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Register a new userA universal probability approximation method: Markov process approach
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We view the classical Lindeberg principle in a Markov process setting to establish a universal probability approximation framework by It^{o}s formula and Markov semigroup. As applications, we consider approximating a family of online stochastic gradient descents (SGDs) by a stochastic differential equation (SDE) driven by additive Brownian motion, and obtain an approximation error with explicit dependence on the dimension which makes it possible to analyse high dimensional models. We also apply our framework to study stable approximation and normal approximation and obtain their optimal convergence rates (up to a logarithmic correction for normal approximation).
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Let $X$ be the constrained random walk on ${mathbb Z}_+^2$ having increments $(1,0)$, $(-1,1)$, $(0,-1)$ with jump probabilities $lambda(M_k)$, $mu_1(M_k)$, and $mu_2(M_k)$ where $M$ is an irreducible aperiodic finite state Markov chain. The process $X$ represents the lengths of two tandem queues with arrival rate $lambda(M_k)$, and service rates $mu_1(M_k)$, and $mu_2(M_k)$. We assume that the average arrival rate with respect to the stationary measure of $M$ is less than the average service rates, i.e., $X$ is assumed stable. Let $tau_n$ be the first time when the sum of the components of $X$ equals $n$ for the first time. Let $Y$ be the random walk on ${mathbb Z} times {mathbb Z}_+$ having increments $(-1,0)$, $(1,1)$, $(0,-1)$ with probabilities $lambda(M_k)$, $mu_1(M_k)$, and $mu_2(M_k)$. Let $tau$ be the first time the components of $Y$ are equal. For $x in {mathbb R}_+^2$, $x(1) + x(2) < 1$, $x(1) > 0$, and $x_n = lfloor nx rfloor$, we show that $P_{(n-x_n(1),x_n(2)),m)}( tau < infty)$ approximates $P_{(x_n,m)}(tau_n < tau_0)$ with exponentially vanishing relative error as $nrightarrow infty$. For the analysis we define a characteristic matrix in terms of the jump probabilities of $(X,M).$ The $0$-level set of the characteristic polynomial of this matrix defines the characteristic surface; conjugate points on this surface and the associated eigenvectors of the characteristic matrix are used to define (sub/super) harmonic functions which play a fundamental role both in our analysis and the computation / approximation of $P_{(y,m)}(tau < infty).$
We build a sequence of empirical measures on the space D(R_+,R^d) of R^d-valued c`adl`ag functions on R_+ in order to approximate the law of a stationary R^d-valued Markov and Feller process (X_t). We obtain some general results of convergence of this sequence. Then, we apply them to Brownian diffusions and solutions to Levy driven SDEs under some Lyapunov-type stability assumptions. As a numerical application of this work, we show that this procedure gives an efficient way of option pricing in stochastic volatility models.
In this note we give a new method for getting a series of approximations for the extinction probability of the one-dimensional contact process by using the Grobner basis.
In this paper we develop the theory of the so-called $mathbf{W}$ and $mathbf{Z}$ scale matrices for (upwards skip-free) discrete-time and discrete-space Markov additive processes, along the lines of the analogous theory for Markov additive processes in continuous-time. In particular, we provide their probabilistic construction, identify the form of the generating function of $mathbf{W}$ and its connection with the occupation mass formula, which provides the tools for deriving semi-explicit expressions for corresponding exit problems for the upward-skip free process and its reflections, in terms the scale matrices.
RNA motifs typically consist of short, modular patterns that include base pairs formed within and between modules. Estimating the abundance of these patterns is of fundamental importance for assessing the statistical significance of matches in genomewide searches, and for predicting whether a given function has evolved many times in different species or arose from a single common ancestor. In this manuscript, we review in an integrated and self-contained manner some basic concepts of automata theory, generating functions and transfer matrix methods that are relevant to pattern analysis in biological sequences. We formalize, in a general framework, the concept of Markov chain embedding to analyze patterns in random strings produced by a memoryless source. This conceptualization, together with the capability of automata to recognize complicated patterns, allows a systematic analysis of problems related to the occurrence and frequency of patterns in random strings. The applications we present focus on the concept of synchronization of automata, as well as automata used to search for a finite number of keywords (including sets of patterns generated according to base pairing rules) in a general text.
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