A Difference Diffusion Model with Two Equilibrium States
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A DIFFERENCE DIFFUSION MODEL WITH TWO EQUILIBRIUM STATES D. V. Koroliuk1 and V. S. Koroliuk2
UDC 519.24
Abstract. A difference diffusion model with two equilibrium states is given by the stochastic equations with two components: a predictable one, defined by increments regression function with two equilibrium states, and a stochastic one, which is a martingale difference. A classification of zones is proposed based on the asymptotic properties of the trajectories of statistical experiments. The asymptotic behavior of statistical experiments defined by sums of N sample values as N ® ¥ is investigated. Keywords: discrete Markov diffusion, evolutionary process, classification of equilibrium states, stochastic approximation. 1. PROBLEM STATEMENT The papers [1, 2] analyze the statistical experiments (SE) specified by averaged sums of sampled quantities that take binary values. The statistical experiments are defined by averaged sums of sampled quantities d n ( k ) , 1 £ n £ N , k ³ 0 , independent in aggregate for fixed k ³ 0 and equally distributed for different n Î[1, N ], which take a finite number of values (for simplicity, ±1):
S N (k ) : =
1 N
N
å d r (k ),
r =1
- 1 £ S N ( k ) £ 1, k ³ 0.
(1)
The respective frequency SE are specified by averaged sums
S N± ( k ) : =
1 N
N
å d ±n (k ),
n =1
d n± ( k ): = I {d n ( k ) = ±1}, k ³ 0 .
(2)
Here, random event indicator I ( A ) = 1 if A happens or I ( A ) = 0 if A does not happen. Parameter k ³ 0 denotes the sequence of observation stages and is considered to be discrete time that parametrizes SE dynamics. Obvious balance condition is S N+ ( k ) + S N- ( k ) º 1 "k ³ 0 . The relationships
1 S N ( k ) = S N+ ( k ) - S N- ( k ), S N± ( k ) = [1 ± S N ( k )] 2
(3)
allow analyzing the evolution of SE that are specified by one of the processes defined by (1) or (2). 1
Institute of Telecommunications and Global Information Space, National Academy of Sciences of Ukraine, Kyiv, Ukraine, [email protected]. 2Institute of Mathematics, National Academy of Sciences of Ukraine, Kyiv, Ukraine, [email protected]. Translated from Kibernetika i Sistemnyi Analiz, No. 6, November–December, 2017, pp. 107–117. Original article submitted June 9, 2017. 914
1060-0396/17/5306-0914 ©2017 Springer Science+Business Media New York
In what follows, we will pay the main attention to SE dynamics, which is defined by positive frequencies S N+ ( k ) ,
k ³ 0 . At the same time, it is also useful to analyze the dynamics of binary SE S N ( k ) , k ³ 0 , that are specified by averaged sums (1). The SE dynamics is defined by evolutionary processes (EP) P± ( k ) , k ³ 0 , specified by conditional expectations P± ( k + 1) : = E [ S N± ( k + 1) | S N± ( k ) = P± ( k )], k ³ 0 ,
(4)
C ( k + 1) : = E [ S N ( k + 1) | S N ( k ) = C ( k )], k ³ 0 .
(5)
Relation (3) generates the relationship between EP (4) and (5): C ( k ) = P+ ( k ) - P- ( k ) . With regard for the obvious balance condition P+ ( k ) + P- ( k ) º 1 "k ³ 0 , frequency EP P± ( k ) , k ³ 0 , have the representation
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