4 research outputs found
Optimal Probability Inequalities for Random Walks related to Problems in Extremal Combinatorics
Let S_n=X_1+...+X_n be a sum of independent symmetric random variables such
that |X_{i}|\leq 1. Denote by W_n=\epsilon_{1}+...+\epsilon_{n} a sum of
independent random variables such that \prob{\eps_i = \pm 1} = 1/2. We prove
that \mathbb{P}{S_{n} \in A} \leq \mathbb{P}{cW_k \in A}, where A is either an
interval of the form [x, \infty) or just a single point. The inequality is
exact and the optimal values of c and k are given explicitly. It improves
Kwapie\'n's inequality in the case of the Rademacher series. We also provide a
new and very short proof of the Littlewood-Offord problem without using
Sperner's Theorem. Finally, an extension to odd Lipschitz functions is given.Comment: 10 pages, submitte
Optimal Littlewood-Offord inequalities in groups
We prove several Littlewood-Offord type inequalities for arbitrary groups. In
groups having elements of finite order the worst case scenario is provided by
the simple random walk on a certain cyclic subgroup. The inequalities we obtain
are optimal if the underlying group contains an element of certain order. It
turns out that for torsion-free groups Erd\H{o}s's bound still holds. Our
results strengthen and generalize some very recent results by Tiep and Vu
A non-uniform Littlewood-Offord inequality
Consider a sum , where
are non-zero vectors in and
are independent Rademacher random variables (i.e.,
). The classical Littlewood-Offord
problem asks for the best possible upper bound for . In this paper we consider a non-uniform version of this problem. Namely,
we obtain the optimal bound for in terms of the length of
the vector
Linear combinations of Rademacher random variables
For a fixed unit vector , we consider the
sign vectors
and the corresponding scalar products . In this paper we will solve for an old
conjecture stating that of the sums of the form it is
impossible that there are more with than there are
with . Although the problem has been solved
completely in case the 's are equal, the more general problem with
possible non-equal 's remains open for values of . The present
method can also be used for , but unfortunately the technical
difficulties seem to grow exponentially with and no "induction type of
argument" has been found. The conjecture has an appealing reformulation in
probability theory and in geometry. In probability theory the results lead to
upper bounds which are much better than for instance Chebyshevnequalities.Comment: 15 page