}}(|V|\geq t)\leq 1/4$, then ${\mathbb{E}}|V|{\mathbf{1}}_{\{|V|\geq t\}}\leq 4t{\mathbb{P}}(|V|\geq t).$

Proof.

Without loss of generality we may assume that ${\mathbb{P}}(|V|\geq t)\leq 1/4$ (otherwise the first estimate is trivial).

Observe that ${\mathbb{P}}(|V|\geq s)=\exp(-N(s))$ where $N\colon[0,\infty)\to[0,\infty]$ is convex and $N(0)=0$. In particular

[TABLE]

and

[TABLE]

We have

[TABLE]

This implies the second part of the lemma.

To conclude the proof of the first bound it is enough to observe that

[TABLE]

Proof of Theorem 3.

By Proposition 1 it is enough to show the lower bound. By Lemma 11 we may assume that $X$ is symmetric. We may also obviously assume that $\|X_{i}\|_{2}^{2}={\mathbb{E}}X_{i}^{2}>0$ for all $i$.

Let $Z=(Z_{1},\ldots,Z_{n})$, where $Z_{i}=X_{i}/\|X_{i}\|_{2}$. Then $Z$ is log-concave, isotropic and, by (7), ${\mathbb{E}}|Z_{i}|\geq 1/(2C_{1})$ for all $i$. Set $Y:=2C_{1}Z$. Then $X_{i}=a_{i}Y_{i}$ and ${\mathbb{E}}|Y_{i}|\geq 1$. Moreover, by the result of Lee and Vempala [13], we know that any $m$-dimensional projection of $Z$ is a log-concave, isotropic $m$-dimensional vector thus it satisfies the exponential concentration with a constants $Cm^{1/4}$. (In fact an easy modification of the proof below shows that for our purposes it would be enough to have exponential concentration with a constant $Cm^{\gamma}$ for some $\gamma<1/2$, so one may also use Eldan’s result [6] which gives such estimates for any $\gamma>1/3$). So any $m$-dimensional projection of $Y$ satisfies exponential concentration with constant $C_{2}m^{1/4}$.

Let us fix $k$ and set $t:=t(k,X)$, then (since $X_{i}$ has no atoms)

[TABLE]

For $l=1,2,\ldots$ define

[TABLE]

where $\beta=2^{-8}$. By (11) there exists $l$ such that

[TABLE]

Let us consider three cases.

(i) $l=1$ and $|I_{1}|\leq k$. Then

[TABLE]

(ii) $l=1$ and $|I_{1}|>k$. Choose $J\subset I_{1}$ of cardinality $k$. Then

[TABLE]

(iii) $l>1$. By Lemma 12 (applied with $\lambda=1/8$) we have

[TABLE]

Moreover for $i\in I_{l}$, ${\mathbb{P}}(|X_{i}|\geq t)\leq\beta^{l-1}\leq 1/4$, so the second part of Lemma 12 yields

[TABLE]

and $|I_{l}|\geq\beta^{1-l}2^{-l-2}k=2^{7l-10}k\geq k$.

Set $k^{\prime}:=\beta^{-7l/8}2^{-l}k=2^{6l}k$. If $k^{\prime}\geq|I_{l}|$ then, using (12), we estimate

[TABLE]

Otherwise set $X^{\prime}=(X_{i})_{i\in I_{l}}$ and $Y^{\prime}=(Y_{i})_{i\in I_{l}}$. By (11) we have

[TABLE]

so $|I_{l}|\leq k\beta^{-l}$ and $Y^{\prime}$ satisfies exponential concentration with constant $\alpha^{\prime}=C_{2}k^{1/4}\beta^{-l/4}$. Estimate (12) yields

[TABLE]

so $t(k^{\prime},X^{\prime})\geq 2^{-12}t$. Moreover, by Proposition 7 we have (since $k^{\prime}\leq|I_{l}|$)

[TABLE]

To conclude observe that

[TABLE]

and since $k^{\prime}\geq k$,

[TABLE]

4. Vectors satisfying condition (3)

Proof of Theorem 2.

By Proposition 1 we need to show only the lower bound. Assume first that variables $X_{i}$ have no atoms and $k\geq 4(1+\alpha)$.

