A Review and perspective on surveillance of Bernoulli processes

2011 ◽  
Vol 27 (6) ◽  
pp. 735-752 ◽  
Author(s):  
John L. Szarka ◽  
William H. Woodall
Keyword(s):  
Bernoulli Processes

scholarly journals How linear reinforcement affects Donsker’s theorem for empirical processes

2020 ◽  
Vol 178 (3-4) ◽  
pp. 1173-1192 ◽  
Author(s):  
Jean Bertoin
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Long Range ◽  
Limit Theorems ◽  
Brownian Bridge ◽  
Empirical Processes ◽  
Constant Factor ◽  
Empirical Measure ◽  
Reinforced Random Walks ◽  
Bernoulli Processes

Abstract A reinforcement algorithm introduced by Simon (Biometrika 42(3/4):425–440, 1955) produces a sequence of uniform random variables with long range memory as follows. At each step, with a fixed probability $$p\in (0,1)$$ p ∈ ( 0 , 1 ) , $${\hat{U}}_{n+1}$$ U ^ n + 1 is sampled uniformly from $${\hat{U}}_1, \ldots , {\hat{U}}_n$$ U ^ 1 , … , U ^ n , and with complementary probability $$1-p$$ 1 - p , $${\hat{U}}_{n+1}$$ U ^ n + 1 is a new independent uniform variable. The Glivenko–Cantelli theorem remains valid for the reinforced empirical measure, but not the Donsker theorem. Specifically, we show that the sequence of empirical processes converges in law to a Brownian bridge only up to a constant factor when $$p<1/2$$ p < 1 / 2 , and that a further rescaling is needed when $$p>1/2$$ p > 1 / 2 and the limit is then a bridge with exchangeable increments and discontinuous paths. This is related to earlier limit theorems for correlated Bernoulli processes, the so-called elephant random walk, and more generally step reinforced random walks.

scholarly journals Spike-Based Winner-Take-All Computation: Fundamental Limits and Order-Optimal Circuits

2019 ◽  
Vol 31 (12) ◽  
pp. 2523-2561 ◽  
Author(s):  
Lili Su ◽  
Chia-Jung Chang ◽  
Nancy Lynch
Keyword(s):  
Lower Bound ◽  
Waiting Time ◽  
Success Probability ◽  
Spike Trains ◽  
Decision Time ◽  
Text Input ◽  
Bernoulli Processes ◽  
Winner Take All ◽  
Input Spike ◽  
Take All

Winner-take-all (WTA) refers to the neural operation that selects a (typically small) group of neurons from a large neuron pool. It is conjectured to underlie many of the brain's fundamental computational abilities. However, not much is known about the robustness of a spike-based WTA network to the inherent randomness of the input spike trains. In this work, we consider a spike-based [Formula: see text]–WTA model wherein [Formula: see text] randomly generated input spike trains compete with each other based on their underlying firing rates and [Formula: see text] winners are supposed to be selected. We slot the time evenly with each time slot of length 1 ms and model the [Formula: see text] input spike trains as [Formula: see text] independent Bernoulli processes. We analytically characterize the minimum waiting time needed so that a target minimax decision accuracy (success probability) can be reached. We first derive an information-theoretic lower bound on the waiting time. We show that to guarantee a (minimax) decision error [Formula: see text] (where [Formula: see text]), the waiting time of any WTA circuit is at least [Formula: see text]where [Formula: see text] is a finite set of rates and [Formula: see text] is a difficulty parameter of a WTA task with respect to set [Formula: see text] for independent input spike trains. Additionally, [Formula: see text] is independent of [Formula: see text], [Formula: see text], and [Formula: see text]. We then design a simple WTA circuit whose waiting time is [Formula: see text]provided that the local memory of each output neuron is sufficiently long. It turns out that for any fixed [Formula: see text], this decision time is order-optimal (i.e., it matches the above lower bound up to a multiplicative constant factor) in terms of its scaling in [Formula: see text], [Formula: see text], and [Formula: see text].

Bernoulli Processes with Non-Positive Correlations

2010 ◽  
Vol 39 (17) ◽  
pp. 3191-3201 ◽  
Author(s):  
Jiří Anděl ◽  
Šárka Došlá
Keyword(s):  
Bernoulli Processes ◽  
Positive Correlations
Sign in / Sign up

Export Citation Format

Share Document