The stochastic equation Yt+1 = AtYt + Bt with non-stationary coefficients

2001 ◽  
Vol 38 (1) ◽  
pp. 80-94 ◽  
Author(s):  
Ulrich Horst
Keyword(s):  
Global Convergence ◽  
Random Environment ◽  
Linear Relation ◽  
Stochastic Equation ◽  
Stationary Process ◽  
Sufficient Conditions ◽  
Convergence Result ◽  
Stationary Case ◽  
Finite Dimensional ◽  
Long Run

In this paper, we consider the stochastic sequence {Yt}t∊ℕ defined recursively by the linear relation Yt+1 = AtYt + Bt in a random environment which is described by the non-stationary process {(At, Bt)}t∊ℕ. We formulate sufficient conditions on the environment which ensure that the finite-dimensional distributions of {Yt}t∊ℕ converge weakly to the finite-dimensional distributions of a unique stationary process. If the driving sequence {(At, Bt)}t∊ℕ becomes stationary in the long run, then we can establish a global convergence result. This extends results of Brandt (1986) and Borovkov (1998) from the stationary to the non-stationary case.

The stochastic equation Y t+1 = A t Y t + B t with non-stationary coefficients

2001 ◽  
Vol 38 (01) ◽  
pp. 80-94
Author(s):  
Ulrich Horst
Keyword(s):  
Global Convergence ◽  
Random Environment ◽  
Linear Relation ◽  
Stochastic Equation ◽  
Stationary Process ◽  
Sufficient Conditions ◽  
Convergence Result ◽  
Stationary Case ◽  
Finite Dimensional ◽  
Long Run

In this paper, we consider the stochastic sequence {Y t } t∊ℕ defined recursively by the linear relation Y t+1 = A t Y t + B t in a random environment which is described by the non-stationary process {(A t , B t )} t∊ℕ. We formulate sufficient conditions on the environment which ensure that the finite-dimensional distributions of {Y t } t∊ℕ converge weakly to the finite-dimensional distributions of a unique stationary process. If the driving sequence {(A t , B t )} t∊ℕ becomes stationary in the long run, then we can establish a global convergence result. This extends results of Brandt (1986) and Borovkov (1998) from the stationary to the non-stationary case.

Stability of linear stochastic difference equations in strategically controlled random environments

2003 ◽  
Vol 35 (4) ◽  
pp. 961-981 ◽  
Author(s):  
Ulrich Horst
Keyword(s):  
Stochastic Process ◽  
Random Environment ◽  
Linear Relation ◽  
Nash Equilibria ◽  
Stochastic Game ◽  
Sufficient Conditions ◽  
Stationary Regime ◽  
Random Environments ◽  
Simultaneous Control ◽  
Markov Strategies

We consider the stochastic sequence {Yt}t∈ℕ defined recursively by the linear relation Yt+1=AtYt+Bt in a random environment. The environment is described by the stochastic process {(At,Bt)}t∈ℕ and is under the simultaneous control of several agents playing a discounted stochastic game. We formulate sufficient conditions on the game which ensure the existence of Nash equilibria in Markov strategies which have the additional property that, in equilibrium, the process {Yt}t∈ℕ converges in distribution to a stationary regime.

scholarly journals Asymptotics of locally-interacting Markov chains with global signals

2002 ◽  
Vol 34 (2) ◽  
pp. 416-440
Author(s):  
Ulrich Horst
Keyword(s):  
Markov Chains ◽  
Random Systems ◽  
Infinite Product ◽  
Sufficient Conditions ◽  
Convergence Result ◽  
Microscopic Level ◽  
Product Spaces ◽  
Macroscopic Level ◽  
Long Run ◽  
Sufficient Conditions For Convergence

We study the long-run behaviour of interactive Markov chains on infinite product spaces. The behaviour at a single site is influenced by the local situation in some neighbourhood and by a random signal about the average situation throughout the whole system. The asymptotic behaviour of such Markov chains is analyzed on the microscopic level and on the macroscopic level of empirical fields. We give sufficient conditions for convergence on the macroscopic level. Combining a convergence result from the theory of random systems with complete connections with a perturbation of the Dobrushin-Vasserstein contraction technique, we show that macroscopic convergence implies that the underlying microscopic process has local asymptotic loss of memory.