Let $t_{k}=t(k,X)$. Then ${\mathbb{E}}\sum_{i=1}^{n}|X_{i}|{\mathbf{1}}_{\{|X_{i}|\geq t_{k}\}}=kt_{k}$. Note, that (3) implies that for all $i\neq j$ we have

[TABLE]

We may assume that ${\mathbb{E}}\max_{|I|=k}\sum_{i\in I}|X_{i}|\leq\frac{1}{6}kt_{k}$, because otherwise the lower bound holds trivially.

Let us define

[TABLE]

Since

[TABLE]

it suffices to bound below the probability that $Y\geq kt_{k}/2$ by a constant depending only on $\alpha$.

We have

[TABLE]

Therefore $A^{2}\leq(1+2\alpha)k^{2}t_{k}^{2}$ and for any $l\geq k/2$ we have

[TABLE]

By Corollary 9 we have (recall definition(6))

[TABLE]

Assumption (3) implies that

[TABLE]

Moreover for $s\geq kt_{k}$ we have

[TABLE]

so

[TABLE]

Thus

[TABLE]

and

[TABLE]

This together with (15) and the assumption that $k\geq 4(1+\alpha)$ implies

[TABLE]

and

[TABLE]

Therefore

[TABLE]

This applied to (14) with $l=(12+24\alpha)k$ gives us ${\mathbb{P}}(Y\geq kt_{k}/2)\geq(144+288\alpha)^{-1}$ and in consequence

[TABLE]

Since $k\mapsto kt(k,X)$ is non-decreasing, in the case $k\leq\lceil 4(1+\alpha)\rceil=:k_{0}\geq 8$ we have

[TABLE]

The last step is to loose the assumption that $X_{i}$ has no atoms. Note that both assumption (3) and the lower bound depend only on $(|X_{i}|)_{i=1}^{n}$, so we may assume that $X_{i}$ are nonnegative almost surely. Consider $X^{\varepsilon}:=(X_{i}+\varepsilon Y_{i})_{i=1}^{n}$, where $Y_{1},\ldots,Y_{n}$ are i.i.d. nonnegative r.v’s with ${\mathbb{E}}Y_{i}<\infty$ and a density $g$, independent of $X$. Then for every $s,t>0$ we have (observe that (3) holds also for $s<0$ or $t<0$).

[TABLE]

Thus $X^{\varepsilon}$ satisfies assumption (3) and has the density function for every $\varepsilon>0$. Therefore for all natural $k$ we have

[TABLE]

Clearly, ${\mathbb{E}}\max_{|I|=k}\sum_{i=1}^{n}X_{i}^{\varepsilon}\to{\mathbb{E}}\max_{|I|=k}\sum_{i=1}^{n}X_{i}$ as $\varepsilon\to 0$, so the lower bound holds in the case of arbitrary $X$ satisfying (3). ∎

We may use Theorem 2 to obtain a comparison of weak and strong moments for the supremum norm:

Corollary 13**.**

Let $X$ be an $n$-dimensional centered random vector satisfying condition (3). Assume that

[TABLE]

Then the following comparison of weak and strong moments for the supremum norm holds: for all $a\in{\mathbb{R}}^{n}$ and all $p\geq 1$,

[TABLE]

where $C(\alpha,\beta)$ is a constant depending only on $\alpha$ and $\beta$.

Proof.

Let $X^{\prime}=(X_{i}^{\prime})_{i\leq n}$ be a decoupled version of $X$. For any $p>0$ a random vector $(|a_{i}X_{i}|^{p})_{i\leq n}$ satisfies condition (3), so by Theorem 2

[TABLE]

for all $p>0$, up to a constant depending only on $\alpha$. The coordinates of $X^{\prime}$ are independent and satisfy condition (16), so due to [11, Theorem 1.1] the comparison of weak and strong moments of $X^{\prime}$ holds, i.e. for $p\geq 1$,

[TABLE]

where $C(\beta)$ depends only on $\beta$. These two observations yield the assertion. ∎

5. Lower estimates for order statistics

The next lemma shows the relation between $t(k,X)$ and $t^{*}(k,X)$ for log-concave vectors $X$.

Lemma 14**.**

Let $X$ be a symmetric log-concave random vector in ${\mathbb{R}}^{n}$. For any $1\leq k\leq n$ we have

[TABLE]

Proof.

