Fluctuation theory in continuous time

1975 ◽  
Vol 7 (4) ◽  
pp. 705-766 ◽  
Author(s):  
N. H. Bingham
Keyword(s):  
Weak Convergence ◽  
Local Time ◽  
Continuous Time ◽  
Regular Variation ◽  
Heavy Traffic ◽  
Fluctuation Theory ◽  
Convergence Theorems ◽  
Time Fluctuation ◽  
Exit Problems ◽  
Time Theory

Our aim here is to give a survey of that part of continuous-time fluctuation theory which can be approached in terms of functionals of Lévy processes, our principal tools being Wiener-Hopf factorisation and local-time theory. Particular emphasis is given to one- and two-sided exit problems for spectrally negative and spectrally positive processes, and their applications to queues and dams. In addition, we give some weak-convergence theorems of heavy-traffic type, and some tail-estimates involving regular variation.

Fluctuation theory in continuous time

1975 ◽  
Vol 7 (04) ◽  
pp. 705-766 ◽  
Author(s):  
N. H. Bingham
Keyword(s):  
Weak Convergence ◽  
Local Time ◽  
Continuous Time ◽  
Regular Variation ◽  
Heavy Traffic ◽  
Fluctuation Theory ◽  
Convergence Theorems ◽  
Time Fluctuation ◽  
Exit Problems ◽  
Time Theory

Our aim here is to give a survey of that part of continuous-time fluctuation theory which can be approached in terms of functionals of Lévy processes, our principal tools being Wiener-Hopf factorisation and local-time theory. Particular emphasis is given to one- and two-sided exit problems for spectrally negative and spectrally positive processes, and their applications to queues and dams. In addition, we give some weak-convergence theorems of heavy-traffic type, and some tail-estimates involving regular variation.

scholarly journals Crossovers induced by discrete-time quantum walks

2011 ◽  
Vol 11 (9&10) ◽  
pp. 741-760
Author(s):  
Kota Chisaki ◽  
Norio Konno ◽  
Etsuo Segawa ◽  
Yutaka Shikano
Keyword(s):  
Weak Convergence ◽  
Random Walks ◽  
Discrete Time ◽  
Convergence Theorem ◽  
Continuous Time ◽  
Quantum Walk ◽  
Convergence Theorems ◽  
Final Time ◽  
Time Quantum ◽  
Weak Convergence Theorem

We consider crossovers with respect to the weak convergence theorems from a discrete-time quantum walk (DTQW). We show that a continuous-time quantum walk (CTQW) and discrete- and continuous-time random walks can be expressed as DTQWs in some limits. At first we generalize our previous study [Phys. Rev. A \textbf{81}, 062129 (2010)] on the DTQW with position measurements. We show that the position measurements per each step with probability $p \sim 1/n^\beta$ can be evaluated, where $n$ is the final time and $0<\beta<1$. We also give a corresponding continuous-time case. As a consequence, crossovers from the diffusive spreading (random walk) to the ballistic spreading (quantum walk) can be seen as the parameter $\beta$ shifts from 0 to 1 in both discrete- and continuous-time cases of the weak convergence theorems. Secondly, we introduce a new class of the DTQW, in which the absolute value of the diagonal parts of the quantum coin is proportional to a power of the inverse of the final time $n$. This is called a final-time-dependent DTQW (FTD-DTQW). The CTQW is obtained in a limit of the FTD-DTQW. We also obtain the weak convergence theorem for the FTD-DTQW which shows a variety of spreading properties. Finally, we consider the FTD-DTQW with periodic position measurements. This weak convergence theorem gives a phase diagram which maps sufficiently long-time behaviors of the discrete- and continuous-time quantum and random walks.

Multi-channel queues in heavy traffic

1973 ◽  
Vol 10 (4) ◽  
pp. 769-777 ◽  
Author(s):  
Richard Loulou
Keyword(s):  
Weak Convergence ◽  
Queue Length ◽  
Heavy Traffic ◽  
Convergence Theory ◽  
Intermediate Result ◽  
Convergence Theorems ◽  
Multi Server ◽  
Channel Systems

In this paper, convergence theorems for heavy traffic queues are extended to multi-channel systems under general assumptions. Whitt (1968), and Iglehart and Whitt (1970) have proved weak convergence of the queue length process and the wait process for one-channel queues. Extensions to multi-server queues were established for the queue length process but only partly for the wait process (ρ = 1 only). We give here convergence theorems for the wait process when ρ > 1. Our approach uses weak convergence theory, but is different from previous ones in that we use the virtual delay as an intermediate result. The class of queues considered is more general than GI/G/m.

