50 research outputs found

### A Berry-Esseen bound with applications to vertex degree counts in the Erd\H{o}s-R\'{e}nyi random graph

Applying Stein's method, an inductive technique and size bias coupling yields
a Berry-Esseen theorem for normal approximation without the usual restriction
that the coupling be bounded. The theorem is applied to counting the number of
vertices in the Erdos-Renyi random graph of a given degree.Comment: Published in at http://dx.doi.org/10.1214/12-AAP848 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org

### Moderate deviations in Poisson approximation: a first attempt

Poisson approximation using Stein's method has been extensively studied in
the literature. The main focus has been on bounding the total variation
distance. This paper is a first attempt on moderate deviations in Poisson
approximation for right-tail probabilities of sums of dependent indicators. We
obtain results under certain general conditions for local dependence as well as
for size-bias coupling. These results are then applied to independent
indicators, 2-runs, and the matching problem.Comment: 21 page

### Poisson convergence in stochastic geometry via generalized size-bias coupling

This dissertation aims to investigate several aspects of the Poisson convergence: Poisson approximation, multivariate Poisson approximation, Poisson process approximation and weak convergence to a Poisson process.
The size-bias coupling is a powerful tool that, when combined with the Chen-Stein method, leads to many general results on Poisson approximation. We define an approximate size-bias coupling for integer-valued random variables by introducing error terms, and we combine it with the Chen-Stein method to compare the distributions of integer-valued random variables and Poisson random variables. In particular, we provide explicit bounds on the pointwise difference between the cumulative distribution functions. By these findings, we show approximation results in the Kolmogorov distance for minimal circumscribed radii and maximal inradii of stationary Poisson-Voronoi tessellations. Moreover, we compare the distributions of Poisson random variables and U-statistics with underlying Poisson processes or binomial point processes, which, in particular, allows us to approximate the rescaled minimum Euclidean distance between pairs of points of a Poisson process with midpoint in an observation window by an exponentially distributed random variable using the Kolmogorov distance.
A multivariate version of the size-bias coupling is employed to investigate the Gaussian approximation for random vectors by L. Goldstein and Y. Rinott. We extend the notion of approximate size-bias coupling for random variables to random vectors, and we combine it with the Chen-Stein method to investigate the multivariate Poisson approximation in the Wasserstein distance and the Poisson process approximation in a new metric defined herein. As an application, we obtain a bound on the Wasserstein distance between the sum of m-dependent Bernoulli random vectors and a Poisson random vector. Moreover, we consider point processes of U-statistic structure, that is, point processes that, once evaluated on a measurable set, become U-statistics. For point processes of U-statistic structure with an underlying Poisson process, we establish a Poisson process approximation result that is the analogue of the one shown by L. Decreusefond, M. Schulte, and C. Thäle with the Kantorovich-Rubinstein distance replaced by our new metric.
General criteria for the weak convergence of locally finite point processes to a Poisson process are derived from the relation between probabilities of two consecutive values of a Poisson random variable. P. Calka and N. Chenavier studied the limiting behavior of characteristic radii of homogeneous Poisson-Voronoi tessellations. By our general results, we extend and improve their findings by showing Poisson process convergence for point processes constructed using inradii and circumscribed radii of inhomogeneous Poisson-Voronoi tessellations

### A Berry-Esseen bound for the uniform multinomial occupancy model

The inductive size bias coupling technique and Stein's method yield a
Berry-Esseen theorem for the number of urns having occupancy $d \ge 2$ when $n$
balls are uniformly distributed over $m$ urns. In particular, there exists a
constant $C$ depending only on $d$ such that \sup_{z \in
\mathbb{R}}|P(W_{n,m} \le z) -P(Z \le z)| \le C \left(
\frac{1+(\frac{n}{m})^3}{\sigma_{n,m}} \right) \quad \mbox{for all $n \ge d$
and $m \ge 2$,} where $W_{n,m}$ and $\sigma_{n,m}^2$ are the standardized
count and variance, respectively, of the number of urns with $d$ balls, and $Z$
is a standard normal random variable. Asymptotically, the bound is optimal up
to constants if $n$ and $m$ tend to infinity together in a way such that $n/m$
stays bounded.Comment: Typo corrected in abstrac

### Central limit theorem for crossings in randomly embedded graphs

We consider the number of crossings in a random embedding of a graph, $G$,
with vertices in convex position. We give explicit formulas for the mean and
variance of the number of crossings as a function of various subgraph counts of
$G$. Using Stein's method and size-bias coupling, we prove an upper bound on
the Kolmogorov distance between the distribution of the number of crossings and
a standard normal random variable. As an application, we establish central
limit theorems, along with convergence rates, for the number of crossings in
random matchings, path graphs, cycle graphs, and the disjoint union of
triangles.Comment: 18 pages, 5 figures. This is a merger of arXiv:2104.01134 and
arXiv:2205.0399

### Quantitative Small Subgraph Conditioning

We revisit the method of small subgraph conditioning, used to establish that
random regular graphs are Hamiltonian a.a.s. We refine this method using new
technical machinery for random $d$-regular graphs on $n$ vertices that hold not
just asymptotically, but for any values of $d$ and $n$. This lets us estimate
how quickly the probability of containing a Hamiltonian cycle converges to 1,
and it produces quantitative contiguity results between different models of
random regular graphs. These results hold with $d$ held fixed or growing to
infinity with $n$. As additional applications, we establish the distributional
convergence of the number of Hamiltonian cycles when $d$ grows slowly to
infinity, and we prove that the number of Hamiltonian cycles can be
approximately computed from the graph's eigenvalues for almost all regular
graphs.Comment: 59 pages, 5 figures; minor changes for clarit