Counting subsets of the random partition and the ‘Brownian Bridge’ process

Let Ω_m be the set of partitions, ω, of a finite m-element set; induce a uniform probability distribution on Ω_m, and define X_ms(ω) as the number of s-element subsets in ω. We alow the existence of an integer-valued function n=n^(m)(t), t?[0, 1], and centering constants b_ms, 0?s? m, such that

$\text{Z}{}^{\mathrm{(m)}}\text{(t)=}\text{∑}\text{s=0}\text{n(m)(t)}\text{(X}{}_{\mathrm{ms}}\text{?b}{}_{\mathrm{ms}}\text{)}\text{√}\text{∑}\text{s=0}\text{m}\text{b}{}_{\mathrm{ms}}$

converges to the ‘Brownian Bridge’ process in terms of its finite-dimensional distributions.     

Stationarity and Self-Similarity Characterization of the Set-Indexed Fractional Brownian Motion

The set-indexed fractional Brownian motion (sifBm) has been defined by Herbin–Merzbach (J. Theor. Probab. 19(2):337–364, 2006) for indices that are subsets of a metric measure space. In this paper, the sifBm is proved to satisfy a strengthened definition of increment stationarity. This new definition for stationarity property allows us to get a complete characterization of this process by its fractal properties: The sifBm is the only set-indexed Gaussian process which is self-similar and has stationary increments. Using the fact that the sifBm is the only set-indexed process whose projection on any increasing path is a one-dimensional fractional Brownian motion, the limitation of its definition for a self-similarity parameter 0<H<1/2 is studied, as illustrated by some examples. When the indexing collection is totally ordered, the sifBm can be defined for 0<H<1.     

On Tightness and Weak Convergence in the Approximation of the Occupation Measure of Fractional Brownian Motion

In this paper we consider approximations of the occupation measure of the Fractional Brownian motion by means of some functionals defined on regularizations of the paths. In a previous article Berzin and León proved a cylindrical convergence to a Wiener process of conveniently rescaled functionals. Here we show the tightness of the approximation in the space of continuous functions endowed with the topology of uniform convergence on compact sets. This allows us to simplify the identification of the limit.     

The Hausdorff Dimension of the Double Points on the Brownian Frontier

The frontier of a planar Brownian motion is the boundary of the unbounded component of the complement of its range. In this paper, we find the Hausdorff dimension of the set of double points on the frontier.     

The First Exit Time of a Brownian Motion from the Minimum and Maximum Parabolic Domains

Consider a Brownian motion starting at an interior point of the minimum or maximum parabolic domains, namely, D_min={(x,y₁,y₂):||x|| < min{(y₁+1)^1/p₁,(y₂+1)^1/p₂}}D_{\min}=\{(x,y_{1},y_{2}):\|x\|< \min\{(y_{1}+1)^{1/p_{1}},(y_{2}+1)^{1/p_{2}}\}\} and D_max={(x,y₁,y₂):||x|| < max{(y₁+1)^1/p₁,\allowbreak(y₂+1)^1/p₂}}D_{\max}=\{(x,y_{1},y_{2}):\|x\|<\max\{(y_{1}+1)^{1/p_{1}},\allowbreak(y_{2}+1)^{1/p_{2}}\}\} in R ^d+2,d≥1, respectively, where ‖⋅‖ is the Euclidean norm in R ^d , y ₁,y ₂≥−1, and p ₁,p ₂>1. Let t_Dmin\tau_{D_{\min}} and t_Dmax\tau_{D_{\max}} denote the first times the Brownian motion exits from D _min and D _max. Estimates with exact constants for the asymptotics of $\log P(\tau_{D_{\min}}>t)$\log P(\tau_{D_{\min}}>t) and $\log P(\tau_{D_{\max}}>t)$\log P(\tau_{D_{\max}}>t) are given as t→∞, depending on the relationship between p ₁ and p ₂, respectively. The proof methods are based on Gordon’s inequality and early works of Li, Lifshits, and Shi in the single general parabolic domain case.     