Let $t_{k}:=t(k,X)$ and $t_{k}^{*}:=t^{*}(k,X)$. We may assume that any $X_{i}$ is not identically equal to [math]. Then $\sum_{i=1}^{n}{\mathbb{P}}(|X_{i}|\geq t_{k}^{*})=k$ and $\sum_{i=1}^{n}{\mathbb{E}}|X_{i}|_{{\{}|X_{i}|\geq t_{k}{\}}}=kt_{k}$.

Obviously $t_{k}^{*}\leq t_{k}$. Also for any $|I|=k$ we have

[TABLE]

To prove the upper bound set

[TABLE]

We have

[TABLE]

so $|I_{1}|\leq 4k$. Hence

[TABLE]

Moreover by the second part of Lemma 12 we get

[TABLE]

so

[TABLE]

Hence if $s=4t_{k}^{*}+\frac{4}{k}\max_{|I|=k}\sum_{i\in I}{\mathbb{E}}|X_{i}|$ then

[TABLE]

that is $t_{k}\leq s$. ∎

To derive bounds for order statistics we will also need a few facts about log-concave vectors.

Lemma 15**.**

Assume that $Z$ is an isotropic one- or two-dimensional log-concave random vector with a density $g$. Then $g(t)\leq C$ for all $t$. If $Z$ is one-dimensional, then also $g(t)\geq c$ for all $|t|\leq t_{0}$, where $t_{0}>0$ is an absolute constant.

Proof.

We will use a classical result (see [4, Theorem 2.2.2, Proposition 3.3.1 and Proposition 2.5.9]): $\|g\|_{\sup}\sim g(0)\sim 1$ (note that here we use the assumption that $Z$ is isotropic, in particular that ${\mathbb{E}}Z=0$, and that the dimension of $Z$ is $1$ or $2$). This implies the upper bound on $g$.

In order to get the lower bound in the one-dimensional case, it suffices to prove that $g(u)\geq c$ for $|u|=\varepsilon{\mathbb{E}}|Z|\geq(2C_{1})^{-1}\varepsilon$, where $1/4>\varepsilon>0$ is fixed and its value will be chosen later (then by the log-concavity we get $g(u)^{s}g(0)^{1-s}\leq g(su)$ for all $s\in(0,1)$). Since $-Z$ is again isotropic we may assume that $u\geq 0$.

If $g(u)\geq g(0)/e$, then we are done. Otherwise by log-concavity of $g$ we get

[TABLE]

On the other hand, $Z$ has mean zero, so ${\mathbb{E}}|Z|=2{\mathbb{E}}Z_{+}$ and by the Paley–Zygmund inequality and (7) we have

[TABLE]

For $\varepsilon<c_{0}/C_{0}$ we get a contradiction. ∎

Lemma 16**.**

Let $Y$ be a mean zero log-concave random variable and let ${\mathbb{P}}(|Y|\geq t)\leq p$ for some $p>0$. Then

[TABLE]

Proof.

By the Grünbaum inequality (8) we have ${\mathbb{P}}(Y\geq 0)\geq 1/e$, hence

[TABLE]

Since $-Y$ satisfies the same assumptions as $Y$ we also have

[TABLE]

Lemma 17**.**

Let $Y$ be a mean zero log-concave random variable and let ${\mathbb{P}}(|Y|\geq t)\geq p$ for some $p>0$. Then there exists a universal constant $C$ such that

[TABLE]

Proof.

Without loss of generality we may assume that ${\mathbb{E}}Y^{2}=1$. Then by Chebyshev’s inequality $t\leq p^{-1/2}$. Let $g$ be the density of $Y$. By Lemma 15 we know that $\|g\|_{\infty}\leq C$ and $g(t)\geq c$ on $[-t_{0},t_{0}]$, where $c,C$ and $t_{0}\in(0,1)$ are universal constants. Thus

[TABLE]

and

[TABLE]

Now we are ready to give a proof of the lower bound in Theorem 4. The next proposition is a key part of it.

Proposition 18**.**

Let $X$ be a mean zero log-concave $n$-dimensional random vector with uncorrelated coordinates and let $\alpha>1/4$. Suppose that

[TABLE]

Then

[TABLE]

Proof.

