Related papers: Maximum Variation of Total Risk

This article deals with the asymptotic behaviour as $t\to +\infty$ of the survival function $P[T > t],$ where $T$ is the first passage time above a non negative level of a random process starting from zero. In many cases of physical…

Let $F(\sigma)=\sum_{n=1}^\infty \frac{X_n}{n^\sigma}$ be a random Dirichlet series where $(X_n)_{n\in\mathbb{N}}$ are independent standard Gaussian random variables. We compute in a quantitative form the expected number of zeros of…

A random vector $X$ with representation $X=\sum_{j\geq0}A_jZ_j$ is considered. Here, $(Z_j)$ is a sequence of independent and identically distributed random vectors and $(A_j)$ is a sequence of random matrices, `predictable' with respect to…

In this article, we consider two forms of shrinkage estimators of the mean $\theta$ of a multivariate normal distribution $X\sim N_{p}\left(\theta, \sigma^{2}I_{p}\right)$ where $\sigma^{2}$ is unknown. We take the prior law $\theta \sim…

Two old conjectures from problem sections, one of which from SIAM Review, concern the question of finding distributions that maximize P(Sn <= t), where Sn is the sum of i.i.d. random variables X1, ..., Xn on the interval [0,1], satisfying…

We consider optimal stopping problems, in which a sequence of independent random variables is drawn from a known continuous density. The objective of such problems is to find a procedure which maximizes the expected reward; this is often…

It is shown that the maximum of $|\zeta(1/2+it)|$ on the interval $T^{1/2}\le t \le T$ is at least $\exp\left((1/\sqrt{2}+o(1)) \sqrt{\log T \log\log\log T/\log\log T}\right)$. Our proof uses Soundararajan's resonance method and a certain…

Motivated by concerns about making online decisions that incur undue amount of risk at each time step, in this paper, we formulate the probably anytime-safe stochastic combinatorial semi-bandits problem. In this problem, the agent is given…

We improve Montgomery's $\Omega$-results for $|\zeta(\sigma+it)|$ in the strip $1/2<\sigma<1$ and give in particular lower bounds for the maximum of $|\zeta(\sigma+it)|$ on $\sqrt{T}\le t \le T$ that are uniform in $\sigma$. We give similar…

We investigate the properties of a discrete-time martingale $\{X_m\}_{m\in \mathbb Z_{\geq 0}}$, where all differences between adjacent random variables are limited to be not more than a constant as a promise. In this situation, it is known…

Results of extensive computations of moments of the Riemann zeta function on the critical line are presented. Calculated values are compared with predictions motivated by random matrix theory. The results can help in deciding between those…

Consider a normal location model $X \mid \theta \sim N(\theta, \sigma^2)$ with known $\sigma^2$. Suppose $\theta \sim G_0$, where the prior $G_0$ has zero mean and variance bounded by $V$. Let $G_1$ be a possibly misspecified prior with…

We consider the problem of estimating the total probability of all symbols that appear with a given frequency in a string of i.i.d. random variables with unknown distribution. We focus on the regime in which the block length is large yet no…

A natural problem in the context of the coupon collector's problem is the behavior of the maximum of independent geometrically distributed random variables (with distinct parameters). This question has been addressed by Brennan et al.…

Several problems involving $E(T)$ and $E_2(T)$, the error terms in the mean square and mean fourth moment formula for $|\zeta(1/2+it)}$ are discussed. In particular, it is proved that $$ \int_0^T E(t)E_2(t)dt \ll_ T^{7/4}(\log…

Let $S_n$ be the sum of independent random variables with distribution $F$. Under the assumption that $-\log(1-F(x))$ is slowly varying, conditions for $$ \lim_{n\to\infty}\sup_{s\ge t_n}\left|{P[S_n>s]\over n(1-F(s))}-1\right| =0 $$ are…

Given a set of independent Poisson random variables with common mean, we study the distribution of their maximum and obtain an accurate asymptotic formula to locate the most probable value of the maximum. We verify our analytic results with…

For a risk vector $V$, whose components are shared among agents by some random mechanism, we obtain asymptotic lower and upper bounds for the individual agents' exposure risk and the aggregated risk in the market. Risk is measured by…

In this paper we revisited the classical problem of max-sum equivalence of randomly weighted sums in two dimensions. In opposite to the most papers in literature, we consider that there exists some interdependence between the primary random…

Let $\{\xi_1,\xi_2,\ldots\}$ be a sequence of independent random variables, and $\eta$ be a counting random variable independent of this sequence. In addition, let $S_0:=0$ and $S_n:=\xi_1+\xi_2+\cdots+\xi_n$ for $n\geqslant1$. We consider…