scholarly journals Asymptotics of locally-interacting Markov chains with global signals

2002 ◽  
Vol 34 (02) ◽  
pp. 416-440 ◽  
Author(s):  
Ulrich Horst
Keyword(s):  
Markov Chains ◽  
Random Systems ◽  
Infinite Product ◽  
Sufficient Conditions ◽  
Convergence Result ◽  
Microscopic Level ◽  
Product Spaces ◽  
Macroscopic Level ◽  
Long Run ◽  
Sufficient Conditions For Convergence

We study the long-run behaviour of interactive Markov chains on infinite product spaces. The behaviour at a single site is influenced by the local situation in some neighbourhood and by a random signal about the average situation throughout the whole system. The asymptotic behaviour of such Markov chains is analyzed on the microscopic level and on the macroscopic level of empirical fields. We give sufficient conditions for convergence on the macroscopic level. Combining a convergence result from the theory of random systems with complete connections with a perturbation of the Dobrushin-Vasserstein contraction technique, we show that macroscopic convergence implies that the underlying microscopic process has local asymptotic loss of memory.

Stability of linear stochastic difference equations in strategically controlled random environments

2003 ◽  
Vol 35 (04) ◽  
pp. 961-981 ◽  
Author(s):  
Ulrich Horst
Keyword(s):  
Stochastic Process ◽  
Random Environment ◽  
Linear Relation ◽  
Nash Equilibria ◽  
Stochastic Game ◽  
Sufficient Conditions ◽  
Stationary Regime ◽  
Random Environments ◽  
Simultaneous Control ◽  
Markov Strategies

We consider the stochastic sequence {Y t } t∈ℕ defined recursively by the linear relation Y t+1=A t Y t +B t in a random environment. The environment is described by the stochastic process {(A t ,B t )} t∈ℕ and is under the simultaneous control of several agents playing a discounted stochastic game. We formulate sufficient conditions on the game which ensure the existence of Nash equilibria in Markov strategies which have the additional property that, in equilibrium, the process {Y t } t∈ℕ converges in distribution to a stationary regime.

scholarly journals Causality Analysis of Google Trends and Dengue Incidence in Bandung, Indonesia With Linkage of Digital Data Modeling: Longitudinal Observational Study

10.2196/17633 ◽  
2020 ◽  
Vol 22 (7) ◽  
pp. e17633 ◽  
Author(s):  
Muhammad Syamsuddin ◽  
Muhammad Fakhruddin ◽  
Jane Theresa Marlen Sahetapy-Engel ◽  
Edy Soewono
Keyword(s):  
Linear Combination ◽  
Stationary Process ◽  
Impulse Response Function ◽  
Google Trends ◽  
Digital Data ◽  
Causality Analysis ◽  
Long Term Effects ◽  
Long Run ◽  
Dengue Outbreaks ◽  
Long Run Equilibrium

Background The popularity of dengue can be inferred from Google Trends that summarizes Google searches of related topics. Both the disease and its Google Trends have a similar source of causation in the dengue virus, leading us to hypothesize that dengue incidence and Google Trends results have a long-run equilibrium. Objective This research aimed to investigate the properties of this long-run equilibrium in the hope of using the information derived from Google Trends for the early detection of upcoming dengue outbreaks. Methods This research used the cointegration method to assess a long-run equilibrium between dengue incidence and Google Trends results. The long-run equilibrium was characterized by their linear combination that generated a stationary process. The Dickey-Fuller test was adopted to check the stationarity of the processes. An error correction model (ECM) was then adopted to measure deviations from the long-run equilibrium to examine the short-term and long-term effects. The resulting models were used to determine the Granger causality between the two processes. Additional information about the two processes was obtained by examining the impulse response function and variance decomposition. Results The Dickey-Fuller test supported an implicit null hypothesis that the dengue incidence and Google Trends results are nonstationary processes (P=.01). A further test showed that the processes were cointegrated (P=.01), indicating that their particular linear combination is a stationary process. These results permitted us to construct ECMs. The model showed the direction of causality of the two processes, indicating that Google Trends results will Granger-cause dengue incidence (not in the reverse order). Conclusions Various hypothesis testing results in this research concluded that Google Trends results can be used as an initial indicator of upcoming dengue outbreaks.

