## Stochastically recursive sequences and their generalizations (1992)

Venue: | Siberian Adv. Math |

Citations: | 41 - 11 self |

### BibTeX

@ARTICLE{Borovkov92stochasticallyrecursive,

author = {A. A. Borovkov and S. G. Foss},

title = {Stochastically recursive sequences and their generalizations},

journal = {Siberian Adv. Math},

year = {1992}

}

### Years of Citing Articles

### OpenURL

### Abstract

The paper deals with the stochastically recursive sequences { X ( n) } defined as the solutions of equations X ( n + 1) = f ( X ( n) , ξn) (where ξn is a given random sequence), and with random sequences of a more general nature, named recursive chains. For those the theorems of existence, ergodicity, stability are established, the stationary majorants are constructed. Continuous-time processes associated with ones studied here are considered as well. Key words and phrases: stochastically recursive sequence; recursive chain; generalized Markov chain; renovating event; coupling-convergence; ergodicity; stability; rate of convergence; stationary majorants; boundedness in probability; processes admitting embedded stochastically recursive sequences. CHAPTER 1.

### Citations

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Citation Context ...g ( Z ( t + L ) ) = ∫ d H ( u ) F ( t − u ) + E ( g ( Z ( t + L ) ) ; γ1 > t ) , o and to prove (2) it suffices to check whether the fact that the function F ( u ) is directly Riemann integrable (see =-=[30]-=-). Condition 2) implies that trajectories of the process ϕ ( u ) are right-continuous with probability one. Thus, by the Lebesgue majorated convergence theorem on limit transition under the integral s... |

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Citation Context ... n ) , X ( 0 ) ≤ M < ∞ be a SRS with driver { ξn }. If there exist a number C and stationary sequences { Ψn } , { ϕn } measurable with respect to F ξ , such that 1) E Ψn < 0 , ϕn ≥ 0 a.s., E ϕn < ∞ , =-=(11)-=- 2) X ( n + 1 ) − X ( n ) ≤ Ψn + ϕn ⋅ I ( X ( n ) ≤ C ) a.s. (12) for all n, then there exists an a.e. finite stationary majorant { L n } for the sequence { X ( n ) }. R e m a r k 1 . The statement of... |

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Citation Context ... characterized by the fact that not the value of X ( n + 1 ) itself but only its conditional distribution with respect to the entire prehistory is a function of ( X ( n ) , ξ n ). SRS were studied in =-=[1]-=--[5] and other works, RC were introduced in [6], but they are treated systematically for the first time in the present paper. The sequences of types (a) and (b) are frequently encountered in applicati... |

103 |
A new approach to the limit theory of recurrent Markov chains
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Citation Context ...oints for the study of SRS and MC were completely different. However, it appears that there are some common features. Moreover, the ideas of artificial regeneration construction for MC, introduced in =-=[7]-=-, [8], can be effectively used in the theory of SRS and RC, so that the general ergodicity conditions for MC and for the processes defined above can be made rather close both formally and in essence. ... |

69 |
Asymtotic Methods in Queueing Theory
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Citation Context ...th types are more general than the Markov chains (MC) (it is evident, that RC belong to the latter class for ξ n ≡ const, and SRS - for independent { ξ n }). Comparing the study of SRS carried out in =-=[2]-=-, [4] with the MC ergodic theory, one can hardly find anything in common from the first glance. The more so that the starting points for the study of SRS and MC were completely different. However, it ... |

59 |
Stationary stochastic models
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Citation Context ... ln αn ) + + + < ∞ , and E ( ln βn ) < ∞ . The asymptotic properties of sequences of the form X ( n + 1 ) = αn X ( n ) + βn and those of the related processes in continuous time were studied in [24], =-=[25]-=-. Denote σn = ln αn . Theorem 6. If E σn < 0 or σn ≡ 0 and E βn < 0 , then a stationary majorant can be constructed for the sequence { X ( n ) }. The last two relations σn ≡ 0, E βn < 0 signify the re... |

