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.

Consider nested subdivisions of a bounded real set into intervals defining the digits $X_1,X_2,\ldots$ of a random variable X with a probability density function f. If f is almost everywhere lower semi-continuous, there is a non-negative integer-valued random variable N such that the distribution of $R=(X_{N+1},X_{N+2},\ldots)$ conditioned on $S=(X_1,\ldots,X_N)$ does not depend on f. If also the lengths of the intervals exhibit a Markovian structure, $R\mid S$ becomes a Markov chain of a certain order $s\ge0$. If $s=0$ then $X_{N+1},X_{N+2},\ldots$ are independent and identically distributed with a known distribution. When $s>0$ and the Markov chain is uniformly geometric ergodic, there is a random time M such that the chain after time $\max\{N,s\}+M-s$ is stationary and M follows a simple known distribution.

We introduce the exponentially preferential recursive tree and study some properties related to the degree profile of nodes in the tree. The definition of the tree involves a radix $a\gt 0$. In a tree of size $n$ (nodes), the nodes are labeled with the numbers $1,2, \ldots ,n$. The node labeled $i$ attracts the future entrant $n+1$ with probability proportional to $a^i$.

We dedicate an early section for algorithms to generate and visualize the trees in different regimes. We study the asymptotic distribution of the outdegree of node $i$, as $n\to \infty$, and find three regimes according to whether $0 \lt a \lt 1$ (subcritical regime), $a=1$ (critical regime), or $a\gt 1$ (supercritical regime). Within any regime, there are also phases depending on a delicate interplay between $i$ and $n$, ramifying the asymptotic distribution within the regime into “early,” “intermediate” and “late” phases. In certain phases of certain regimes, we find asymptotic Gaussian laws. In certain phases of some other regimes, small oscillations in the asymototic laws are detected by the Poisson approximation techniques.

Motivated by the investigation of probability distributions with finite variance but heavy tails, we study infinitely divisible laws whose Lévy measure is characterized by a radial component of geometric (tempered) stable type. We closely investigate the univariate case: characteristic exponents and cumulants are calculated, as well as spectral densities; absolute continuity relations are shown, and short- and long-time scaling limits of the associated Lévy processes analyzed. Finally, we derive some properties of the involved probability density functions.

We consider a superprocess $\{X_t\colon t\geq 0\}$ in a random environment described by a Gaussian field $\{W(t,x)\colon t\geq 0,x\in \mathbb{R}^d\}$. First, we set up a representation of $\mathbb{E}[\langle g, X_t\rangle\mathrm{e}^{-\langle \,f,X_t\rangle }\mid\sigma(W)\vee\sigma(X_r,0\leq r\leq s)]$ for $0\leq s < t$ and some functions f,g, which generalizes the result in Mytnik and Xiong (2007, Theorem 2.15). Next, we give a uniform upper bound for the conditional log-Laplace equation with unbounded initial values. We then use this to establish the corresponding conditional entrance law. Finally, the excursion representation of $\{X_t\colon t\geq 0\}$ is given.

Considering a double-indexed array $(Y_{n,i:\,n\ge 1,i\ge 1})$ of non-negative regularly varying random variables, we study the random-length weighted sums and maxima from its ‘row’ sequences. These sums and maxima may have the same tail and extremal indices (Markovich and Rodionov 2020). The main constraints of the latter results are that there exists a unique series in a scheme of series with the minimum tail index and the tail of the term number is lighter than the tail of the terms. Here, a bounded random number of series are allowed to have the minimum tail index and the tail of the term number may be heavier than the tail of the terms. We derive the tail and extremal indices of the weighted non-stationary random-length sequences under a broader set of conditions than in Markovich and Rodionov (2020). We provide examples of random sequences for which the assumptions are valid. Perspectives in adopting the results in different application areas are formulated.