scholarly journals Functional ARCH and GARCH Models: A Yule-Walker Approach

2020 ◽  
Author(s):  
Sebastian Kühnert
Keyword(s):  
Financial Time Series ◽  
Sufficient Conditions ◽  
Dimensional Subspace ◽  
Stationary Solutions ◽  
Finite Dimensional Subspace ◽  
Estimation Errors ◽  
Linear Processes ◽  
Finite Dimensional ◽  
Garch Processes ◽  
Asymptotic Upper Bound

Conditional heteroskedastic financial time series are commonly modelled by ARCH and GARCH. ARCH(1) and GARCH processes were recently extended to the function spaces C[0,1] and L2[0,1], their probabilistic features were studied and their parameters were estimated. The projections of the operators on finite-dimensional subspace were estimated, as were the complete operators in GARCH(1,1). An explicit asymptotic upper bound of the estimation errors was stated in ARCH(1). This article provides sufficient conditions for the existence of strictly stationary solutions, weak dependence and finite moments of ARCH and GARCH processes in various Lp[0,1] spaces, C[0,1] and other spaces. In L2[0,1] we deduce explicit asymptotic upper bounds of the estimation errors for the shift term and the complete operators in ARCH and GARCH and for the projections of the operators on a finite-dimensional subspace in ARCH. The operator estimaton is based on Yule-Walker equations. The estimation of the GARCH operators also involves a result concerning the estimation of the operators in invertible, linear processes which is valid beyond the scope of ARCH and GARCH. Through minor modifications, all results in this article regarding functional ARCH and GARCH can be transferred to functional ARMA.

Alternating projections and interpolation of stationary processes

1992 ◽  
Vol 29 (4) ◽  
pp. 921-931 ◽  
Author(s):  
Mohsen Pourahmadi
Keyword(s):  
Missing Values ◽  
Stationary Process ◽  
Dimensional Space ◽  
Stationary Processes ◽  
Finite Dimensional Space ◽  
Alternating Projections ◽  
Von Neumann ◽  
Finite Dimensional ◽  
Explicit Formulae ◽  
Linear Interpolator

By using the alternating projection theorem of J. von Neumann, we obtain explicit formulae for the best linear interpolator and interpolation error of missing values of a stationary process. These are expressed in terms of multistep predictors and autoregressive parameters of the process. The key idea is to approximate the future by a finite-dimensional space.

On the Integral Extensions of Quadratic Forms Over Local Fields

1970 ◽  
Vol 22 (2) ◽  
pp. 297-307 ◽  
Author(s):  
Melvin Band
Keyword(s):  
Prime Ideal ◽  
Local Field ◽  
Quadratic Forms ◽  
Sufficient Conditions ◽  
Local Fields ◽  
Ring Of Integers ◽  
Finite Dimensional ◽  
Integral Analogue ◽  
Necessary And Sufficient ◽  
Special Case

Let F be a local field with ring of integers and unique prime ideal (p). Suppose that V a finite-dimensional regular quadratic space over F, W and W′ are two isometric subspaces of V (i.e. τ: W → W′ is an isometry from W to W′). By the well-known Witt's Theorem, τ can always be extended to an isometry σ ∈ O(V).The integral analogue of this theorem has been solved over non-dyadic local fields by James and Rosenzweig [2], over the 2-adic fields by Trojan [4], and partially over the dyadics by Hsia [1], all for the special case that W is a line. In this paper we give necessary and sufficient conditions that two arbitrary dimensional subspaces W and W′ are integrally equivalent over non-dyadic local fields.

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