The negative multinomial distribution appears in many areas of applications such as polarimetric image processing and the analysis of longitudinal count data. In previous studies, Mosimann (1963) derived general formulas for the falling factorial moments of the negative multinomial distribution, while Withers & Nadarajah (2014) obtained expressions for the cumulants. Despite the availability of the moment generating function, no comprehensive formulas for the moments have been calculated thus far. This paper addresses this gap by presenting general formulas for both central and non-central moments of the negative multinomial distribution. These formulas are expressed in terms of binomial coefficients and Stirling numbers of the second kind. Utilizing these formulas, we provide explicit expressions for all central moments up to the 4th order and all non-central moments up to the 8th order.
We present the first general formulas for the central and non-central moments of the multinomial distribution, using a combinatorial argument and the factorial moments previously obtained in Mosimann (1962). We use the formulas to give explicit expressions for all the non-central moments up to order 8 and all the central moments up to order 4. These results expand significantly on those in Newcomer (2008) and Newcomer et al. (2008), where the non-central moments were calculated up to order 4.
We give the first explicit formulas for the joint third and fourth central moments of the multinomial distribution, by differentiating the moment generating function. A general formula for the joint factorial moments was previously given in Mosimann (1962).
This article provides the domain of existence $Ω$ of the Laplace transform of infinitely divisible negative multinomial distributions. We define a negative multinomial distribution on $\mathbb{N}^{n},$ where $\mathbb{N}$ is the set of nonnegative integers, by its probability generating function which will be of the form $\left( A\left( a_{1}z_{1} ,\ldots,a_{n}z_{n}\right) /A\left( a_{1},\ldots,a_{n}\right) \right) ^{-λ}$ where $A\left( \mathbf{z}\right) = {\displaystyle\sum\limits_{T\subset\left\{ 1,2,\ldots,n\right\} }} a_{T}\prod\limits_{i\in T}z_{i},$ where $a_{\emptyset}\neq0,$ and where $…
We show that the moment generating function of the Kullback-Leibler divergence (relative entropy) between the empirical distribution of $n$ independent samples from a distribution $P$ over a finite alphabet of size $k$ (i.e. a multinomial distribution) and $P$ itself is no more than that of a gamma distribution with shape $k - 1$ and rate $n$. The resulting exponential concentration inequality becomes meaningful (less than 1) when the divergence $\varepsilon$ is larger than $(k-1)/n$, whereas the standard method of types bound requires $\varepsilon > \frac{1}{n} \cdot \log{\binom{n+k-1}{k-1}} …
Considering discrete models, the univariate framework has been studied in depth compared to the multivariate one. This paper first proposes two criteria to define a sensu stricto multivariate discrete distribution. It then introduces the class of splitting distributions that encompasses all usual multivariate discrete distributions (multinomial, negative multinomial, multivariate hypergeometric, multivariate neg- ative hypergeometric, etc . . . ) and contains several new. Many advantages derive from the compound aspect of split- ting distributions. It simplifies the study of their characteris-…
Let $μ$ be a probability measure (or corresponding random variable) such that all moments $μ_n$ exist. Knowledge of the moments is not sufficient to determine infinite divisibility of the measure; we show also that infinitely divisible, and in particular lognormal, distributions lose infinitely divisibilty when censored in certain ways even if all moments are arbitrarily close to those of the uncensored distribution. The moments of a composition of k copies of $μ$ are expressed as combinatorial compositions of the $μ_n$. We express the moments of the compositions in the context of occupancy pr…
This note examines linear combinations of multi-indexed sequences and derives the multivariate generating function of such a linear combination in terms of the original sequence's m.g.f. Applications include finding distributions and moments of non-negative discrete random variables conditioned on non-negative linear combinations of the original variables. Examples include independent Poisson r.v.'s and a $d$-variate multinomial distribution.
