Related papers: A Maximal Inequality for Supermartingales

In this note, we are interested in obtaining uniform upper bounds for the number of powerful numbers in short intervals $(x, x + y]$. We obtain unconditional upper bounds $O(\frac{y}{\log y})$ and $O(y^{11/12})$ for all powerful numbers and…

The main result of the article reads: the distribution of a continuous starting from zero local martingale whose quadratic characteristic is almost surely absolutely continuous with respect to some non-random increasing continuous function…

This note considers a variation of the full-information secretary problem where the random variables to be observed are independent and identically distributed. Consider $X_1,\dots,X_n$ to be an independent sequence of random variables, let…

This letter derives some new exponential bounds for discrete time, real valued, conditionally symmetric martingales with bounded jumps. The new bounds are extended to conditionally symmetric sub/ supermartingales, and they are compared to…

Multi-round competitions often double or triple the points awarded in the final round, calling it a bonus, to maximize spectators' excitement. In a two-player competition with $n$ rounds, we aim to derive the optimal bonus size to maximize…

Programs with randomization constructs is an active research topic, especially after the recent introduction of martingale-based analysis methods for their termination and runtimes. Unlike most of the existing works that focus on proving…

In this paper, we derive new properties of the Mertens function and discuss a likely upper bound of the absolute value of the Mertens function $\sqrt{\log{x!}}>|M(x)|$ when $x>1$. Using this likely bound we show that we have a sufficient…

Consider a discrete-time martingale $\{X_t\}$ taking values in a Hilbert space $\mathcal H$. We show that if for some $L \geq 1$, the bounds $\mathbb{E} \left[\|X_{t+1}-X_t\|_{\mathcal H}^2 \mid X_t\right]=1$ and $\|X_{t+1}-X_t\|_{\mathcal…

We show somewhat unexpectedly that whenever a general Bernstein-type maximal inequality holds for partial sums of a sequence of random variables, a maximal form of the inequality is also valid.

In this paper we show that the so-called array Fr\'echet problem in Probability/Statistics is (max, +)-linear. The upper bound of Fr\'echet is obtained using simple arguments from residuation theory and lattice distributivity. The lower…

We provide sufficient conditions for polynomial rate of convergence in the weak law of large numbers for supercritical general indecomposable multi-type branching processes. The main result is derived by investigating the embedded…

Let $M_n$ be the maximum of $n$ zero-mean gaussian variables $X_1,..,X_n$ with covariance matrix of minimum eigenvalue $\lambda$ and maximum eigenvalue $\Lambda$. Then, for $n \ge 70$, $$\Pr\{M_n \ge \lambda \left (2 \log n - 2.5 - \log(2…

The Principle of Maximum Entropy is a rigorous technique for estimating an unknown distribution given partial information while simultaneously minimizing bias. However, an important requirement for applying the principle is that the…

In an incomplete model, where under an appropriate num\'eraire, the stock price process is driven by a sigma-bounded semimartingale, we investigate the behavior of the expected utility maximization problem under small perturbations of the…

This paper does not suppose a priori that the evolution of the price of a financial asset is a semimartingale. Since possible strategies of investors are self-financing, previous prices are forced to be finite quadratic variation processes.…

Score matching is an alternative to maximum likelihood (ML) for estimating a probability distribution parametrized up to a constant of proportionality. By fitting the ''score'' of the distribution, it sidesteps the need to compute this…

We prove that for a so-called sticky process $S$ there exists an equivalent probability $Q$ and a $Q$-martingale $\tilde{S}$ that is arbitrarily close to $S$ in $L^p(Q)$ norm. For continuous $S$, $\tilde{S}$ can be chosen arbitrarily close…

In this paper, we obtain almost sure invariance principles with rate of order $n^{1/p}\log^\beta n$, $2< p\le 4$, for sums associated to a sequence of reverse martingale differences. Then, we apply those results to obtain similar…

In this note we introduce a new kind of augmentation of filtrations along a sequence of stopping times. This augmentation is suitable for the construction of new probability measures associated to a positive strict local martingale as done…

A novel, non-trivial, probabilistic upper bound on the entropy of an unknown one-dimensional distribution, given the support of the distribution and a sample from that distribution, is presented. No knowledge beyond the support of the…