53 |
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Citation Context ...so the embedded MC X (1) . Therefore we may assume, without loss of generality, that the process Z admits an embedded chain X satisfying Conditions (I) - (III) for m = 0. It was noted above (see [7], =-=[8]-=- and also Chapters 2, 4) that the MC X ~ = { X ( n ) , δn } possessing a "positive" atom can be defined on an extended probability space. If we define in addition a random variable δ ( t ) assuming va... |

43 |
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Citation Context ...s on X ( n ) only. Note also that if the process Z admits an embedded MC, it is not necessarily a Markov process. Apparently, for the first time the notion of an embedded MC was introduced by Kendall =-=[32]-=-. The up-to-date literature employs various definitions of processes admitting embedded MC (semi-Markov or, according to Asmussen, regenerative processes, etc.; see [29], [33], and the references in t... |

35 |
Ergodicity and Stability of Stochastic Processes
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Citation Context ..., 0 ) = x ∈ X. Fairly general ergodicity conditions of MC were established in [7]-[11]. There exist several closely resembling versions of these conditions. We dwell on one of them introduced in [6], =-=[12]-=-, [13].Stochastically Recursive Sequences 18 For some set V ∈ B X denote τV ( x ) = min { i ≥ 1 : X ( x , i ) ∈ V }. Suppose that there exist a set V ∈ B X , a probability measure ϕ on ( X , BX ) , a... |

33 | Probability - Shiryaev - 1995 |

31 |
probabilities and stationary queues
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Citation Context ...is a stationary marked point process (SMPP) (viz., ( T ) is a point process and ( ηn ) is a sequence of corresponding stationary "marks"). Definition of SMPP can be encountered, for instance, in [3], =-=[5]-=-, [25]. Theorem 4. If Condition (A) is fulfilled and there exists a stationary sequence of "positive" ξ renovating events An ∈ Fn+ m for SRS X, then there exists a probability measure P ( ⋅ ) on ( X ,... |

18 |
Lecture notes on limit theorems for Markov chain transition probabilities
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Citation Context ...) then this chain satisfies Conditions (I)-(III) and, by Theorem 1, MC X is ergodic. Condition (9) coincides with following one: P ( X ( n + 1 ) ∈ B | X ( n ) , ξn− 1 , ξn− 2 , … ) ≥ p ⋅ ϕ ( B ) a.s. =-=(10)-=- Let now X be a SRS defined by the relations X ( n + 1 ) = f ( X ( n ) , ξ n ), X ( 0 ) = const, where the driver { ξ n } is stationary and metrically transitive. Does (10) imply ergodicity of X in th... |

11 |
Mathematical Methods for Construction for Queueing Models
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(Show Context)
Citation Context ...} , { ζn } of real-valued random variables and an increasing sequence of σ − algebras F n be defined on the same probability space so that 1) X ( n + 1 ) − X ( n ) ≤ ψn + ζn + C1 ⋅ I ( X ( n ) ≤ C2 ) =-=(16)-=- for some C1 , C2 a.s. for all n ≥ 0 ; 2) { ψn } is a stationary metrically transitive sequence; E ψn < 0 ; 3) F n ⊇ σ { ζk ; k ≤ n } ; E ( ζn+ 1 | F n ) ≤ 0 a.s.; 4) sup E ( | ζn | ⋅ g ( | ζn | ) ) ≡... |

10 |
On ergodicity conditions in multi-server queues
- FOSS
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(Show Context)
Citation Context ...deed, P ( µ 0 ≤ n ) = P ( X k ( n ) = X n for all k ≥ 0 ) = = P ( X k+ n ( 0 ) = X 0 for all k ≥ 0 ) = = P ( X l ( 0 ) = X 0 for all l ≥ n ) = P ( ν ≤ n ) . So the following theorem is true (see [4], =-=[18]-=-). Theorem 4. The conditions of Theorem 3 (viz., existence of the stationary sequence of renovating events { A n } with P ( A n ) > 0 ) are necessary and sufficient for sc- convergence sc n X ( n ) → ... |

