Abstract: We give a probabilistic introduction to determinantal and permanental point processes. Determinantal processes arise in physics (fermions, eigenvalues of random matrices) and in combinatorics (nonintersecting paths, random spanning trees). They have the striking property that the number of points in a region D is a sum of independent Bernoulli random variables, with parameters which a...

The Bernoulli polynomials Bk restricted to 0, 1 and extended by periodicity have nth sine and cosine Fourier coefficients of the formCk/n . In general, the Fourier coefficients of any polynomial restricted to 0, 1 are linear combinations of terms of the form 1/n . If we can make this linear combination explicit for a specific family of polynomials, then by uniqueness of Fourier series, we get a...

A sequence of random variables, each taking values 0 or 1, is called a Bernoulli sequence. We say that a string of length d occurs in a Bernoulli sequence if a success is followed by exactly (d − 1) failures before the next success. The counts of such d-strings are of interest, and in specific independent Bernoulli sequences are known to correspond to asymptotic d-cycle counts in random permuta...

This paper continues our quest to develop a rigorous analytical theory of 1-D cellular automata via a nonlinear dynamics perspective. The 18 yet uncharacterized local rules are henceforth partitioned into ten complex Bernoulli στ -shift rules and eight hyper Bernoulli στ -shift rules, the latter including such famous rules 30 and 110 . All exhibit a bizarre composite wave dynamics with arbitrar...

In this paper, we consider the multivariate Bernoulli distribution as a model to estimate the structure of graphs with binary nodes. This distribution is discussed in the framework of the exponential family, and its statistical properties regarding independence of the nodes are demonstrated. Importantly the model can estimate not only the main effects and pairwise interactions among the nodes b...

Property (iv) is called boundedly finite. So, when we use (iv) instead of (iii), we are interested in simple, boundedly finite spatial point processes with no fixed atoms. When we use (iii), the domain A can be an arbitrary set. When we use (iv), the domain A must have some notion of boundedness. This is no problem when A is a subset of Rd for some d. We just use the usual notion of boundedness...

Arithmetic properties of integer sequences counting periodic points are studied, and applied to the case of linear recurrence sequences, Bernoulli numerators, and Bernoulli denominators.

In this paper, a new acoustic model called time-inhomogeneous hidden Bernoulli model (TI-HBM) is introduced as an alternative to hidden Markov model (HMM) in continuous speech recognition. Contrary to HMM, the state transition process in TI-HBM is not a Markov process, rather it is an independent (generalized Bernoulli) process. This difference leads to elimination of dynamic programming at sta...

A Bernoulli random walk is a random trajectory starting from 0 and having i.i.d. increments, each of them being +1 or −1, equally likely. The other families quoted in the title are Bernoulli random walks under various conditions. A peak in a trajectory is a local maximum. In this paper, we condition the families of trajectories to have a given number of peaks. We show that, asymptotically, the ...