Let $(X_t)_{t \geqslant 0}$ be the solution of the stochastic differential equation

\[\mathrm{d} X_t = b(X_t) \,\mathrm{d} t+A\,\mathrm{d} Z_t, \quad X_{0}=x,\]

$b\,:\, \mathbb{R}^d \rightarrow \mathbb{R}^d$

$A \in \mathbb{R}^{d \times d}$

$(Z_t)_{t\geqslant 0}$

$\alpha$

$\alpha \in (1,2)$

$x\in\mathbb{R}^{d}$

$\Gamma = (\gamma_n)_{n\in \mathbb{N}}$

$(X_t)_{t \geqslant 0}$

$\alpha$

$\gamma^{\frac{1}{\alpha}}_n$

$\gamma^{1+\frac {1}{\alpha}-\frac{1}{\kappa}}_n$

$\kappa \in [1,\alpha)$

$\gamma^{\frac{2-\alpha}{\alpha}}_n$

$\alpha$

We establish a sample path moderate deviation principle for the integrated shot noise process with Poisson arrivals and non-stationary noises. As in Pang and Taqqu (2019), we assume that the noise is conditionally independent given the arrival times, and the distribution of each noise depends on its arrival time. As applications, we derive moderate deviation principles for the workload process and the running maximum process for a stochastic fluid queue with the integrated shot noise process as the input; we also show that a steady-state distribution exists and derive the exact tail asymptotics.

In a recent paper, the authors studied the distribution properties of a class of exchangeable processes, called measure-valued Pólya sequences (MVPSs), which arise as the observation process in a generalized urn sampling scheme. Here we present several results in the form of ‘sufficientness’ postulates that characterize their predictive distributions. In particular, we show that exchangeable MVPSs are the unique exchangeable models whose predictive distributions are a mixture of the marginal distribution and the average of a probability kernel evaluated at past observations. When the latter coincides with the empirical measure, we recover a well-known result for the exchangeable model with a Dirichlet process prior. In addition, we provide a ‘pure’ sufficientness postulate for exchangeable MVPSs that does not assume a particular analytic form for the predictive distributions. Two other sufficientness postulates consider the case when the state space is finite.

The Wiener–Hopf factors of a Lévy process are the maximum and the displacement from the maximum at an independent exponential time. The majority of explicit solutions assume the upward jumps to be either phase-type or to have a rational Laplace transform, in which case the traditional expressions are lengthy expansions in terms of roots located by means of Rouché’s theorem. As an alternative, compact matrix formulas are derived, with parameters computable by iteration schemes.

We consider the count of subgraphs with an arbitrary configuration of endpoints in the random-connection model based on a Poisson point process on ${\mathord{\mathbb R}}^d$. We present combinatorial expressions for the computation of the cumulants and moments of all orders of such subgraph counts, which allow us to estimate the growth of cumulants as the intensity of the underlying Poisson point process goes to infinity. As a consequence, we obtain a central limit theorem with explicit convergence rates under the Kolmogorov distance and connectivity bounds. Numerical examples are presented using a computer code in SageMath for the closed-form computation of cumulants of any order, for any type of connected subgraph, and for any configuration of endpoints in any dimension $d{\geq} 1$. In particular, graph connectivity estimates, Gram–Charlier expansions for density estimation, and correlation estimates for joint subgraph counting are obtained.

This paper defines and studies a broad class of shock models by assuming that a Markovian arrival process models the arrival pattern of shocks. Under the defined class, we show that the system’s lifetime follows the well-known phase-type distribution. Further, we examine the age replacement policy for systems with a continuous phase-type distribution, identifying sufficient conditions for determining the optimal replacement time. Since phase-type distributions are dense in the class of lifetime distributions, our findings for the age replacement policy are widely applicable. We include numerical examples and graphical illustrations to support our results.

In this note, we formulate a ‘one-sided’ version of Wormald’s differential equation method. In the standard ‘two-sided’ method, one is given a family of random variables that evolve over time and which satisfy some conditions, including a tight estimate of the expected change in each variable over one-time step. These estimates for the expected one-step changes suggest that the variables ought to be close to the solution of a certain system of differential equations, and the standard method concludes that this is indeed the case. We give a result for the case where instead of a tight estimate for each variable’s expected one-step change, we have only an upper bound. Our proof is very simple and is flexible enough that if we instead assume tight estimates on the variables, then we recover the conclusion of the standard differential equation method.