9 |
Introduction to Queueing Theory
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Citation Context ...as introduced by Kendall [32]. The up-to-date literature employs various definitions of processes admitting embedded MC (semi-Markov or, according to Asmussen, regenerative processes, etc.; see [29], =-=[33]-=-, and the references in these books). Suppose that a MC X = { X ( n ) } satisfies Conditions (I) - (II) (see Chapter 2). Let a number n1 > 0 be such that P ( τV ( ϕ ) = n1 ) ≡ q > 0 . Define a probabi... |

6 |
Ergodicity and stability theorems for a class of stochastic equations and their applications. Theory Prob
- Borovkov
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(Show Context)
Citation Context ... not be considered conventional. The study of SRS started apparently from [1], where the case X = R d was considered, and the function f was assumed to be monotone in the first variable. In [2], [4], =-=[14]-=- the general ergodicity and stability theorems for SRS were proved, which were based on the notion of the so-called renovating (renewing) events. A series of general constructions and assertions for S... |

4 |
Qualitative Analysis of Complex Systems Behaviour by the Test Function Method
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Citation Context ...ressions (2) and (3) converges, for r → ∞ , to the corresponding expression with superscript r omitted. Thus the theorem is proved. Other approaches to the stability problems are considered in [26] - =-=[27]-=-. CHAPTER 7. ERGODICITY OF THE PROCESSES ADMITTING EMBEDDED RECURSIVE CHAINS 1. The main definitions Let Z = { Z ( t ) = Z ( t , x ) , t ∈ T } , Z ( x , 0 ) = x be arbitrary X-valued random processes.... |

3 |
Ergodic Theory of Random Transformations
- Yu
- 1986
(Show Context)
Citation Context ...ed to study multi-server queueing systems. As already mentioned, SRS is a more general object than MC. Namely, any MC can be represented as a SRS with independent { ξn } (for details see Chapter 3 or =-=[19]-=-). ξ ξ ξ Define the σ - algebras Fl , n = σ { ξk ; l ≤ k ≤ n }; Fn = σ { ξk ; k ≤ n } = F− ∞ ,n ; F ξ ξ = σ { ξk ; − ∞ < k < ∞ } = F− ∞ ,∞ . ξ Definition 2. We shall say that an event A ∈ Fn+ m , m ≥ ... |

3 | Regeneration and renovation in queues
- Foss, Kalashnikov
- 1991
(Show Context)
Citation Context ...ochastically Recursive Sequences 80 It is easy to see that the process Z (t) defined by (7) admits the embedded RC X . Ergodicity conditions for the processes of the form (7) were considered in [25], =-=[34]-=-. The processes of virtual waiting times, studied in queueing theory, may be considered as examples of the processes of the form (7). In particular, for systems G ⁄ G ⁄ 1 the virtual waiting time is d... |

2 |
Ergodicity and stability of Markov chains and of their generalizations. Multidimensional chains
- Borovkov
- 1990
(Show Context)
Citation Context ...f X ( n + 1 ) itself but only its conditional distribution with respect to the entire prehistory is a function of ( X ( n ) , ξ n ). SRS were studied in [1]-[5] and other works, RC were introduced in =-=[6]-=-, but they are treated systematically for the first time in the present paper. The sequences of types (a) and (b) are frequently encountered in applications (see, e.g., [1]-[5]), and both types are mo... |

2 |
Ergodicity and stability of multidimensional Markov chains, Theor. Probab
- BOROVKOV
- 1990
(Show Context)
Citation Context ...= x ∈ X. Fairly general ergodicity conditions of MC were established in [7]-[11]. There exist several closely resembling versions of these conditions. We dwell on one of them introduced in [6], [12], =-=[13]-=-.Stochastically Recursive Sequences 18 For some set V ∈ B X denote τV ( x ) = min { i ≥ 1 : X ( x , i ) ∈ V }. Suppose that there exist a set V ∈ B X , a probability measure ϕ on ( X , BX ) , a numbe... |

