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Let 
$\mathcal{C}$ denote the family of all coherent distributions on the unit square 
$[0,1]^2$, i.e. all those probability measures 
$\mu$ for which there exists a random vector 
$(X,Y)\sim \mu$, a pair 
$(\mathcal{G},\mathcal{H})$ of 
$\sigma$-fields, and an event E such that 
$X=\mathbb{P}(E\mid\mathcal{G})$, 
$Y=\mathbb{P}(E\mid\mathcal{H})$ almost surely. We examine the set 
$\mathrm{ext}(\mathcal{C})$ of extreme points of 
$\mathcal{C}$ and provide its general characterisation. Moreover, we establish several structural properties of finitely-supported elements of 
$\mathrm{ext}(\mathcal{C})$. We apply these results to obtain the asymptotic sharp bound 
$\lim_{\alpha \to \infty}\alpha\cdot(\sup_{(X,Y)\in \mathcal{C}}\mathbb{E}|X-Y|^{\alpha}) = {2}/{\mathrm{e}}$.
We show that an estimate by de la Peña, Ibragimov, and Jordan for 
${\mathbb{E}}(X-c)^+$, with c a constant and X a random variable of which the mean, the variance, and 
$\mathbb{P}(X \leqslant c)$ are known, implies an estimate by Scarf on the infimum of 
${\mathbb{E}}(X \wedge c)$ over the set of positive random variables X with fixed mean and variance. This also shows, as a consequence, that the former estimate implies an estimate by Lo on European option prices.
It is proven that a conjecture of Tao (2010) holds true for log-concave random variables on the integers: For every 
$n \geq 1$, if 
$X_1,\ldots,X_n$ are i.i.d. integer-valued, log-concave random variables, thenas 
\begin{equation*} H(X_1+\cdots +X_{n+1}) \geq H(X_1+\cdots +X_{n}) + \frac {1}{2}\log {\Bigl (\frac {n+1}{n}\Bigr )} - o(1) \end{equation*}
$H(X_1) \to \infty$, where 
$H(X_1)$ denotes the (discrete) Shannon entropy. The problem is reduced to the continuous setting by showing that if 
$U_1,\ldots,U_n$ are independent continuous uniforms on 
$(0,1)$, thenas 
\begin{equation*} h(X_1+\cdots +X_n + U_1+\cdots +U_n) = H(X_1+\cdots +X_n) + o(1), \end{equation*}
$H(X_1) \to \infty$, where 
$h$ stands for the differential entropy. Explicit bounds for the 
$o(1)$-terms are provided.
The switch process alternates independently between 1 and 
$-1$, with the first switch to 1 occurring at the origin. The expected value function of this process is defined uniquely by the distribution of switching times. The relation between the two is implicitly described through the Laplace transform, which is difficult to use for determining if a given function is the expected value function of some switch process. We derive an explicit relation under the assumption of monotonicity of the expected value function. It is shown that geometric divisible switching time distributions correspond to a non-negative decreasing expected value function. Moreover, an explicit relation between the expected value of a switch process and the autocovariance function of the switch process stationary counterpart is obtained, leading to a new interpretation of the classical Pólya criterion for positive-definiteness.
In the classical gambler’s ruin problem, the gambler plays an adversary with initial capitals z and 
$a-z$, respectively, where 
$a>0$ and 
$0< z < a$ are integers. At each round, the gambler wins or loses a dollar with probabilities p and 
$1-p$. The game continues until one of the two players is ruined. For even a and 
$0<z\leq {a}/{2}$, the family of distributions of the duration (total number of rounds) of the game indexed by 
$p \in [0,{\frac{1}{2}}]$ is shown to have monotone (increasing) likelihood ratio, while for 
${a}/{2} \leq z<a$, the family of distributions of the duration indexed by 
$p \in [{\frac{1}{2}}, 1]$ has monotone (decreasing) likelihood ratio. In particular, for 
$z={a}/{2}$, in terms of the likelihood ratio order, the distribution of the duration is maximized over 
$p \in [0,1]$ by 
$p={\frac{1}{2}}$. The case of odd a is also considered in terms of the usual stochastic order. Furthermore, as a limit, the first exit time of Brownian motion is briefly discussed.
Suppose that a system is affected by a sequence of random shocks that occur over certain time periods. In this paper we study the discrete censored 
$\delta$-shock model, 
$\delta \ge 1$, for which the system fails whenever no shock occurs within a 
$\delta$-length time period from the last shock, by supposing that the interarrival times between consecutive shocks are described by a first-order Markov chain (as well as under the binomial shock process, i.e., when the interarrival times between successive shocks have a geometric distribution). Using the Markov chain embedding technique introduced by Chadjiconstantinidis et al. (Adv. Appl. Prob. 32, 2000), we study the joint and marginal distributions of the system’s lifetime, the number of shocks, and the number of periods in which no shocks occur, up to the failure of the system. The joint and marginal probability generating functions of these random variables are obtained, and several recursions and exact formulae are given for the evaluation of their probability mass functions and moments. It is shown that the system’s lifetime follows a Markov geometric distribution of order 
$\delta$ (a geometric distribution of order 
$\delta$ under the binomial setup) and also that it follows a matrix-geometric distribution. Some reliability properties are also given under the binomial shock process, by showing that a shift of the system’s lifetime random variable follows a compound geometric distribution. Finally, we introduce a new mixed discrete censored 
$\delta$-shock model, for which the system fails when no shock occurs within a 
$\delta$-length time period from the last shock, or the magnitude of the shock is larger than a given critical threshold 
$\gamma >0$. Similarly, for this mixed model, we study the joint and marginal distributions of the system’s lifetime, the number of shocks, and the number of periods in which no shocks occur, up to the failure of the system, under the binomial shock process.
The Bruss–Robertson–Steele (BRS) inequality bounds the expected number of items of random size which can be packed into a given suitcase. Remarkably, no independence assumptions are needed on the random sizes, which points to a simple explanation; the inequality is the integrated form of an 
$\omega$-by-
$\omega$ inequality, as this note proves.