We introduce and study a game-theoretic model to understand the spread of an epidemic in a homogeneous population. A discrete-time stochastic process is considered where, in each epoch, first, a randomly chosen agent updates their action trying to maximize a proposed utility function, and then agents who have viral exposures beyond their immunity get infected. Our main results discuss asymptotic limiting distributions of both the cardinality of the subset of infected agents and the action profile, considered under various values of two parameters (initial action and immunity profile). We also show that the theoretical distributions are almost always achieved in the first few epochs.

We prove a large deviation principle for the slow-fast rough differential equations (RDEs) under the controlled rough path (RP) framework. The driver RPs are lifted from the mixed fractional Brownian motion (FBM) with Hurst parameter $H\in (1/3,1/2)$. Our approach is based on the continuity of the solution mapping and the variational framework for mixed FBM. By utilizing the variational representation, our problem is transformed into a qualitative property of the controlled system. In particular, the fast RDE coincides with Itô stochastic differential equation (SDE) almost surely, which possesses a unique invariant probability measure with frozen slow component. We then demonstrate the weak convergence of the controlled slow component by averaging with respect to the invariant measure of the fast equation and exploiting the continuity of the solution mapping.

Let $\{\omega _n\}_{n\geq 1}$ be a sequence of independent and identically distributed random variables on a probability space $(\Omega , \mathcal {F}, \mathbb {P})$, each uniformly distributed on the unit circle $\mathbb {T}$, and let $\ell _n=cn^{-\tau }$ for some $c>0$ and $0<\tau <1$. Let $I_{n}=(\omega _n,\omega _n+\ell _n)$ be the random interval with left endpoint $\omega _n$ and length $\ell _n$. We study the asymptotic property of the covering time $N_n(x)=\sharp \{1\leq k\leq n: x\in I_k\}$ for each $x\in \mathbb {T}$. We prove the quenched central limit theorem for the covering time, that is, $\mathbb {P}$-almost surely,

$$ \begin{align*}\frac{N_n(x)-\mathbb{E}_{\mathbb{P}}(N_n(x))}{\sqrt{\sum_{k=1}^n \ell_k(1-\ell_k)}}\end{align*} $$

converges in law to the standard normal distribution.

We study the last exit time that a spectrally negative Lévy process is below zero until it reaches a positive level b, denoted by $g_{\tau_b^+}$. We generalize the results of the infinite-horizon last exit time explored by Chiu and Yin (2005) by incorporating a random horizon $\tau_b^+$, which represents the first passage time above b. We derive an explicit expression for the joint Laplace transform of $g_{\tau_b^+}$ and $\tau_b^+$ by utilizing a hybrid observation scheme approach proposed by Li, Willmot, and Wong (2018). We further study the optimal prediction of $g_{\tau_b^+}$ in the $L_1$ sense, and find that the optimal stopping time is the first passage time above a level $y_b^{\ast}$, with an explicit characterization of the stopping boundary $y_b^{\ast}$. As examples, Brownian motion with drift and the Cramér–Lundberg model with exponential jumps are considered.

We consider stochastic differential equations (SDEs) driven by a fractional Brownian motion with a drift coefficient that is allowed to be arbitrarily close to criticality in a scaling sense. We develop a comprehensive solution theory that includes strong existence, path-by-path uniqueness, existence of a solution flow of diffeomorphisms, Malliavin differentiability and $\rho $-irregularity. As a consequence, we can also treat McKean-Vlasov, transport and continuity equations.

Let $G$ be a group. The notion of linear sofic approximations of $G$ over an arbitrary field $F$ was introduced and systematically studied by Arzhantseva and Păunescu [3]. Inspired by one of the results of [3], we introduce and study the invariant $\kappa _F(G)$ that captures the quality of linear sofic approximations of $G$ over $F$. In this work, we show that when $F$ has characteristic zero and $G$ is linear sofic over $F$, then $\kappa _F(G)$ takes values in the interval $[1/2,1]$ and $1/2$ cannot be replaced by any larger value. Further, we show that under the same conditions, $\kappa _F(G)=1$ when $G$ is torsion-free. These results answer a question posed by Arzhantseva and Păunescu [3] for fields of characteristic zero. One of the new ingredients of our proofs is an effective non-concentration estimates for random walks on finitely generated abelian groups, which may be of independent interest.