Gene flow across geographical barriers — scaling limits of random walks with obstacles

We study a class of random walks which behave like simple random walks outside of a bounded region around the origin and which are subject to a partial reflection near the origin. We obtain a non trivial scaling limit which behaves like reflected Brownian motion until its local time at zero reaches an exponential variable. It then follows reflected Brownian motion on the other side of the origin until its local time at zero reaches another exponential level, etc. These random walks are used in population genetics to trace the position of ancestors in the past near geographical barriers.     

The application of spectral distribution of product of two random matrices in the factor analysis

In the factor analysis model with large cross-section and time-series dimensions,we pro- pose a new method to estimate the number of factors.Specially if the idiosyncratic terms satisfy a linear time series model,the estimators of the parameters can be obtained in the time series model. The theoretical properties of the estimators are also explored.A simulation study and an empirical analysis are conducted.     

Estimates on the Speedup and Slowdown for a Diffusion in a Drifted Brownian Potential

We study a model of diffusion in a Brownian potential. This model was first introduced by T. Brox (Ann. Probab. 14:1206–1218, 1986) as a continuous time analogue of random walk in random environment. We estimate the deviations of this process above or under its typical behavior. Our results rely on different tools such as a representation introduced by Y. Hu, Z. Shi and M. Yor, Kotani’s lemma, introduced at first by K. Kawazu and H. Tanaka (J. Math. Soc. Jpn. 49:189–211, 1997), and a decomposition of hitting times developed in a recent article by A. Fribergh, N. Gantert and S. Popov (Preprint, 2008). Our results are in agreement with their results in the discrete case.     

The Exact Hausdorff Measure of the Zero Set of Fractional Brownian Motion

Let {X(t), t∈? ^N } be a fractional Brownian motion in ? ^d of index H. If L(0,I) is the local time of X at 0 on the interval I?? ^N , then there exists a positive finite constant c(=c(N,d,H)) such that

$m_\phi\bigl(X^{-1}(0)\cap I\bigr)=cL(0,I),$

where \(\phi(t)=t^{N-dH}(\log\log\frac{1}{t})^{dH/N}\), and m _φ (E) is the Hausdorff φ-measure of E. This refines a previous result of Xiao (Probab. Theory Relat. Fields 109: 126–197, 1997) on the relationship between the local time and the Hausdorff measure of zero set for d-dimensional fractional Brownian motion on ? ^N .

     

Type-II matrices in weighted Bose–Mesner algebras of ranks 2 and 3

Type-II matrices are nonzero complex matrices that were introduced in connection with spin models for link invariants. Type-II matrices have been found in connection with symmetric designs, sets of equiangular lines, strongly regular graphs, and some distance regular graphs. We investigate weighted complete and strongly regular graphs, and show that type-II matrices arise in this setting as well.     

Reflected Brownian motion in the quadrant: tail behavior of the stationary distribution

Let Z be a two-dimensional Brownian motion confined to the non-negative quadrant by oblique reflection at the boundary. Such processes arise in applied probability as diffusion approximations for two-station queueing networks. The parameters of Z are a drift vector, a covariance matrix, and a “direction of reflection” for each of the quadrant’s two boundary rays. Necessary and sufficient conditions are known for Z to be a positive recurrent semimartingale, and they are the only restrictions imposed on the process data in our study. Under those assumptions, a large deviations principle (LDP) is conjectured for the stationary distribution of Z, and we recapitulate the cases for which it has been rigorously justified. For sufficiently regular sets B, the LDP says that the stationary probability of xB decays exponentially as x→∞, and the asymptotic decay rate is the minimum value achieved by a certain function I(?) over the set B. Avram, Dai and Hasenbein  (Queueing Syst.: Theory Appl. 37, 259–289, 2001) provided a complete and explicit solution for the large deviations rate function I(?). In this paper we re-express their solution in a simplified form, showing along the way that the computation of I(?) reduces to a relatively simple problem of least-cost travel between a point and a line.     

