Recovery of the correlation function for a stationary case of a discrete-time stochastic process

1995 ◽  
Vol 74 (5) ◽  
pp. 1260-1262
Author(s):  
V. G. Nikitin ◽  
L. G. Chubakov
Keyword(s):  
Stochastic Process ◽  
Correlation Function ◽  
Discrete Time ◽  
Stationary Case

scholarly journals Markov Stochastic Processes in Biology and Mathematics -- the Same, and yet Different

2018 ◽  
Vol 14 (1) ◽  
pp. 7540-7559
Author(s):  
MI lOS lAWA SOKO
Keyword(s):  
Stochastic Process ◽  
Stochastic Processes ◽  
Discrete Time ◽  
Stochastic Matrix ◽  
Random Number Generator ◽  
Point Of View ◽  
State Transitions ◽  
Similar Principle ◽  
Probabilistic Space ◽  
And Mathematics

Virtually every biological model utilising a random number generator is a Markov stochastic process. Numerical simulations of such processes are performed using stochastic or intensity matrices or kernels. Biologists, however, define stochastic processes in a slightly different way to how mathematicians typically do. A discrete-time discrete-value stochastic process may be defined by a function p : X0 × X → {f : Î¥ → [0, 1]}, where X is a set of states, X0 is a bounded subset of X, Î¥ is a subset of integers (here associated with discrete time), where the function p satisfies 0 < p(x, y)(t) < 1 and  EY p(x, y)(t) = 1. This definition generalizes a stochastic matrix. Although X0 is bounded, X may include every possible state and is often infinite. By interrupting the process whenever the state transitions into the X −X0 set, Markov stochastic processes defined this way may have non-quadratic stochastic matrices. Similar principle applies to intensity matrices, stochastic and intensity kernels resulting from considering many biological models as Markov stochastic processes. Class of such processes has important properties when considered from a point of view of theoretical mathematics. In particular, every process from this class may be simulated (hence they all exist in a physical sense) and has a well-defined probabilistic space associated with it.

scholarly journals Chain-referral sampling on stochastic block models

2020 ◽  
Vol 24 ◽  
pp. 718-738
Author(s):  
Thi Phuong Thuy Vo
Keyword(s):  
Stochastic Process ◽  
Markov Chain ◽  
Social Network ◽  
Discrete Time ◽  
Precise Description ◽  
Stochastic Block Model ◽  
Hidden Population ◽  
The People ◽  
A Chain ◽  
Block Models

The discovery of the “hidden population”, whose size and membership are unknown, is made possible by assuming that its members are connected in a social network by their relationships. We explore these groups by a chain-referral sampling (CRS) method, where participants recommend the people they know. This leads to the study of a Markov chain on a random graph where vertices represent individuals and edges connecting any two nodes describe the relationships between corresponding people. We are interested in the study of CRS process on the stochastic block model (SBM), which extends the well-known Erdös-Rényi graphs to populations partitioned into communities. The SBM considered here is characterized by a number of vertices N, a number of communities (blocks) m, proportion of each community π = (π1, …, πm) and a pattern for connection between blocks P = (λkl∕N)(k,l)∈{1,…,m}2. In this paper, we give a precise description of the dynamic of CRS process in discrete time on an SBM. The difficulty lies in handling the heterogeneity of the graph. We prove that when the population’s size is large, the normalized stochastic process of the referral chain behaves like a deterministic curve which is the unique solution of a system of ODEs.

Criteria for the non-ergodicity of stochastic processes: application to the exponential back-off protocol

1987 ◽  
Vol 24 (02) ◽  
pp. 347-354 ◽  
Author(s):  
Guy Fayolle ◽  
Rudolph Iasnogorodski
Keyword(s):  
Stochastic Process ◽  
Markov Chain ◽  
Stochastic Processes ◽  
State Space ◽  
Discrete Time ◽  
Random Variables ◽  
Countable State Space ◽  
Countable State ◽  
New Criteria

In this paper, we present some simple new criteria for the non-ergodicity of a stochastic process (Yn ), n ≧ 0 in discrete time, when either the upward or downward jumps are majorized by i.i.d. random variables. This situation is encountered in many practical situations, where the (Yn ) are functionals of some Markov chain with countable state space. An application to the exponential back-off protocol is described.

