The law of large numbers is one of the most fundamental results in Probability Theory. In the case of independent sequences, there are some known characterizations; for instance, in the independent and identically distributed setting it is known that the law of large numbers is equivalent to integrability. In the case of dependent sequences, there are no known general characterizations -- to the best of our knowledge. We provide such a characterization for identically distributed Bernoulli sequences in terms of a product disintegration.
In this paper we mainly discuss sharp lower and upper bounds for the length of longest consecutive switches in IID Bernoulli sequences. This work is an extension of results in Erdős and Révész (1975) for longest head-run and Hao et al. (2021) for longest consecutive switches in unbiased coin-tossing, and might be applied to reliability theory, biology, quality control, pattern recognition, finance, etc.
We provide a sufficient condition for the bounded law of the iterated logarithms for strictly stationary random fields expressable as a functional of i.i.d. random fields when the summation is done on rectangles. The study is done via the control of the moments of an appropriated maximal function. Applications to functionals of linear random fields, functions of a Gaussian linear random field and Volterra process are given.
We study the sequence alignment problem and its independent version, the discrete Hammersley process with an exploration penalty. We obtain rigorous upper bounds for the number of optimality regions in both models near the soft edge. At zero penalty the independent model becomes an exactly solvable model and we identify cases for which the law of the last passage time converges to a Tracy-Widom law.
Feller (1945) provided a coupling between the counts of cycles of various sizes in a uniform random permutation of $[n]$ and the spacings between successes in a sequence of $n$ independent Bernoulli trials with success probability $1/n$ at the $n$th trial. Arratia, Barbour and Tavaré (1992) extended Feller's coupling, to associate cycles of random permutations governed by the Ewens $(θ)$ distribution with spacings derived from independent Bernoulli trials with success probability $θ/(n-1+θ)$ at the $n$th trial, and to conclude that in an infinite sequence of such trials, the numbers of spacing…
A result of Chebyshev (1864) and Hoeffding1956}, on bounding an expectation of a given function with respect to a Bernoulli convolution (also called Poisson binomial law, or law of the number of successes in independent trials) with any given first moment, is here generalised to the case of any given first few moments, as indicated in the title. A nonprobabilistic, and perhaps more obvious, reformulation is: Every permutation invariant and separately affine-linear function of $n$ real variables $x_i\in[a,b]$ assumes its extremal values given the power sums $\sum_{i=1}^nx_i^1,\ldots, \sum_{i=1}…
In this paper, we consider U-statistics whose data is a strictly stationary sequence which can be expressed as a functional of an i.i.d. one. We establish a strong law of large numbers, a bounded law of the iterated logarithms and a central limit theorem under a dependence condition. The main ingredients for the proof are an approximation by U-statistics whose data is a functional of $\ell$ i.i.d. random variables and an analogue of the Hoeffding's decomposition for U-statistics of this type.
We propose an approach to construct Bernoulli trials $\{X_i, i\ge 1\}$ combining dependence and independence periods, and call it Bernoulli sequence with random dependence (BSRD). The structure of dependence, on the past $S_i = X_1 + \ldots + X_i$, {defines} a class of non-Markovian random walks of recent interest in the literature. In this paper, the dependence is activated by an auxiliary collection of Bernoulli trials $\{Y_i, i\ge 1\}$, called {\it memory switch sequence}. We introduce the concept of {\it memory lapses property}, which {is} characterized by intervals of consecutive independ…
Infinite Bernoulli convolutions generated by multigeometric series and their properties
Dragomanov Ukrainian State University（德拉戈马诺夫乌克兰国立大学）；Institute of Mathematics of NAS of Ukraine（乌克兰国家科学院数学研究所）；Central Ukrainian State University（中央乌克兰国立大学）
The Bernoulli convolution $ν_λ$ with parameter $λ\in(0,1)$ is the probability measure supported on $\mathbf{R}$ that is the law of the random variable $\sum\pmλ^n$, where the $\pm$ are independent fair coin-tosses. We prove that $\dimν_λ=1$ for all transcendental $λ\in(1/2,1)$.
The Euclidean first-passage percolation model of Howard and Newman is a rotationally invariant percolation model built on a Poisson point process. It is known that the passage time between 0 and $ne_1$ obeys a diffusive upper bound: $\mbox{Var}\, T(0,ne_1) \leq Cn$, and in this paper we improve this inequality to $Cn/\log n$. The methods follow the strategy used for sublinear variance proofs on the lattice, using the Falik-Samorodnitsky inequality and a Bernoulli encoding, but with substantial technical difficulties. To deal with the different setup of the Euclidean model, we represent the pas…
An iterative randomness extraction algorithm which generalized the Von Neumann's extraction algorithm is detailed, analyzed and implemented in standard C++. Given a sequence of independently and identically distributed biased Bernoulli random variables, to extract randomness from the aforementioned sequence pertains to produce a new sequence of independently and identically distributed unbiased Bernoulli random variables. The iterative construction here is inspired from the work of Stout and Warren 1984 who modified appropriately the tree of probabilities produced by recursively repeating the …
The Bernoulli convolution with parameter $λ\in(0,1)$ is the probability measure $μ_λ$ that is the law of the random variable $\sum_{n\ge0}\pmλ^n$, where the signs are independent unbiased coin tosses. We prove that each parameter $λ\in(1/2,1)$ with $\dimμ_λ<1$ can be approximated by algebraic parameters $ξ\in(1/2,1)$ within an error of order $\exp(-deg(ξ)^{A})$ for any number $A$, such that $\dimμ_ξ<1$. As a corollary, we conclude that $\dimμ_λ=1$ for each of $λ=\ln 2, e^{-1/2}, π/4$. These are the first explicit examples of such transcendental parameters. Moreover, we show that Lehmer's conje…