Branching rules in the ring of superclass functions of unipotent upper-triangular matrices

It is becoming increasingly clear that the supercharacter theory of the finite group of unipotent upper-triangular matrices has a rich combinatorial structure built on set-partitions that is analogous to the partition combinatorics of the classical representation theory of the symmetric group. This paper begins by exploring a connection to the ring of symmetric functions in non-commuting variables that mirrors the symmetric group’s relationship with the ring of symmetric functions. It then also investigates some of the representation theoretic structure constants arising from the restriction, tensor products and superinduction of supercharacters in this context.     

On the Maximum of a Brownian Motion and Its Location

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Adiabatic limits of η-invariants and the Meyer functions

We construct a function on the orbifold fundamental group of the moduli space of smooth theta divisors, which we call the Meyer function for smooth theta divisors. In the construction, we use the adiabatic limits of the η-invariants of the mapping torus of theta divisors. We shall prove that the Meyer function for smooth theta divisors cobounds the signature cocycle, and we determine the values of the Meyer function for the Dehn twists. In particular, we give an analytic construction of the Meyer function of genus two.     

Diffusive limits for “true” (or myopic) self-avoiding random walks and self-repellent Brownian polymers in d?≥?3

The problems considered in the present paper have their roots in two different cultures. The `true’ (or myopic) self-avoiding walk model (TSAW) was introduced in the physics literature by Amit et?al. (Phys Rev B 27:1635–1645, 1983). This is a nearest neighbor non-Markovian random walk in ${{\mathbb Z}^d}$ which prefers to jump to those neighbors which were less visited in the past. The self-repelling Brownian polymer model (SRBP), initiated in the probabilistic literature by Durrett and Rogers (Probab Theory Relat Fields 92:337–349, 1992) (independently of the physics community), is the continuous space–time counterpart: a diffusion in ${{\mathbb R}^d}$ pushed by the negative gradient of the (mollified) occupation time measure of the process. In both cases, similar long memory effects are caused by a path-wise self-repellency of the trajectories due to a push by the negative gradient of (softened) local time. We investigate the asymptotic behaviour of TSAW and SRBP in the non-recurrent dimensions. First, we identify a natural stationary (in time) and ergodic distribution of the environment (the local time profile) as seen from the moving particle. The main results are diffusive limits. In the case of TSAW, for a wide class of self-interaction functions, we establish diffusive lower and upper bounds for the displacement and for a particular, more restricted class of interactions, we prove full CLT for the finite dimensional distributions of the displacement. In the case of SRBP, we prove full CLT without restrictions on the interaction functions. These results settle part of the conjectures, based on non-rigorous renormalization group arguments (equally ‘valid’ for the TSAW and SRBP cases), in Amit et?al. (1983). The proof of the CLT follows the non-reversible version of Kipnis–Varadhan theory. On the way to the proof, we slightly weaken the so-called graded sector condition.     

Maxima and Minima of the Hosoya Index and the Merrifield-Simmons Index

The Hosoya index and the Merrifield-Simmons index are typical examples of graph invariants used in mathematical chemistry for quantifying relevant details of molecular structure. In recent years, quite a lot of work has been done on the extremal problem for these two indices, i.e., the problem of determining the graphs within certain prescribed classes that maximize or minimize the index value. This survey collects and classifies these results, and also provides some useful auxiliary results, tools and techniques that are frequently used in the study of this type of problem.     

Rates of convergence for the homogenization of fully nonlinear uniformly elliptic pde in random media

We establish a logarithmic-type rate of convergence for the homogenization of fully nonlinear uniformly elliptic second-order pde in strongly mixing media with similar, i.e., logarithmic, decorrelation rate. The proof consists of two major steps. The first, which is actually the only place in the paper where probability plays a role, establishes the rate for special (quadratic) data using the methodology developed by the authors and Wang to study the homogenization of nonlinear uniformly elliptic pde in general stationary ergodic random media. The second is a general argument, based on the new notion of δ-viscosity solutions which is introduced in this paper, that shows that rates known for quadratic can be extended to general data. As an application of this we also obtain here rates of convergence for the homogenization in periodic and almost periodic environments. The former is algebraic while the latter depends on the particular equation.