Parameter estimation and asymptotic stability in stochastic filtering

In this paper, we study the problem of estimating a Markov chain X$X$ (signal) from its noisy partial information Y$Y$, when the transition probability kernel depends on some unknown parameters. Our goal is to compute the conditional distribution process P{_Xn∣_Yn,…,_Y1}$\mathbb{P}\{X_n\mid Y_n,\dots ,Y_1\}$, referred to hereafter as the optimal filter. Following a standard Bayesian technique, we treat the parameters as a non-dynamic component of the Markov chain. As a result, the new Markov chain is not going to be mixing, even if the original one is. We show that, under certain conditions, the optimal filters are still going to be asymptotically stable with respect to the initial conditions. Thus, by computing the optimal filter of the new system, we can estimate the signal adaptively.     

A minimal repair replacement model with two types of failure and a safety constraint

Consider a system subject to two types of failures. If the failure is of type 1, the system is minimally repaired, and a cost C₁ is incurred. If the failure is of type 2, the system is minimally repaired with probability p and replaced with probability 1−p  . The associated costs are _C2,_m$C_{2\text{,}m}$ and _C2,_r$C_{2\text{,}r}$, respectively. Failures of type 2 are safety critical and to control the risk, management has specified a requirement that the probability of at least one such failure occurring in the interval [0, A] should not exceed a fixed probability limit ω. The problem is to determine an optimal planned replacement time T, minimizing the expected discounted costs under the safety constraint. A cost C_r is incurred whenever a planned replacement is performed. Conditions are established for when the safety constraint affects the optimal replacement time and causes increased costs.     

Reduced limit for semilinear boundary value problems with measure data

We study boundary value problems for semilinear elliptic equations of the form −Δu+g°u=μ$-\mathrm{\Delta }u+g°u=\mu$ in a smooth bounded domain Ω⊂R^N$\Omega \subset R^N$. Let {_μn}$\{{\mu }_n\}$ and {_νn}$\{{\nu }_n\}$ be sequences of measure in Ω and ∂Ω   respectively. Assume that there exists a solution _un$u_n$ with data (_μn,_νn)$({\mu }_n,{\nu }_n)$, i.e., _un$u_n$ satisfies the equation with μ=_μn$\mu ={\mu }_n$ and has boundary trace _νn${\nu }_n$. Further assume that the sequences of measures converge in a weak sense to μ and ν   respectively while {_un}$\{u_n\}$ converges to u   in L¹(Ω)$L^1(\Omega )$. In general u   is not a solution of the boundary value problem with data (μ,ν)$(\mu ,\nu )$. However there exists a pair of measures (μ^?,ν^?)$({\mu }^{?},{\nu }^{?})$ such that u   is a solution of the boundary value problem with this data. The pair (μ^?,ν^?)$({\mu }^{?},{\nu }^{?})$ is called the reduced limit of the sequence {(_μn,_νn)}$\{({\mu }_n,{\nu }_n)\}$. We investigate the relation between the weak limit and the reduced limit and the dependence of the latter on the sequence. A closely related problem was studied by Marcus and Ponce [3].     

Coarse differentiation and quantitative nonembeddability for Carnot groups

We give lower bound estimates for the macroscopic scale of coarse differentiability of Lipschitz maps from a Carnot group with the Carnot–Carathéodory metric (G,_dcc)$(G,d_{cc})$ to a few different classes of metric spaces. Using this result, we derive lower bound estimates for quantitative nonembeddability of Lipschitz embeddings of G   into a metric space (X,_dX)$(X,d_X)$ if X is either an Alexandrov space with nonpositive or nonnegative curvature, a superreflexive Banach space, or another Carnot group that does not admit a biLipschitz homomorphic embedding of G  . For the same targets, we can further give lower bound estimates for the biLipschitz distortion of every embedding f:B(n)→X$f:B(n)\to X$, where B(n)$B(n)$ is the ball of radius n of a finitely generated nonabelian torsion-free nilpotent group G. We also prove an analogue of Bourgain's discretization theorem for Carnot groups and show that Carnot groups have nontrivial Markov convexity. These give the first examples of metric spaces that have nontrivial Markov convexity but cannot biLipschitzly embed into Banach spaces of nontrivial Markov convexity.     