2 |
Stability estimates for renovative processes
- Kalashnikov
- 1980
(Show Context)
Citation Context ...riables Ψn = F1 ( ξn , ξn− 1 , … ) , ζn ( x ) = F2 ( x , ξn , ξn− 1 , … ) , ϕn = F3 ( ξn , ξn− 1 , … ) satisfy the relations: 1) X ( n + 1 ) − X ( n ) ≤ Ψn + ζn ( X ( n ) ) + ϕn ⋅ I ( X ( n ) ≤ C ) ; =-=(15)-=- 2) E ϕn < ∞ , δ > 0 ; 3) for some δ > 0 sup x E { | ζ0 ( x ) | 2+δ } < ∞ ; ξ 4) for all n ≥ 0 and x E { ζn ( x ) | Fn− 1 } ≤ 0 a.s. If in addition the set V = [ 0 , C ] satisfies Condition (NV) , the... |

2 |
On the ergodicity and stability of the sequence wn+1 = f(wn,ξn): Applications to communication networks
- BOROVKOV
- 1988
(Show Context)
Citation Context ...ural sense, L n ≥ X ( n ) a.s. for all n ≥ 0 , (6) and to indicating such a value of N that P ( L n ≤ N ) > 1 − P ( ( ξ0 , … , ξm ) ∈ Z ). The following sufficient condition for (6) was introduced in =-=[22]-=- (see also Section 5): Theorem 1. Assume that there exist a number N > 0 and a function g1 : Y → R possessing the properties 1) E g1 ( ξ1 ) < 0 , (7) 2) h ( x , y ) ≤ ⎧ g1 ( y ) for x > N , (8) ⎨ ⎩ g1... |

1 |
Asymptotic Methods in Queueing Theory. J.Wiley, ChichesterNew York-Toronto. (Revised translation of [2
- BOROVKOV
- 1984
(Show Context)
Citation Context ...pes are more general than the Markov chains (MC) (it is evident, that RC belong to the latter class for ξ n ≡ const, and SRS - for independent { ξ n }). Comparing the study of SRS carried out in [2], =-=[4]-=- with the MC ergodic theory, one can hardly find anything in common from the first glance. The more so that the starting points for the study of SRS and MC were completely different. However, it appea... |

1 |
Recurrent Markov processes
- HARRIS
- 1955
(Show Context)
Citation Context ...( 0 ) = const, where { ξn } is a sequence of i.i.d. random variables. If we suppose that the MC X satisfies Condition (II) for V = X , m = 0, i.e., P ( X ( n + 1 ) ∈ B | X ( n ) ) ≥ p ⋅ ϕ ( B ) a.s., =-=(9)-=- then this chain satisfies Conditions (I)-(III) and, by Theorem 1, MC X is ergodic. Condition (9) coincides with following one: P ( X ( n + 1 ) ∈ B | X ( n ) , ξn− 1 , ξn− 2 , … ) ≥ p ⋅ ϕ ( B ) a.s. (... |

1 |
On certain method of estimation of the rate of convergence in ergodicity and stability theorems for multiserver queues
- FOSS
- 1985
(Show Context)
Citation Context ... ) } and τ the lapse between two successive realizations of An : P ( τ = k ) = P ( Ak ∩ A _ k− 1 ∩ … ∩ A_ 1 | A0 ). Then we can estimate the rate of convergence in terms of the distribution of τ (see =-=[17]-=-). Theorem 10. The following inequality is valid: Proof. Indeed, P ( µ 0 > n ) ≤ P ( A _ = P ( A 0 ∩ A _ ∞ = ∑ P ( A0 ) P ( A i= n− m _ 1 1 P ( µ 0 > n ) ≤ P ( A 0 ) E ( τ − n + m ) + . 0 ∩ … ∩ A_ n− ... |

1 |
Ergodicity of the synchronized ALOHA
- TSYBAKOV, MIKHAILOV
- 1979
(Show Context)
Citation Context ...erest is a recursive chain. Consider a communication network which is a "random multiple access broadcast channel" [19]; such systems are treated in a fairly extensive collection of works (see, e.g., =-=[20]-=- for more detailed references). Let there be given a transmission channel of the messages (packages) connecting many users. The arrival times are discrete (integer-valued), theStochastically Recursiv... |

