We develop explicit bounds for the tail of the distribution of the all-time supremum of a random walk with negative drift, where the increments have a truncated heavy-tailed distribution. As an application, we consider a ruin problem in the presence of reinsurance.

The payoff in the Chow–Robbins coin-tossing game is the proportion of heads when you stop. Stopping to maximize expectation was addressed by Chow and Robbins (1965), who proved there exist integers ${k_n}$ such that it is optimal to stop at n tosses when heads minus tails is ${k_n}$. Finding ${k_n}$ was unsolved except for finitely many cases by computer. We prove an $o(n^{-1/4})$ estimate of the stopping boundary of Dvoretsky (1967), which then proves ${k_n} = \left\lceil {\alpha \sqrt n \,\, - 1/2\,\, + \,\,\frac{{\left( { - 2\zeta (\! -1/2)} \right)\sqrt \alpha }}{{\sqrt \pi }}{n^{ - 1/4}}} \right\rceil $ except for n in a set of density asymptotic to 0, at a power law rate. Here, $\alpha$ is the Shepp–Walker constant from the Brownian motion analog, and $\zeta$ is Riemann’s zeta function. An $n^{ - 1/4}$ dependence was conjectured by Christensen and Fischer (2022). Our proof uses moments involving Catalan and Shapiro Catalan triangle numbers which appear in a tree resulting from backward induction, and a generalized backward induction principle. It was motivated by an idea of Häggström and Wästlund (2013) to use backward induction of upper and lower Value bounds from a horizon, which they used numerically to settle a few cases. Christensen and Fischer, with much better bounds, settled many more cases. We use Skorohod’s embedding to get simple upper and lower bounds from the Brownian analog; our upper bound is the one found by Christensen and Fischer in another way. We use them first for yet many more examples and a conjecture, then algebraically in the tree, with feedback to get much sharper Value bounds near the border, and analytic results. Also, we give a formula that gives the exact optimal stop rule for all n up to about a third of a billion; it uses the analytic result plus terms arrived at empirically.

We investigate the tail behavior of the first-passage time for Sinai’s random walk in a random environment. Our method relies on the connection between Sinai’s walk and branching processes with immigration in a random environment, and the analysis on some important quantities of these branching processes such as extinction time, maximum population, and total population.

In this paper, we study random walks on groups that contain superlinear-divergent geodesics, in the line of thoughts of Goldsborough and Sisto. The existence of a superlinear-divergent geodesic is a quasi-isometry invariant which allows us to execute Gouëzel’s pivoting technique. We develop the theory of superlinear divergence and establish a central limit theorem for random walks on these groups.

We prove that the local time of random walks conditioned to stay positive converges to the corresponding local time of three-dimensional Bessel processes by proper scaling. Our proof is based on Tanaka’s pathwise construction for conditioned random walks and the derivation of asymptotics for mixed moments of the local time.

In water-rich smectite gels, bound or less mobile H2O layers exist near negatively-charged clay platelets. These bound H2O layers are obstacles to the diffusion of unbound H2O molecules in the porespace, and therefore reduce the H2O self-diffusion coefficient, D, in the gel system as a whole. In this study, the self-diffusion coefficients of H2O molecules in water-rich gels of Na-rich smectites (montmorillonite, stevensite and hectorite) were measured by pulsed-gradient spin-echo proton nuclear magnetic resonance (NMR) to evaluate the effects of obstruction on D. The NMR results were interpreted using random-walk computer simulations which show that unbound H2O diffuses in the gels while avoiding randomly-placed obstacles (clay platelets sandwiched in immobilized bound H2O layers). A ratio (volume of the clay platelets and immobilized H2O layers)/(volume of clay platelets) was estimated for each water-rich gel. The results showed that the ratio was 8.92, 16.9, 3.32, 3.73 and 3.92 for Wyoming montmorillonite (⩽ 5.74 wt.% clay), Tsukinuno montmorillonite (⩽ 3.73 wt.% clay), synthetic stevensite (⩽ 8.97 wt.% clay), and two synthetic hectorite samples (⩽ 11.0 wt.% clay), respectively. The ratios suggest that the thickness of the immobilized H2O layers in the gels is 4.0, 8.0, 1.2, 1.4 and 1.5 nm, respectively, assuming that each clay particle in the gels consists of a single 1 nm-thick platelet. The present study confirmed that the obstruction effects of immobilized H2O layers near the clay surfaces are important in restricting the self-diffusion of unbound H2O in water-rich smectite gels.

We investigate the translation lengths of group elements that arise in random walks on the isometry groups of Gromov hyperbolic spaces. In particular, without any moment condition, we prove that non-elementary random walks exhibit at least linear growth of translation lengths. As a corollary, almost every random walk on mapping class groups eventually becomes pseudo-Anosov, and almost every random walk on $\mathrm {Out}(F_n)$ eventually becomes fully irreducible. If the underlying measure further has finite first moment, then the growth rate of translation lengths is equal to the drift, the escape rate of the random walk.

We then apply our technique to investigate the random walks induced by the action of mapping class groups on Teichmüller spaces. In particular, we prove the spectral theorem under finite first moment condition, generalizing a result of Dahmani and Horbez.

We consider an SIR (susceptible $\to$ infective $\to$ recovered) epidemic in a closed population of size n, in which infection spreads via mixing events, comprising individuals chosen uniformly at random from the population, which occur at the points of a Poisson process. This contrasts sharply with most epidemic models, in which infection is spread purely by pairwise interaction. A sequence of epidemic processes, indexed by n, and an approximating branching process are constructed on a common probability space via embedded random walks. We show that under suitable conditions the process of infectives in the epidemic process converges almost surely to the branching process. This leads to a threshold theorem for the epidemic process, where a major outbreak is defined as one that infects at least $\log n$ individuals. We show further that there exists $\delta \gt 0$, depending on the model parameters, such that the probability that a major outbreak has size at least $\delta n$ tends to one as $n \to \infty$.

Higher-order networks aim at improving the classical network representation of trajectories data as memory-less order $1$ Markov models. To do so, locations are associated with different representations or “memory nodes” representing indirect dependencies between visited places as direct relations. One promising area of investigation in this context is variable-order network models as it was suggested by Xu et al. that random walk-based mining tools can be directly applied on such networks. In this paper, we focus on clustering algorithms and show that doing so leads to biases due to the number of nodes representing each location. To address them, we introduce a representation aggregation algorithm that produces smaller yet still accurate network models of the input sequences. We empirically compare the clustering found with multiple network representations of real-world mobility datasets. As our model is limited to a maximum order of $2$, we discuss further generalizations of our method to higher orders.

In this article we introduce a simple tool to derive polynomial upper bounds for the probability of observing unusually large maximal components in some models of random graphs when considered at criticality. Specifically, we apply our method to a model of a random intersection graph, a random graph obtained through p-bond percolation on a general d-regular graph, and a model of an inhomogeneous random graph.