Multi-channel queues in heavy traffic

1973 ◽  
Vol 10 (04) ◽  
pp. 769-777 ◽  
Author(s):  
Richard Loulou
Keyword(s):  
Weak Convergence ◽  
Queue Length ◽  
Heavy Traffic ◽  
Convergence Theory ◽  
Intermediate Result ◽  
Convergence Theorems ◽  
Multi Server ◽  
Channel Systems

In this paper, convergence theorems for heavy traffic queues are extended to multi-channel systems under general assumptions. Whitt (1968), and Iglehart and Whitt (1970) have proved weak convergence of the queue length process and the wait process for one-channel queues. Extensions to multi-server queues were established for the queue length process but only partly for the wait process (ρ = 1 only). We give here convergence theorems for the wait process when ρ &gt; 1. Our approach uses weak convergence theory, but is different from previous ones in that we use the virtual delay as an intermediate result. The class of queues considered is more general than GI/G/m.

Point processes, regular variation and weak convergence

1986 ◽  
Vol 18 (01) ◽  
pp. 66-138 ◽  
Author(s):  
Sidney I. Resnick
Keyword(s):  
Weak Convergence ◽  
Function Space ◽  
Regular Variation ◽  
Point Processes ◽  
Sufficient Condition ◽  
Space Setting

A method is reviewed for proving weak convergence in a function-space setting when regular variation is a sufficient condition. Point processes and weak convergence techniques involving continuity arguments play a central role. The method is dimensionless and holds computations to a minimum. Many applications of the methods to processes derived from sums and maxima are given.

scholarly journals An iterative process for a hybrid pair of generalized I-asymptotically nonexpansive single-valued mappings and generalized nonexpansive multi-valued mappings in Banach spaces

2018 ◽  
Vol 34 (1) ◽  
pp. 31-45
Author(s):  
ALI FARAJZADEH ◽  
◽  
PREEYANUCH CHUASUK ◽  
ANCHALEE KAEWCHAROEN ◽  
MOHAMMAD MURSALEEN ◽  
...  
Keyword(s):  
Weak Convergence ◽  
Strong Convergence ◽  
Banach Spaces ◽  
Iterative Process ◽  
Finite Family ◽  
Convergence Theorems ◽  
Asymptotically Nonexpansive ◽  
Hybrid Pair

In this paper, an iterative process for a hybrid pair of a finite family of generalized I-asymptotically nonexpansive single-valued mappings and a finite family of generalized nonexpansive multi-valued mappings is established. Moreover, the weak convergence theorems and strong convergence theorems of the proposed iterative process in Banach spaces are proven. The examples are established for supporting our main results. The obtained results can be viewed as an improvement and extension of the several results in the literature.

scholarly journals Conditioning an additive functional of a Markov chain to stay nonnegative. I. Survival for a long time

2005 ◽  
Vol 37 (4) ◽  
pp. 1015-1034 ◽  
Author(s):  
Saul D. Jacka ◽  
Zorana Lazic ◽  
Jon Warren
Keyword(s):  
Markov Chain ◽  
Weak Convergence ◽  
State Space ◽  
Continuous Time ◽  
Additive Functional ◽  
Finite State ◽  
Long Time ◽  
Negative Drift ◽  
Irreducible Markov Chain ◽  
Finite State Space

Let (Xt)t≥0 be a continuous-time irreducible Markov chain on a finite state space E, let v be a map v: E→ℝ\{0}, and let (φt)t≥0 be an additive functional defined by φt=∫0tv(Xs)d s. We consider the case in which the process (φt)t≥0 is oscillating and that in which (φt)t≥0 has a negative drift. In each of these cases, we condition the process (Xt,φt)t≥0 on the event that (φt)t≥0 is nonnegative until time T and prove weak convergence of the conditioned process as T→∞.

Some diffusion models for the mabinogion sheep problem of williams

1996 ◽  
Vol 28 (03) ◽  
pp. 763-783 ◽  
Author(s):  
Terence Chan
Keyword(s):  
Local Time ◽  
Discrete Time ◽  
Continuous Time ◽  
Programming Approach ◽  
Value Functions ◽  
Stochastic Version ◽  
Optimal Boundary ◽  
Dynamic Programming Approach ◽  
Time Problem ◽  
A Free Boundary Problem

The ‘Mabinogion sheep’ problem, originally due to D. Williams, is a nice illustration in discrete time of the martingale optimality principle and the use of local time in stochastic control. The use of singular controls involving local time is even more strikingly highlighted in the context of continuous time. This paper considers a class of diffusion versions of the discrete-time Mabinogion sheep problem. The stochastic version of the Bellman dynamic programming approach leads to a free boundary problem in each case. The most surprising feature in the continuous-time context is the existence of diffusion versions of the original discrete-time problem for which the optimal boundary is different from that in the discrete-time case; even when the optimal boundary is the same, the value functions can be very different.

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