Let $t^{*}=t^{*}(\alpha,X)$, $k:=\lfloor 4\alpha\rfloor$ and $L=\lfloor\frac{\sqrt{C_{3}}}{4\sqrt{e}}\rfloor$. We will choose $C_{3}$ in such a way that $L$ is large, in particular we may assume that $L\geq 2$. Observe also that $\alpha=\sum_{i=1}^{n}{\mathbb{P}}(|X_{i}|\geq t^{*}(\alpha,X))\leq nC_{3}^{-1}$, thus $Lk\leq C_{3}^{1/2}e^{-1/2}\alpha\leq e^{-1/2}C_{3}^{-1/2}n\leq n$ if $C_{3}\geq 1$. Hence

[TABLE]

Lemma 16 and the definition of $t^{*}(\alpha,X)$ yield

[TABLE]

This yields $t(Lk,X)\geq t^{*}(Lk,X)\geq\frac{t^{*}}{2}$ and by Theorem 3 we have

[TABLE]

Since for any norm ${\mathbb{P}}(\|X\|\leq t{\mathbb{E}}\|X\|)\leq Ct$ for $t>0$ (see [10, Corollary 1]) we have

[TABLE]

By the Paley-Zygmund inequality and (7), ${\mathbb{P}}(|X_{i}|\geq\frac{1}{2}{\mathbb{E}}|X_{i}|)\geq\frac{({\mathbb{E}}|X_{i}|)^{2}}{4{\mathbb{E}}|X_{i}|^{2}}>\frac{1}{C_{3}}$ if $C_{3}>4C_{1}^{2}$, so $\frac{1}{2}{\mathbb{E}}|X_{i}|\leq t^{*}$. Moreover it is easy to verify that $k=\lfloor 4\alpha\rfloor>\alpha$ for $\alpha>1/4$, thus $t^{*}(k,X)\leq t^{*}(\alpha,X)=t^{*}$. Hence Proposition 1 and Lemma 14 yield

[TABLE]

and therefore

[TABLE]

Estimates (17)-(19) yield

[TABLE]

so it is enough to choose $C_{3}$ in such a way that $L\geq 400/c_{2}$. ∎

Proof of the first part of Theorem 4.

Let $t^{*}=t^{*}(k-1/2,X)$ and $C_{3}$ be as in Proposition 18. It is enough to consider the case when $t^{*}>0$, then ${\mathbb{P}}(|X_{i}|=t^{*})=0$ for all $i$ and $\sum_{i=1}^{n}{\mathbb{P}}(|X_{i}|\geq t^{*})=k-1/2$. Define

[TABLE]

[TABLE]

If $\beta=0$ then $\alpha=k-1/2$, $|I_{1}|=[n]$, and the assertion immediately follows by Proposition 18 since $4\alpha\geq k$.

Otherwise define

[TABLE]

We have by Lemma 17 applied with $p=1/C_{3}$

[TABLE]

Thus

[TABLE]

Therefore

[TABLE]

If $\alpha<1/2$ then $\lceil\beta\rceil=k$ and the assertion easily follows. Otherwise Proposition 18 yields

[TABLE]

Observe that for $\alpha\geq 1/2$ we have $\lfloor 4\alpha\rfloor+\lceil\beta\rceil\geq 4\alpha-1+\beta\geq\alpha+1/2+\beta=k$, so

[TABLE]

Remark 19**.**

A modification of the proof above shows that under the assumptions of Theorem 4 for any $p<1$ there exists $c(p)>0$ such that

[TABLE]

6. Upper estimates for order statistics

We will need a few more facts concerning log-concave vectors.

Lemma 20**.**

Suppose that $X$ is a mean zero log-concave random vector with uncorrelated coordinates. Then for any $i\neq j$ and $s>0$,

[TABLE]

Proof.

Let $C_{7},c_{3}$ and $t_{0}$ be the constants from Lemma 15. If $s>t_{0}\|X_{i}\|_{2}$ then, by Lemma 15, ${\mathbb{P}}(|X_{i}|\leq s)\geq 2c_{3}t_{0}$ and the assertion is obvious (with any $C_{6}\geq(2c_{3}t_{0})^{-1}$). Thus we will assume that $s\leq t_{0}\min\{\|X_{i}\|_{2},\|X_{j}\|_{2}\}$.