Rational functions with identical measure of maximal entropy

We discuss when two rational functions f and g   can have the same measure of maximal entropy. The polynomial case was completed by Beardon, Levin, Baker–Eremenko, Schmidt–Steinmetz, etc., 1980s–1990s, and we address the rational case following Levin and Przytycki (1997). We show: _μf=_μg${\mu }_f={\mu }_g$ implies that f and g   share an iterate (fⁿ=g^m$f^n=g^m$ for some n and m) for general f   with degree d≥3$d\ge 3$. And for generic f∈_Ratd≥3$f\in {\mathrm{Rat}}_{d\ge 3}$, _μf=_μg${\mu }_f={\mu }_g$ implies g=fⁿ$g=f^n$ for some n≥1$n\ge 1$. For generic f∈_Rat2$f\in {\mathrm{Rat}}_2$, _μf=_μg${\mu }_f={\mu }_g$ implies that g=fⁿ$g=f^n$ or _σf°fⁿ${\sigma }_f°f^n$ for some n≥1$n\ge 1$, where _σf∈_PSL2(C)${\sigma }_f\in {\mathit{PSL}}_2(\mathbb{C})$ permutes two points in each fiber of f. Finally, we construct examples of f and g   with _μf=_μg${\mu }_f={\mu }_g$ such that fⁿ≠σ°g^m$f^n\ne \sigma °g^m$ for any σ∈_PSL2(C)$\sigma \in {\mathit{PSL}}_2(\mathbb{C})$ and m,n≥1$m,n\ge 1$.     

Extremes of space–time Gaussian processes

Let Z={_Zt(h);h∈R^d,t∈R}$Z=\{Z_t\left(h\right);h\in {\mathbb{R}}^d,t\in \mathbb{R}\}$ be a space–time Gaussian process which is stationary in the time variable t$t$. We study _Mn(h)=_{supt∈[0,n]Zt}(_snh)$M_n\left(h\right)={sup}_{t\in [0,n]}{Z}_t\left(s_{n}h\right)$, the supremum of Z$Z$ taken over t∈[0,n]$t\in [0,n]$ and rescaled by a properly chosen sequence _sn→0$s_n\to 0$. Under appropriate conditions on Z$Z$, we show that for some normalizing sequence _bn→∞$b_n\to \infty$, the process _bn(_Mn−_bn)$b_n(M_n-b_n)$ converges as n→∞$n\to \infty$ to a stationary max-stable process of Brown–Resnick type. Using strong approximation, we derive an analogous result for the empirical process.     

Nonparametric estimation of the stationary density and the transition density of a Markov chain

In this paper, we study first the problem of nonparametric estimation of the stationary density f$f$ of a discrete-time Markov chain (_Xi)$\left(X_i\right)$. We consider a collection of projection estimators on finite dimensional linear spaces. We select an estimator among the collection by minimizing a penalized contrast. The same technique enables us to estimate the density g$g$ of (_Xi,_Xi+1)$(X_i,X_{i+1})$ and so to provide an adaptive estimator of the transition density π=g/f$\pi =g/f$. We give bounds in L²$L^2$ norm for these estimators and we show that they are adaptive in the minimax sense over a large class of Besov spaces. Some examples and simulations are also provided.     