Non-asymptotic approximations for Pearson's chi-square statistic and its application to confidence intervals for strictly convex functions of the probability weights of discrete distributions
An $(n,k)$-Poisson Multinomial Distribution (PMD) is the distribution of the sum of $n$ independent random vectors supported on the set ${\cal B}_k=\{e_1,\ldots,e_k\}$ of standard basis vectors in $\mathbb{R}^k$. We show that any $(n,k)$-PMD is ${\rm poly}\left({k\over σ}\right)$-close in total variation distance to the (appropriately discretized) multi-dimensional Gaussian with the same first two moments, removing the dependence on $n$ from the Central Limit Theorem of Valiant and Valiant. Interestingly, our CLT is obtained by bootstrapping the Valiant-Valiant CLT itself through the structura…
In the paper we develop an approach to asymptotic normality through factorial cumulants. Factorial cumulants arise in the same manner from factorial moments, as do (ordinary) cumulants from (ordinary) moments. Another tool we exploit is a new identity for "moments" of partitions of numbers. The general limiting result is then used to (re-)derive asymptotic normality for several models including classical discrete distributions, occupancy problems in some generalized allocation schemes and two models related to negative multinomial distribution.
A novel multinomial theorem for commutative idempotents is shown to lead to new results about the moments, central moments, factorial moments, and their generating functions for any random variable $X = \sum_{i} Y_i $ expressible as a sum of Bernoulli indicator random variables $Y_i$. The resulting expressions are functions of the expectation of products of the Bernoulli indicator random variables. These results are used to derive novel expressions for the various moments in a large number of familiar examples and classic problems including: the binomial, hypergeometric, Poisson limit of the b…
The Mahonian statistic is the number of inversions in a permutation of a multiset with $a_i$ elements of type $i$, $1\le i\le m$. The counting function for this statistic is the $q$ analog of the multinomial coefficient $\binom{a_1+...+a_m}{a_1,... a_m}$, and the probability generating function is the normalization of the latter. We give two proofs that the distribution is asymptotically normal. The first is {\it computer-assisted}, based on the method of moments. The Maple package {\tt MahonianStat}, available from the webpage of this article, can be used by the reader to perform experiments …
We establish a sharp moment comparison inequality between an arbitrary negative moment and the second moment for sums of independent uniform random variables, which extends Ball's cube slicing inequality.
The existence of moments of first downward passage times of a spectrally negative Lévy process is governed by the general dynamics of the Lévy process, i.e. whether the Lévy process is drifting to $+\infty$, $-\infty$ or oscillates. Whenever the Lévy process drifts to $+\infty$, we prove that the $κ$-th moment of the first passage time (conditioned to be finite) exists if and only if the $(κ+1)$-th moment of the Lévy jump measure exists. This generalises a result shown earlier by Delbaen for Cramér-Lundberg risk processes \cite{Delbaen1990}. Whenever the Lévy process drifts to $-\infty$, we pr…
The objective of this note is to study the probability that the total mass of a sub-critical Gaussian multiplicative chaos (GMC) with arbitrary base measure $σ$ is small. When $σ$ has some continuous density w.r.t Lebesgue measure, a scaling argument shows that the logarithm of the total GMC mass is sub-Gaussian near $-\infty$. However, when $σ$ has no scaling properties, the situation is much less clear. In this paper, we prove that for any base measure $σ$, the total GMC mass has negative moments of all orders.
We consider Markov processes in continuous time with state space $\posint^N$ and provide two sufficient conditions and one necessary condition for the existence of moments $E(\|X(t)\|^r)$ of all orders $r \in \nat$ for all $t \geq 0$. The sufficient conditions also guarantee an exponential in time growth bound for the moments. The class of processes studied have finitely many state independent jumpsize vectors $ν_1,\dots,ν_M$. This class of processes arise naturally in many applications such as stochastic models of chemical kinetics, population dynamics and queueing theory for example. We also…
Suppose a sequence of random variables {X_n} has negative drift when above a certain threshold and has increments bounded in L^p. When p>2 this implies that EX_n is bounded above by a constant independent of n and the particular sequence {X_n}. When p=<2 there are counterexamples showing this does not hold. In general, increments bounded in L^p lead to a uniform L^r bound on X_n^+ for any r<p-1, but not for r>=p-1. These results are motivated by questions about stability of queueing networks.
Very recently, Junk [11] showed that for directed polymers in bounded random environments, the weak disorder (uniform integrable) phase implies that the polymer martingale is bounded in $L^p$ for some $p>1$ and also in $L^q$ for some $q<0$. Here, we establish this characterization of the weak disorder phase without requiring the boundedness assumption on the environments.
