Preservation of log-concavity on summation

In this paper, we establish some general forms of the law of the iterated logarithm for independent random variables in a sub-linear expectation space, where the random variables are not necessarily identically distributed. Exponential inequalities for the maximum sum of independent random variables and Kolmogorov's converse exponential inequalities are established as tools for showing the law of the iterated logarithm. As an application, the sufficient and necessary conditio...

We prove limit theorems for sums of randomly chosen random variables conditioned on the summands. We consider several versions of the corner growth setting, including specific cases of dependence amongst the summands and summands with heavy tails. We also prove a version of Hoeffding's combinatorial central limit theorem and results for summands taken uniformly from a random sample. These results are proved with concentration of measure techniques.

We expand on the remark by Andrews on the importance of infinite sums and products in combinatorics. Let $\{g_d(n)\}_{d\geq 0,n \geq 1}$ be the double sequences $\sigma_d(n)= \sum_{\ell \mid n} \ell^d$ or $\psi_d(n)= n^d$. We associate double sequences $\left\{ p^{g_{d} }\left( n\right) \right\}$ and $\left\{ q^{g_{d} }\left( n\right) \right\} $, defined as the coefficients of \begin{eqnarray*} \sum_{n=0}^{\infty} p^{g_{d} }\left( n\right) \, t^{n} & := & \prod_{n=1}^{\infty}...

We review and formulate results concerning log-concavity and strong-log-concavity in both discrete and continuous settings. We show how preservation of log-concavity and strongly log-concavity on $\mathbb{R}$ under convolution follows from a fundamental monotonicity result of Efron (1969). We provide a new proof of Efron's theorem using the recent asymmetric Brascamp-Lieb inequality due to Otto and Menz (2013). Along the way we review connections between log-concavity and oth...

Let $\mathcal{A}=\left(a_i\right)_{i=1}^\infty$ be a weakly increasing sequence of positive integers and let $k$ be a fixed positive integer. For an arbitrary integer $n$, the restricted partition $p_\mathcal{A}(n,k)$ enumerates all the partitions of $n$ whose parts belong to the multiset $\{a_1,a_2,\ldots,a_k\}$. In this paper we investigate some generalizations of the log-concavity of $p_\mathcal{A}(n,k)$. We deal with both some basic extensions like, for instance, the stro...

Every symmetric polynomial $h(x) = h_0 + h_1\,x + \cdots + h_n\,x^n$, where $h_i = h_{n-i}$ for each $i$, can be expressed as a linear combination in the basis $\{x^i(1+x)^{n-2i}\}_{i=0}^{\lfloor n/2\rfloor}$. The polynomial $\gamma_h(x) = \gamma_0 + \gamma_1 \,x+ \cdots + \gamma_{\lfloor n/2\rfloor}\, x^{\lfloor n/2\rfloor}$, commonly referred to as the $\gamma$-polynomial of $h(x)$, records the coefficients of the aforementioned linear combination. Two decades ago, Br\"and\...

We formulate conditions on a set of log-concave sequences, under which any linear combination of those sequences is log-concave, and further, of conditions under which linear combinations of log-concave sequences that have been transformed by convolution are log-concave. These conditions involve relations on sequences called \textit{synchronicity} and \textit{ratio-dominance}, and a characterization of some bivariate sequences as \textit{lexicographic}. We are motivated by th...

Euler's gamma function is logarithmically convex on positive semi-axis. Additivity of logarithmic convexity implies that the function sum of gammas with non-negative coefficients is also log-convex. In this paper we investigate the series in reciprocal gamma functions, where each term is clearly log-concave. Log-concavity is not preserved by addition, so that non-negativity of the coefficients is now insufficient to draw any conclusions about the sum. We demonstrate that the ...

We prove a lemma, which we call the Order Ideal Lemma, that can be used to demonstrate a wide array of log-concavity and log-convexity results in a combinatorial manner using order ideals in distributive lattices. We use the Order Ideal Lemma to prove log-concavity and log-convexity of various sequences involving lattice paths (Catalan, Motzkin and large Schr\"oder numbers), intervals in Young's lattice, order polynomials, specializations of Schur and Schur Q-functions, Lucas...

The contents are divided into two papers "The Monotone Cumulants" (arXiv:0907.4896) and "Conditionally monotone independence" (arXiv:0907.5473).