Let $\widetilde{X}_{i}=X_{i}/\|X_{i}\|_{2}$ and let $g_{ij}$ be the density of $(\widetilde{X}_{i},\widetilde{X}_{j})$. By Lemma 15 we know that $\|g_{i,j}\|_{\infty}\leq C_{7}$, so

[TABLE]

On the other hand the second part of Lemma 15 yields

[TABLE]

Lemma 21**.**

Let $Y$ be a log-concave random variable. Then

[TABLE]

Proof.

We may assume that $Y$ is non-degenerate (otherwise the statement is obvious), in particular $Y$ has no atoms. Log-concavity of $Y$ yields

[TABLE]

Hence

[TABLE]

Since $-Y$ satisfies the same assumptions as $Y$, we also have

[TABLE]

Adding both estimates we get

[TABLE]

Lemma 22**.**

Suppose that $Y$ is a log-concave random variable and ${\mathbb{P}}(|Y|\leq t)\leq\frac{1}{10}$. Then ${\mathbb{P}}(|Y|\leq 21t)\geq 5{\mathbb{P}}(|Y|\leq t)$.

Proof.

Let ${\mathbb{P}}(|Y|\leq t)=p$ then by Lemma 21

[TABLE]

Let us now prove (4) and see how it implies the second part of Theorem 4. Then we give a proof of (5).

Proof of (4).

Fix $k$ and set $t^{*}:=t^{*}(k-1/2,X)$. Then $\sum_{i=1}^{n}{\mathbb{P}}(|X_{i}|\geq t^{*})=k-1/2$. Define

[TABLE]

Observe that for $u>3$ and $1\leq l\leq|I_{1}|$ we have by Lemma 21

[TABLE]

Consider two cases.

Case 1. $\beta>|I_{2}|-1/2$. Then $|I_{2}|<\beta+1/2\leq k$, so $k-|I_{2}|\geq 1$ and

[TABLE]

Therefore by (22)

[TABLE]

Case 2. $\beta\leq|I_{2}|-1/2$. Observe that for any disjoint sets $J_{1}$, $J_{2}$ and integers $l,m$ such that $l\leq|J_{1}|$, $m\leq|J_{2}|$ we have

[TABLE]

Since

[TABLE]

we have $\lceil\alpha\rceil+\lceil\beta\rceil\leq k+1$ and, by (23),

[TABLE]

Estimate (22) yields

[TABLE]

To estimate $\lceil\beta\rceil\text{-}\max_{i\in I_{2}}|X_{i}|=(|I_{2}|+1-\lceil\beta\rceil)\text{-}\min_{i\in I_{2}}|X_{i}|$ observe that by Lemma 22, the definition of $I_{2}$ and assumptions on $\beta$,

[TABLE]

Set $l:=(|I_{2}|+1-\lceil\beta\rceil)$ and

[TABLE]

Note that we know already that ${\mathbb{E}}\tilde{N}(21t^{*})\geq 2l$. Thus the Paley-Zygmund inequality implies

[TABLE]

However Lemma 20 yields

[TABLE]

Therefore

[TABLE]

for sufficiently large $u$. ∎

The unconditionality assumption plays a crucial role in the proof of the next lemma, which allows to derive the second part of Theorem 4 from estimate (4).

Lemma 23**.**

Let $X$ be an unconditional log-concave $n$-dimensional random vector. Then for any $1\leq k\leq n$,

[TABLE]

Proof.

Let $\nu$ be the law of $(|X_{1}|,\ldots,|X_{n}|)$. Then $\nu$ is log-concave on ${\mathbb{R}}_{n}^{+}$. Define for $t>0$,

[TABLE]

It is easy to check that $\frac{1}{u}A_{ut}+(1-\frac{1}{u}){\mathbb{R}}_{+}^{n}\subset A_{t}$, hence

[TABLE]

Proof of the second part of Theorem 4.

Estimate (4) together with Lemma 23 yields

[TABLE]

and the assertion follows by integration by parts. ∎

Proof of (5).

Define $I_{1}$, $I_{2}$, $\alpha$ and $\beta$ by (20) and (21), where this time $t^{*}=t^{*}(k-k^{5/6}/2,X)$. Estimate (22) is still valid so integration by parts yields

[TABLE]

Set

[TABLE]

Observe that

[TABLE]

Hence $\lceil\alpha\rceil+k_{\beta}\leq k+1$.

