Large deviations for the empirical process of a symmetric measure: a lower bound

2004 ◽  
Vol 66 (2) ◽  
pp. 197-204 ◽  
Author(s):  
Tryfon Daras
Keyword(s):  
Lower Bound ◽  
Large Deviations ◽  
Empirical Process ◽  
Symmetric Measure

scholarly journals Large deviations for acyclic networks of queues with correlated Gaussian inputs

2021 ◽  
Author(s):  
Martin Zubeldia ◽  
Michel Mandjes
Keyword(s):  
Lower Bound ◽  
Large Deviations ◽  
Decay Rate ◽  
Exponential Decay ◽  
Gaussian Processes ◽  
Sample Path ◽  
Decay Rates ◽  
Single Server ◽  
Exponential Decay Rate ◽  
Overflow Probability

AbstractWe consider an acyclic network of single-server queues with heterogeneous processing rates. It is assumed that each queue is fed by the superposition of a large number of i.i.d. Gaussian processes with stationary increments and positive drifts, which can be correlated across different queues. The flow of work departing from each server is split deterministically and routed to its neighbors according to a fixed routing matrix, with a fraction of it leaving the network altogether. We study the exponential decay rate of the probability that the steady-state queue length at any given node in the network is above any fixed threshold, also referred to as the ‘overflow probability’. In particular, we first leverage Schilder’s sample-path large deviations theorem to obtain a general lower bound for the limit of this exponential decay rate, as the number of Gaussian processes goes to infinity. Then, we show that this lower bound is tight under additional technical conditions. Finally, we show that if the input processes to the different queues are nonnegatively correlated, non-short-range dependent fractional Brownian motions, and if the processing rates are large enough, then the asymptotic exponential decay rates of the queues coincide with the ones of isolated queues with appropriate Gaussian inputs.

scholarly journals Asymptotic Bahavior of the Moran Particle System

2013 ◽  
Vol 45 (2) ◽  
pp. 379-397
Author(s):  
Shui Feng ◽  
Jie Xiong
Keyword(s):  
Asymptotic Behavior ◽  
Large Deviations ◽  
Random Sampling ◽  
Limit Theorems ◽  
Empirical Process ◽  
Particle System ◽  
Sampling Rate ◽  
Interacting Particle ◽  
Independent Motion ◽  
Number Of Particles

The asymptotic behavior is studied for an interacting particle system that involves independent motion and random sampling. For a fixed sampling rate, the empirical process of the particle system converges to the Fleming-Viot process when the number of particles approaches ∞. If the sampling rate approaches 0 as the number of particles becomes large, the corresponding empirical process will converge to the deterministic flow of the motion. In the main results of this paper, we study the corresponding central limit theorems and large deviations. Both the Gaussian limits and the large deviations depend on the sampling scales explicitly.

Sample mean based index policies by O(log n) regret for the multi-armed bandit problem

1995 ◽  
Vol 27 (04) ◽  
pp. 1054-1078 ◽  
Author(s):  
Rajeev Agrawal
Keyword(s):  
Lower Bound ◽  
Large Deviations ◽  
Probability Distributions ◽  
Infinite Horizon ◽  
Exponential Families ◽  
Contraction Principle ◽  
Bandit Problem ◽  
Sample Mean ◽  
Seminal Work ◽  
Index Policies

We consider a non-Bayesian infinite horizon version of the multi-armed bandit problem with the objective of designing simple policies whose regret increases slowly with time. In their seminal work on this problem, Lai and Robbins had obtained a O(log n) lower bound on the regret with a constant that depends on the Kullback–Leibler number. They also constructed policies for some specific families of probability distributions (including exponential families) that achieved the lower bound. In this paper we construct index policies that depend on the rewards from each arm only through their sample mean. These policies are computationally much simpler and are also applicable much more generally. They achieve a O(log n) regret with a constant that is also based on the Kullback–Leibler number. This constant turns out to be optimal for one-parameter exponential families; however, in general it is derived from the optimal one via a ‘contraction' principle. Our results rely entirely on a few key lemmas from the theory of large deviations.

Sample mean based index policies by O(log n) regret for the multi-armed bandit problem

1995 ◽  
Vol 27 (4) ◽  
pp. 1054-1078 ◽  
Author(s):  
Rajeev Agrawal
Keyword(s):  
Lower Bound ◽  
Large Deviations ◽  
Probability Distributions ◽  
Infinite Horizon ◽  
Exponential Families ◽  
Contraction Principle ◽  
Bandit Problem ◽  
Sample Mean ◽  
Seminal Work ◽  
Index Policies

We consider a non-Bayesian infinite horizon version of the multi-armed bandit problem with the objective of designing simple policies whose regret increases slowly with time. In their seminal work on this problem, Lai and Robbins had obtained a O(log n) lower bound on the regret with a constant that depends on the Kullback–Leibler number. They also constructed policies for some specific families of probability distributions (including exponential families) that achieved the lower bound. In this paper we construct index policies that depend on the rewards from each arm only through their sample mean. These policies are computationally much simpler and are also applicable much more generally. They achieve a O(log n) regret with a constant that is also based on the Kullback–Leibler number. This constant turns out to be optimal for one-parameter exponential families; however, in general it is derived from the optimal one via a ‘contraction' principle. Our results rely entirely on a few key lemmas from the theory of large deviations.

scholarly journals Asymptotic Bahavior of the Moran Particle System

2013 ◽  
Vol 45 (02) ◽  
pp. 379-397
Author(s):  
Shui Feng ◽  
Jie Xiong
Keyword(s):  
Asymptotic Behavior ◽  
Large Deviations ◽  
Random Sampling ◽  
Limit Theorems ◽  
Empirical Process ◽  
Particle System ◽  
Sampling Rate ◽  
Interacting Particle ◽  
Independent Motion ◽  
Number Of Particles

The asymptotic behavior is studied for an interacting particle system that involves independent motion and random sampling. For a fixed sampling rate, the empirical process of the particle system converges to the Fleming-Viot process when the number of particles approaches∞. If the sampling rate approaches 0 as the number of particles becomes large, the corresponding empirical process will converge to the deterministic flow of the motion. In the main results of this paper, we study the corresponding central limit theorems and large deviations. Both the Gaussian limits and the large deviations depend on the sampling scales explicitly.

Fundamental Research into Thin Films in a Developing Country

1972 ◽  
Vol 30 ◽  
pp. 500-501
Author(s):  
J.A. Eades ◽  
E. Grünbaum
Keyword(s):  
Thin Films ◽  
Growth Process ◽  
Empirical Process ◽  
Nucleation And Growth ◽  
Research Groups ◽  
Film Research ◽  
Fundamental Research ◽  
Controlled Deposition ◽  
The One ◽  
The University

In the last decade and a half, thin film research, particularly research into problems associated with epitaxy, has developed from a simple empirical process of determining the conditions for epitaxy into a complex analytical and experimental study of the nucleation and growth process on the one hand and a technology of very great importance on the other. During this period the thin films group of the University of Chile has studied the epitaxy of metals on metal and insulating substrates. The development of the group, one of the first research groups in physics to be established in the country, has parallelled the increasing complexity of the field.The elaborate techniques and equipment now needed for research into thin films may be illustrated by considering the plant and facilities of this group as characteristic of a good system for the controlled deposition and study of thin films.

Sign in / Sign up

Export Citation Format

Share Document