Non-homogeneous random walks on a semi-infinite strip

We study the asymptotic behaviour of Markov chains (_Xn,_ηn)$(X_n,{\eta }_n)$ on _Z+×S${\mathbb{Z}}_{+}\times S$, where _Z+${\mathbb{Z}}_{+}$ is the non-negative integers and S$S$ is a finite set. Neither coordinate is assumed to be Markov. We assume a moments bound on the jumps of _Xn$X_n$, and that, roughly speaking, _ηn${\eta }_n$ is close to being Markov when _Xn$X_n$ is large. This departure from much of the literature, which assumes that _ηn${\eta }_n$ is itself a Markov chain, enables us to probe precisely the recurrence phase transitions by assuming asymptotically zero drift for _Xn$X_n$ given _ηn${\eta }_n$. We give a recurrence classification in terms of increment moment parameters for _Xn$X_n$ and the stationary distribution for the large- X$X$ limit of _ηn${\eta }_n$. In the null case we also provide a weak convergence result, which demonstrates a form of asymptotic independence between _Xn$X_n$ (rescaled) and _ηn${\eta }_n$. Our results can be seen as generalizations of Lamperti’s results for non-homogeneous random walks on _Z+${\mathbb{Z}}_{+}$ (the case where S$S$ is a singleton). Motivation arises from modulated queues or processes with hidden variables where _ηn${\eta }_n$ tracks an internal state of the system.     

On the numerical radius of Lipschitz operators in Banach spaces

We study the numerical radius of Lipschitz operators on Banach spaces via the Lipschitz numerical index, which is an analogue of the numerical index in Banach space theory. We give a characterization of the numerical radius and obtain a necessary and sufficient condition for Banach spaces to have Lipschitz numerical index 1. As an application, we show that real lush spaces and C  -rich subspaces have Lipschitz numerical index 1. Moreover, using the Gâteaux differentiability of Lipschitz operators, we characterize the Lipschitz numerical index of separable Banach spaces with the RNP. Finally, we prove that the Lipschitz numerical index has the stability properties for the _c0$c_0$-, _l1$l_1$-, and _l∞$l_{\infty }$-sums of spaces and vector-valued function spaces. From this, we show that the C(K)$C(K)$ spaces, _L1(μ)$L_1(\mu )$-spaces and _L∞(ν)$L_{\infty }(\nu )$-spaces have Lipschitz numerical index 1.     

A class of spectral self-affine measures with four-element digit sets

The self-affine measure _μM,D${\mu }_{M,D}$ associated with an expanding matrix M∈_Mn(Z)$M\in M_n(\mathbb{Z})$ and a finite digit set D⊂Zⁿ$D\subset {\mathbb{Z}}^n$ is uniquely determined by the self-affine identity with equal weight. In this paper we construct a class of self-affine measures _μM,D${\mu }_{M,D}$ with four-element digit sets in the higher dimensions (n≥3$n\ge 3$) such that the Hilbert space L²(_μM,D)$L^2({\mu }_{M,D})$ possesses an orthogonal exponential basis. That is, _μM,D${\mu }_{M,D}$ is spectral. Such a spectral measure cannot be obtained from the condition of compatible pair. This extends the corresponding result in the plane.     

Polynomial identities for the Jordan algebra of a degenerate symmetric bilinear form

Let _Jn$J_n$ be the Jordan algebra of a degenerate symmetric bilinear form. In the first section we classify all possible G  -gradings on _Jn$J_n$ where G   is any group, while in the second part we restrict our attention to a degenerate symmetric bilinear form of rank n−1$n-1$, where n is the dimension of the vector space V   defining _Jn$J_n$. We prove that in this case the algebra _Jn$J_n$ is PI-equivalent to the Jordan algebra of a nondegenerate bilinear form.     

Bishop–Phelps–Bollobás property for certain spaces of operators

We characterize the Banach spaces Y   for which certain subspaces of operators from _L1(μ)$L_1(\mu )$ into Y have the Bishop–Phelps–Bollobás property in terms of a geometric property of Y, namely AHSP. This characterization applies to the spaces of compact and weakly compact operators. New examples of Banach spaces Y with AHSP are provided. We also obtain that certain ideals of Asplund operators satisfy the Bishop–Phelps–Bollobás property.