If $k_{\beta}>|I_{2}|$, then $k-|I_{2}|\geq\lceil\alpha\rceil+k_{\beta}-1-|I_{2}|\geq\lceil\alpha\rceil$, so

[TABLE]

Therefore it suffices to consider case $k_{\beta}\leq|I_{2}|$ only.

Since $\lceil\alpha\rceil+k_{\beta}-1\leq k$ and $k_{\beta}\leq|I_{2}|$, we have by (23),

[TABLE]

Since $\beta\leq k-\frac{1}{2}k^{5/6}$ and $x\rightarrow x-\frac{1}{2}x^{5/6}$ is increasing for $x\geq 1/2$ we have

[TABLE]

Therefore, considering $(X_{i})_{i\in I_{2}}$ instead of $X_{i}$ and $k_{\beta}$ instead of $k$ it is enough to show the following claim:

Let $s>0$, $n\geq k$ and let $X$ be an $n$-dimensional log-concave vector. Suppose that

[TABLE]

then

[TABLE]

We will show the claim by induction on $k$. For $k=1$ the statement is obvious (since the assumptions are contradictory). Suppose now that $k\geq 2$ and the assertion holds for $k-1$.

Case 1. ${\mathbb{P}}(|X_{i_{0}}|\geq s)\geq 1-\frac{5}{12}k^{-1/6}$ for some $1\leq i_{0}\leq n$. Then

[TABLE]

where to get the last inequality we used that $x^{5/6}$ is concave on ${\mathbb{R}}_{+}$, so $(1-t)^{5/6}\leq 1-\frac{5}{6}t$ for $t=1/k$. Therefore by the induction assumption applied to $(X_{i})_{i\neq i_{0}}$,

[TABLE]

Case 2. ${\mathbb{P}}(|X_{i}|\leq s)\geq\frac{5}{12}k^{-1/6}$ for all $i$. Applying Lemma 15 we get

[TABLE]

so $\max_{i}\|X_{i}\|_{2}\leq Ck^{1/6}s$. Moreover $n\leq\frac{10}{9}k$. Therefore by the result of Lee and Vempala [13] $X$ satisfies the exponential concentration with $\alpha\leq C_{9}k^{5/12}s$.

Let $l=\lceil k-\frac{1}{2}(k^{5/6}-1)\rceil$ then $s\geq t_{*}(l-1/2,X)$ and $k-l+1\geq\frac{1}{2}(k^{5/6}-1)\geq\frac{1}{9}k^{5/6}$. Let

[TABLE]

By (4) (applied with $l$ instead of $k$) we have ${\mathbb{P}}(X\in A)\geq c_{4}$. Observe that

[TABLE]

Therefore by Lemma 10 we get

[TABLE]

Integration by parts yields

[TABLE]

and the induction step is shown in this case provided that $C_{8}\geq C_{10}+3C_{9}(1-\ln c_{4})$. ∎

To obtain Corollary 6 we used the following lemma.

Lemma 24**.**

Assume that $X$ is a symmetric isotropic log-concave vector in ${\mathbb{R}}^{n}$. Then

[TABLE]

and

[TABLE]

Proof.

Observe that

[TABLE]

Thus Lemma 15 implies that for $p\geq c_{5}n$ (with $c_{5}\in(\frac{1}{2},1)$) we have $t^{*}(p,X)\sim\frac{n-p}{n}$. Moreover, by the Markov inequality

[TABLE]

so $t^{*}(n/4,X)\leq 4$. Since $p\mapsto t^{*}(p,X)$ is non-increasing, we know that $t^{*}(p,X)\sim 1$ for $n/4\leq p\leq c_{5}n$.

Now we will prove (25). We have

[TABLE]

so it suffices to show that $t^{*}(k,X)\geq ct(k,X)$. To this end we fix $k\leq n/2$. By (24) we know that $t:=C_{11}t^{*}(k,X)\geq C_{11}t^{*}(n/2,X)\geq e$, so the isotropicity of $X$ and Markov’s inequality yield ${\mathbb{P}}(|X_{i}|\geq t)\leq e^{-2}$ for all $i$. We may also assume that $t\geq t^{*}(k,X)$. Integration by parts and Lemma 21 yield

[TABLE]

Therefore

[TABLE]

so $t(k,X)\leq 4C_{11}t^{*}(k,X)$. ∎