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\title{Example based on my paper about rank sequences} 
\author{Klas Markstr\"om\thanks{Department of Mathematics and Mathematical Statistics, 
Ume\aa\  University , SE-901 87  Ume\aa, Sweden} }

\maketitle

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\begin{abstract}
    We present a minimum counterexample to the conjecture that a
    negatively associated random variable has an ultra-logconcave
    rank  sequence. The rank sequence does in fact not
    even need to be unimodal.
\end{abstract}

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\section{Introduction}
Random variables with positive association are so well known that
their usefulness hardly needs promotion. The properties of 
random variables which are negatively associated, such as Fermi-Dirac 
statistics and the uniform spanning tree of a graph, are on the other
far less understood, but potentially highly useful too. In \cite{pem}
Pemantle started an exploration of negatively associated random
variables and gave a number of results and conjectures about their 
structure.  Our aim here is to prove that one of these  conjectures is 
false. After the first version of this note appeared the author was
also informed that counterexamples to several of these conjectures has been
found independently by Borcea, Br\"and\'en and Ligett, and are available 
in the preprint \cite{BBL}.


Let us recall some of the relevant definitions. We let $B_{n}$ denote
the boolean lattices of subsets of $1,\ldots,n$.
\begin{definition}
    A probability measure $\mu$ on $B_{n}$ is negatively associated if every pair
    of increasing functions $f:A\rightarrow\mathbb{R}^{+}$, 
    $g:(B_{n}\setminus A)\rightarrow\mathbb{R}^{+}$ satisfies
\begin{equation} \label{NA}
    C(f,g):=  \int fg \, d\mu - \int f \, d\mu \int g \, d\mu \leq 0
\end{equation}
Likewise we say that $\mu$ is strictly negatively associated if we can
take strict inequality above.
\end{definition}
Since any increasing function can be written as a linear combination of indicator
functions of upsets we see that it suffices for condition \ref{NA} to hold for
all such pairs  of indicator functions.
\begin{definition}
    The rank sequence of a measure $\mu$ on $B_{n}$ is the sequence
    $$a_{k}=\mu(|X|=k),\ \ k=0\ldots n,$$
    i.e. $a_{k}$ is the probability that $X$ has size $k$.
\end{definition}
Recall that a positive sequence is log-concave if $a_{i}^{2}\geq
a_{i-1}a_{i+1}$. In \cite{pem} a sequence $a_{0},\ldots a_{n}$ is defined to be
\emph{ultra-log-concave} (ULC) if the sequence $\frac{a_{k}}{{n \choose
k}}$ is log-concave.


In \cite{pem} Pemantle proved that an exchangeable measure on
$B_{n}$ is conditionally negatively associated, meaning that all
measures obtained by conditioning on the value of one variable are
negatively associated, if and only if it's rank sequence is
ULC and gave several conjectures on the connection between negative
association and ULC rank sequences. One of these conjecture was
verified in \cite{lig} where Ligett proved that the convolution of
two ULC sequences is also ULC.

The conjecture which concerns us here is
\begin{conjecture}[4\cite{pem}]\label{co}
    If $\mu$ is negatively associated and the set of indices for which
    the rank sequence is non-zero forms an interval then the rank
    sequence is ULC. 
\end{conjecture}
It is easy to check that the conjecture holds for measures defined on $B_{2}$, but we will now 
give examples which show that this conjecture is false on $B_{3}$.

\begin{example}
    Our first example is a measure $\mu_{1}$ on $B_{3}$ which is
    negatively associated and has $a_{1}=a_{4}=0$ while $a_{2}$ and
    $a_{3}$ are non-zero. Here the non-zero elements of the rank sequence 
    does not form an internal and so does  not contradict Pemantle's
    conjecture.
    
    Our measure is defined as :
    $$\mu_{1}\left(\emptyset\right)=\frac{6}{32}$$
    $$\mu_{1}\left(\{1,2,3\}\right)=\mu_{1}\left(\{1\}\right)=
    \mu_{1}\left(\{2\}\right)=\mu_{1}\left(\{3\}\right)=0$$
     
    $$\mu_{1}\left(\{1,2\}\right)=\mu_{1}\left(\{1,3\}\right)=\frac{9}{32}, 
    \   \mu_{1}\left(\{2,3\}\right)=\frac{8}{32}$$ 
    Thanks to the symmetries of this measure it is easy to check that
    the measure is strictly negatively associated.
\end{example}
This example would have been a counterexample to Conjecture \ref{co}
had it not been for the caveat that the non-zero elements of the rank sequence  
must form an interval. However since the example is strictly
negatively associated this can easily be circumvented. 
\begin{theorem}
    If $\mu_{1}$ is strictly negatively associated then there exists a
    $\delta>0$ such that if $\sup_{A\in
    B_{n}}|\mu_{1}(A)-\mu_{2}(A)|<\delta$ 
    then $\mu_{2}$ is strictly negatively associated.
\end{theorem}
\begin{proof}
    We immediately see that each $C(f,g)$ is changes continuously in
    $\mu$, in fact they are polynomials in the $\mu(X_{i})$, for $X_{i}\in
    B_{n}$. As noted above we only need to check that $C(f,g)$ is negative
    for the  set of pairs of indicator functions of upsets and since this
    set of pairs is finite all $C(f,g)$ will remain negative for small enough
    $\delta$.
\end{proof}
By this theorem we can disturb our measure $\mu_{1}$ to get a measure
with only non-zero elements in the rank sequence, while still being
negatively associated. An explicit such example is the following
measure. 
\begin{example}
    $$\mu_{2}\left(\emptyset\right)=\frac{6}{32}$$
    $$\mu_{2}\left(\{1,2,3\}\right)=\mu_{2}\left(\{1\}\right)=\mu_{2}\left(\{2\}\right)=
	 \mu_{2}\left(\{3\}\right)=\frac{1}{32}$$ 
	 $$\mu_{2}\left(\{1,2\}\right)=\mu_{2}\left(\{1,3\}\right)=\frac{7}{32},\	 
    \mu_{2}\left(\{2,3\}\right)=\frac{8}{32}$$
    Again it is easy to explicitly  verify that this measure
    negatively associated.
\end{example}

The caveat that the rank sequence should form an interval was
introduced in \cite{pem} to get rid of trivial examples, but since the set of
strictly negatively associated measures forms an open set this does
not give the wanted consequences.


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\begin{thebibliography}{BBL07}

\bibitem[BBL07]{BBL}
J.~Borcea, P.~Br\"and\'en, and T.~M. Liggett.
\newblock Negative dependence and the geometry of polynomials, 2007.
\newblock http://www.arxiv.org/abs/0707.2340.

\bibitem[Lig97]{lig}
Thomas~M. Liggett.
\newblock Ultra logconcave sequences and negative dependence.
\newblock {\em J. Combin. Theory Ser. A}, 79(2):315--325, 1997.

\bibitem[Pem00]{pem}
Robin Pemantle.
\newblock Towards a theory of negative dependence.
\newblock {\em J. Math. Phys.}, 41(3):1371--1390, 2000.
\newblock Probabilistic techniques in equilibrium and nonequilibrium
  statistical physics.

\end{thebibliography}

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