Related papers: A Sharp Tail Bound for the Expander Random Sampler
We give explicit bounds for the tail probabilities for sums of independent geometric or exponential variables, possibly with different parameters.
A graph whose edges only appear at certain points in time is called a temporal graph (among other names). Such a graph is temporally connected if each ordered pair of vertices is connected by a path which traverses edges in chronological…
We study the large deviation probabilities of infinite weighted sums of independent random variables that have stretched exponential tails. This generalizes Kiesel and Stadtm\"uller (2000), who study the same objects under the assumption of…
If the edges of the complete graph $K_n$ are totally ordered, a simple path whose edges are in ascending order is called increasing. The worst-case length of the longest increasing path has remained an open problem for several decades, with…
We develop tail estimates for the number of edges in a Chung-Lu random graph with regularly varying weight distribution. Our results show that the most likely way to have an unusually large number of edges is through the presence of one or…
We consider sums of $n$ i.i.d. random variables with tails close to $\exp\{-x^\beta\}$ for some $\beta>1$. Asymptotics developed by Rootz\'en (1987) and Balkema, Kl\"uppelberg & Resnick (1993) are discussed from the point of view of tails…
We establish and generalise several bounds for various random walk quantities including the mixing time and the maximum hitting time. Unlike previous analyses, our derivations are based on rather intuitive notions of local expansion…
Let $X_{1},\ldots ,X_{n}$ be $n$ real-valued dependent random variables. With motivation from Mitra and Resnick (2009), we derive the tail asymptotic expansion for the weighted sum of order statistics $X_{1:n}\leq \cdots \leq X_{n:n}$ of…
We prove a large deviation principle for the sum of n independent heavy-tailed random variables, which are subject to a moving cut-off boundary at location n. Conditional on the sum being large at scale n, we show that a finite number of…
We establish scaling limits for the random walk whose state space is the range of a simple random walk on the four-dimensional integer lattice. These concern the asymptotic behaviour of the graph distance from the origin and the spatial…
This paper explores a conditional Gibbs theorem for a random walkinduced by i.i.d. (X_{1},..,X_{n}) conditioned on an extreme deviation of its sum (S_{1}^{n}=na_{n}) or (S_{1}^{n}>na_{n}) where a_{n}\rightarrow\infty. It is proved that when…
We provide Monte Carlo estimates of the scaling of the length $L_{n}$ of the longest increasing subsequences of $n$-steps random walks for several different distributions of step lengths, short and heavy-tailed. Our simulations indicate…
We propose a variational tail bound for norms of random vectors under moment assumptions on their one-dimensional marginals. A simplified version of the bound that parametrizes the ``aggregating distribution'' using a certain pushforward of…
The task for a general and useful classification of the tail behaviors of probability distributions still has no satisfactory solution. Due to lack of information outside the range of the data the tails of the distribution should be…
We study the distribution of the maximum $M$ of a random walk whose increments have a distribution with negative mean and belonging, for some $\gamma>0$, to a subclass of the class $\mathcal{S}_\gamma$--see, for example, Chover, Ney, and…
Benjamini and Schramm introduced the notion of distributional limit of a sequence of graphs with uniformly bounded valence and studied such limits in the case that the involved graphs are planar. We investigate distributional limits of…
The extremal tail probabilities of moving sums in a marked Poisson random field is examined here. These sums are computed by adding up the weighted occurrences of events lying within a scanning set of fixed shape and size. Change of measure…
We study a discrete time self interacting random process on graphs, which we call Greedy Random Walk. The walker is located initially at some vertex. As time evolves, each vertex maintains the set of adjacent edges touching it that have not…
Consider the following computational problem: given a regular digraph $G=(V,E)$, two vertices $u,v \in V$, and a walk length $t\in \mathbb{N}$, estimate the probability that a random walk of length $t$ from $u$ ends at $v$ to within $\pm…
We consider a class of growing random graphs obtained by creating vertices sequentially one by one: at each step, we choose uniformly the neighbours of the newly created vertex; its degree is a random variable with a fixed but arbitrary…