1 |
The phenomenon of asymptotic stabilization for decentralized ALOHA protocol. Diffusion approximation
- BOROVKOV
- 1989
(Show Context)
Citation Context ...eous SRS with driver ⎧ ⎨ ⎩ ( ξ n , α n ) ⎫ ⎬ ⎭ . Note that for the case when ⎧ ⎨ ⎩ X ( n ) ⎫ ⎬ ⎭ assume values on the real line (i.e., X = R ), RC were introduced and constructively reduced to SRS in =-=[21]-=-. In the sequel we shall consider only homogeneous RC and SRS. Therefore we omit the word "homogeneous".Stochastically Recursive Sequences 42 CHAPTER 4. ERGODICITY OF RECURSIVE CHAINS 1. General crit... |

1 |
On convergence of semi-Markov multiplication processes with drift to a diffusion process. Theory Probab
- Sh
- 1972
(Show Context)
Citation Context ...1, E ( ln αn ) + + + < ∞ , and E ( ln βn ) < ∞ . The asymptotic properties of sequences of the form X ( n + 1 ) = αn X ( n ) + βn and those of the related processes in continuous time were studied in =-=[24]-=-, [25]. Denote σn = ln αn . Theorem 6. If E σn < 0 or σn ≡ 0 and E βn < 0 , then a stationary majorant can be constructed for the sequence { X ( n ) }. The last two relations σn ≡ 0, E βn < 0 signify ... |

1 |
On the continuity of stochastic sequences generated by recurrent processes. Theory Probab
- ZOLOTAREV
- 1975
(Show Context)
Citation Context ... of expressions (2) and (3) converges, for r → ∞ , to the corresponding expression with superscript r omitted. Thus the theorem is proved. Other approaches to the stability problems are considered in =-=[26]-=- - [27]. CHAPTER 7. ERGODICITY OF THE PROCESSES ADMITTING EMBEDDED RECURSIVE CHAINS 1. The main definitions Let Z = { Z ( t ) = Z ( t , x ) , t ∈ T } , Z ( x , 0 ) = x be arbitrary X-valued random pro... |

1 |
A method for solving a class of recursive stochastic equations
- LISEK
- 1982
(Show Context)
Citation Context .... for n → ∞ (1) be some random sequence. It is natural to expect the ergodicity of the process Z to follow from the ergodicity of the sequence X ( n ) ≡ Z ( Tn ) under fairly general assumptions (see =-=[28]-=- for another approach to ergodicity studies of the processes in continuous time). We shall assume Tn to be stopping times, i.e., for any n , t the event { Tn ≤ t } belongs to the σ − algebra F (t) = σ... |

1 |
Applied Probability and Queues. J.Wiley, Chichester-New York-Toronto
- ASMUSSEN
- 1987
(Show Context)
Citation Context ... L − u ) ) I ( γ1 > t − u ) | D0 ) , 0 where the point w ∈ V is arbitrary. In the sequel we use the same argumentation as in the proof of the ergodicity theorem for regenerative processes (see, e.g., =-=[29]-=-). Introduce the random process ϕ ( u ) = g ( Z ( w , L + u ) ) ⋅ I ( γ 1 > u ) ⋅ I ( D 0 ) ⋅ [ P ( D 0 ) ] − 1 , u ≥ 0, and denote F ( u ) = E ϕ ( u ). Here t E g ( Z ( t + L ) ) = ∫ d H ( u ) F ( t ... |

1 |
Stochastic Processes in Queueing Theory. J.Wiley, ChichesterNew York-Toronto
- BOROVKOV
- 1976
(Show Context)
Citation Context ...ixed b < ∞ . Set R ( u ) = H ( u ) − u ⁄ a . Condition 3) implies that the random variables { ψi } have absolutely continuous components. Thus t ∫ | d R ( u ) | → 0 as t → ∞ for any b < ∞ (see, e.g., =-=[31]-=-), . As for any B ∈ B X t− b I b = | b ∫ 0 F 1 ( u ) d R ( t − u ) | ≤ b ∫ 0 t | d R ( t − u ) | = ∫ | d R ( u ) |, t− b the second statement of the theorem is also proved. In the case of discrete tim... |