scholarly journals On reverberation mapping lag uncertainties

2019 ◽  
Vol 491 (4) ◽  
pp. 6045-6064 ◽  
Author(s):  
Z Yu ◽  
C S Kochanek ◽  
B M Peterson ◽  
Y Zu ◽  
W N Brandt ◽  
...  
Keyword(s):  
Stochastic Process ◽  
Correlation Function ◽  
Light Curve ◽  
Cross Correlation ◽  
Systematic Errors ◽  
Light Curves ◽  
Reverberation Mapping ◽  
Uncertainty Estimates ◽  
Gaussian Statistics ◽  
The Continuum

ABSTRACT We broadly explore the effects of systematic errors on reverberation mapping lag uncertainty estimates from javelin and the interpolated cross-correlation function (ICCF) method. We focus on simulated light curves from random realizations of the light curves of five intensively monitored AGNs. Both methods generally work well even in the presence of systematic errors, although javelin generally provides better error estimates. Poorly estimated light-curve uncertainties have less effect on the ICCF method because, unlike javelin , it does not explicitly assume Gaussian statistics. Neither method is sensitive to changes in the stochastic process driving the continuum or the transfer function relating the line light curve to the continuum. The only systematic error we considered that causes significant problems is if the line light curve is not a smoothed and shifted version of the continuum light curve but instead contains some additional sources of variability.

Optional Correlation Function Method in Kriging for Stochastic Optimal Design

2008 ◽  
Vol 580-582 ◽  
pp. 609-612
Author(s):  
Jong Bin Im ◽  
Young Hee Ro ◽  
Soo Yong Lee ◽  
Jung Sun Park
Keyword(s):  
Stochastic Process ◽  
Correlation Function ◽  
Function Method ◽  
Optimization Models ◽  
Space Filling ◽  
Adaptation Algorithm ◽  
Localization Method ◽  
Classical Design ◽  
Space Filling Design ◽  
Sequential Adaptation

This paper is focused on the three strategies to improve efficiency and accuracy of approximate optimization models using Kriging. The strategies are performed by the stochastic process which is called stochastic-localization method as the criterion to move the local domains and the design of experiments, the classical design and space-filling design. We also propose the methodology conducted by the max-min reused sampling and a sequential adaptation algorithm of correlation functions. The proposed strategies are applied to the known analytical function such as Sandgren’s pressure vessel and three-bar truss for practical examples.

The behavior of the ratio of a small-noise Markov chain to its deterministic approximation

1985 ◽  
Vol 17 (4) ◽  
pp. 731-747
Author(s):  
Norman Kaplan ◽  
Thomas Darden
Keyword(s):  
Stochastic Process ◽  
Markov Chain ◽  
Discrete Time ◽  
Small Noise ◽  
Fisher Model ◽  
Image Position

For each N≧1, let {XN(t, x), t≧0} be a discrete-time stochastic process with XN(0) = x. Let FN(y) = E(XN(t + 1) | XN(t) = y), and define YN(t, x) = FN(YN(t – 1, x)), t≧1 and YN(0, x) = x. Assume that in a neighborhood of the origin FN(y) = mNy(l + O(y)) where mN> 1, and define for δ> 0 and x> 0, υN(δ, x) = inf{t:xmtN>δ}. Conditions are given under which, for θ> 0 and ε> 0, there exist constants δ > 0 and L <∞, depending on εand 0, such that This result together with a result of Kurtz (1970), (1971) shows that, under appropriate conditions, the time needed for the stochastic process {XN(t, 1/N), t≧0} to escape a δ -neighborhood of the origin is of order log Νδ /log mN. To illustrate the results the Wright-Fisher model with selection is considered.

Generalized Levinson–Durbin Sequences and Binomial Coefficients

Integers ◽  
2009 ◽  
Vol 9 (2) ◽  
Author(s):  
Paul Shaman
Keyword(s):  
Stochastic Process ◽  
Sample Size ◽  
Mean Square Error ◽  
Discrete Time ◽  
Minimum Mean Square Error ◽  
Stationary Stochastic Process ◽  
Binomial Coefficients ◽  
Stationary Processes ◽  
Mean Square ◽  
Special Case

AbstractThe Levinson–Durbin recursion is used to construct the coefficients which define the minimum mean square error predictor of a new observation for a discrete time, second-order stationary stochastic process. As the sample size varies, the coefficients determine what is called a Levinson–Durbin sequence. A generalized Levinson–Durbin sequence is also defined, and we note that binomial coefficients constitute a special case of such a sequence. Generalized Levinson–Durbin sequences obey formulas which generalize relations satisfied by binomial coefficients. Some of these results are extended to vector stationary processes.

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