The necessary and sufficient condition for a function $f:(0,1)\to [0,1]$ to be Borel measurable (given by Theorem stated below) provides a technique to prove (in Corollary 2) the existence of a Borel measurable map $H:{\{0,1\}}^{\mathbb{N}}\to {\{0,1\}}^{\mathbb{N}}$ such that $ℒ\left(H\left({\mathbf{\text{X}}}^p\right)\right)=ℒ\left({\mathbf{\text{X}}}^{1/2}\right)$ holds for each $p\in (0,1)$, where ${\mathbf{\text{X}}}^p=(X_1^p,X_2^p,...)$ denotes Bernoulli sequence of random variables with $P[X_i^p=1]=p$.

The autoregressive process takes an important part in predicting problems leading to decision making. In practice, we use the least squares method to estimate the parameter θ̃ of the first-order autoregressive process taking values in a real separable Banach space B (ARB(1)), if it satisfies the following relation: $X{̃}_t=\theta ̃X{̃}_{t-1}+\epsilon {̃}_t$. In this paper we study the convergence in distribution of the linear operator $I(\theta {̃}_T,\theta ̃)=(\theta {̃}_T-\theta ̃)\theta {̃}^{T-2}$ for ||θ̃|| > 1 and so we construct inequalities of Bernstein type for this operator.

In this paper, we study particular examples of the intertwining relationQtΛ = ΛPtbetween two Markov semi-groups (Pt, t ≥ 0) defined respectively on (E,ε) and (F,F), via the Markov kernelΛ: (E,ε) → (F,F).

We asymptotically estimate from above the expected Betti numbers of random real hypersurfaces in smooth real projective manifolds. Our upper bounds grow as the square root of the degree of the hypersurfaces as the latter grows to infinity, with a coefficient involving the Kählerian volume of the real locus of the manifold as well as the expected determinant of random real symmetric matrices of given index. In particular, for large dimensions, these coefficients get exponentially small away from...

Logic and Probability, as theories, have been developed quite independently and, with a few exceptions (like Boole's), have largely ignored each other. And nevertheless they share a lot of similarities, as well a considerable common ground. The exploration of the shared concepts and their mathematical treatment and unification is here attempted following the lead of illustrious researchers (Reichenbach, Carnap, Popper, Gaifman, Scott & Krauss, Fenstad, Miller, David Lewis, Stalnaker, Hintikka...

We introduce the function $Z(x;\xi ,\nu ):={\int }_{-\infty }^x\varphi (t-\xi )·\Phi \left(\nu t\right)\text{d}t$, where $\varphi$ and $\Phi$ are the pdf and cdf of $N(0,1)$, respectively. We derive two recurrence formulas for the effective computation of its values. We show that with an algorithm for this function, we can efficiently compute the second-order terms of Bonferroni-type inequalities yielding the upper and lower bounds for the distribution of a max-type binary segmentation statistic in the case of small samples (where asymptotic results do not work), and in general for max-type random variables...

This paper describes a new method for generating stationary integer-valued time series from renewal processes. We prove that if the lifetime distribution of renewal processes is nonlattice and the probability generating function is rational, then the generated time series satisfy causal and invertible ARMA type stochastic difference equations. The result provides an easy method for generating integer-valued time series with ARMA type autocovariance functions. Examples of generating binomial ARMA(p,p-1)...

This article provides entropic inequalities for binomial-Poisson distributions, derived from the two point space. They appear as local inequalities of the M/M/∞ queue. They describe in particular the exponential dissipation of Φ-entropies along this process. This simple queueing process appears as a model of “constant curvature”, and plays for the simple Poisson process the role played by the Ornstein-Uhlenbeck process for Brownian Motion. Some of the inequalities are recovered by semi-group ...