scholarly journals Bit flipping and time to recover

2016 ◽  
Vol 53 (3) ◽  
pp. 650-666 ◽  
Author(s):  
Anton Muratov ◽  
Sergei Zuyev
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Ground State ◽  
Sufficient Conditions ◽  
Return Time ◽  
Time Step ◽  
Transient Behaviour ◽  
Fractional Moments ◽  
Positive Integers

AbstractWe call `bits' a sequence of devices indexed by positive integers, where every device can be in two states: 0 (idle) and 1 (active). Start from the `ground state' of the system when all bits are in 0-state. In our first binary flipping (BF) model the evolution of the system behaves as follows. At each time step choose one bit from a given distribution P on the positive integers independently of anything else, then flip the state of this bit to the opposite state. In our second damaged bits (DB) model a `damaged' state is added: each selected idling bit changes to active, but selecting an active bit changes its state to damaged in which it then stays forever. In both models we analyse the recurrence of the system's ground state when no bits are active. We present sufficient conditions for both the BF and DB models to show recurrent or transient behaviour, depending on the properties of the distribution P. We provide a bound for fractional moments of the return time to the ground state for the BF model, and prove a central limit theorem for the number of active bits for both models.

scholarly journals The central limit theorem for trigonometric series

1982 ◽  
Vol 87 ◽  
pp. 79-94 ◽  
Author(s):  
Takafumi Murai
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Trigonometric Series ◽  
Central Limit ◽  
Cardinal Number ◽  
Semi Group ◽  
The Central Limit Theorem ◽  
Positive Integers

We denote by Z+ the semi-group of positive integers. For a subset E of Z+, we denote by |E| (≦ + ∞) its cardinal number and by E(n) the intersection of E and an interval [1, n) (n ≧ 1). We shall identify a subset E of Z+ with a sequence, arranging elements of E according to their order.

scholarly journals A martingale central limit theorem without negligibility conditions

1978 ◽  
Vol 18 (1) ◽  
pp. 13-19 ◽  
Author(s):  
Robert J. Adler
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Sufficient Conditions ◽  
Infinitely Divisible ◽  
Martingale Central Limit Theorem ◽  
Triangular Arrays ◽  
Infinitely Divisible Laws ◽  
Martingale Case

We obtain sufficient conditions for the convergence of martingale triangular arrays to infinitely divisible laws with finite variances, without making the usual assumptions of uniform asymptotic negligibility. Our results generalise known results for both the martingale case under a negligibility assumption and the classical (independence) case without such assumptions.

Multiple comparisons and sums of dissociated random variables

1985 ◽  
Vol 17 (1) ◽  
pp. 147-162 ◽  
Author(s):  
A. D. Barbour ◽  
G. K. Eagleson
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Multiple Comparisons ◽  
The Central Limit Theorem ◽  
Normally Distributed

Sufficient conditions for a sum of dissociated random variables to be approximately normally distributed are derived. These results generalize the central limit theorem for U-statistics and provide conditions which can be verified in a number of applications. The method of proof is that due to Stein (1970).

The central limit theorem for m-dependent variables

1955 ◽  
Vol 51 (1) ◽  
pp. 92-95 ◽  
Author(s):  
P. H. Diananda
Keyword(s):  
Standard Deviation ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Limit Theorems ◽  
Sufficient Conditions ◽  
The Central Limit Theorem ◽  
Lindeberg Condition ◽  
Dependent Variables ◽  
Basic Ideas

In a previous paper (4) central limit theorems were obtained for sequences of m-dependent random variables (r.v.'s) asymptotically stationary to second order, the sufficient conditions being akin to the Lindeberg condition (3). In this paper similar theorems are obtained for sequences of m-dependent r.v.'s with bounded variances and with the property that for large n, where s′n is the standard deviation of the nth partial sum of the sequence. The same basic ideas as in (4) are used, but the proofs have been simplified. The results of this paper are examined in relation to earlier ones of Hoeffding and Robbins(5) and of the author (4). The cases of identically distributed r.v.'s and of vector r.v.'s are mentioned.

scholarly journals A Central Limit Theorem for Non-Overlapping Return Times

2006 ◽  
Vol 43 (1) ◽  
pp. 32-47
Author(s):  
Oliver Johnson
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Random Process ◽  
Central Limit ◽  
Return Time ◽  
Negative Association ◽  
Return Times ◽  
Entropy Estimation

Define the non-overlapping return time of a block of a random process to be the number of blocks that pass by before the block in question reappears. We prove a central limit theorem based on these return times. This result has applications to entropy estimation, and to the problem of determining if digits have come from an independent, equidistributed sequence. In the case of an equidistributed sequence, we use an argument based on negative association to prove convergence under conditions weaker than those required in the general case.

Bounded normal approximation in simulations of highly reliable Markovian systems

1999 ◽  
Vol 36 (4) ◽  
pp. 974-986 ◽  
Author(s):  
Bruno Tuffin
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Relative Error ◽  
Normal Approximation ◽  
Sufficient Conditions ◽  
The Central Limit Theorem ◽  
Normal Law ◽  
Markovian Systems ◽  
Necessary And Sufficient

In this paper, we give necessary and sufficient conditions to ensure the validity of confidence intervals, based on the central limit theorem, in simulations of highly reliable Markovian systems. We resort to simulations because of the frequently huge state space in practical systems. So far the literature has focused on the property of bounded relative error. In this paper we focus on ‘bounded normal approximation’ which asserts that the approximation of the normal law, suggested by the central limit theorem, does not deteriorate as the reliability of the system increases. Here we see that the set of systems with bounded normal approximation is (strictly) included in the set of systems with bounded relative error.

A note on the central limit theorem for doubly stochastic Poisson processes

1976 ◽  
Vol 13 (04) ◽  
pp. 809-813
Author(s):  
Holger Rootzén
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Poisson Process ◽  
Central Limit ◽  
Sufficient Conditions ◽  
Poisson Processes ◽  
Doubly Stochastic ◽  
The Central Limit Theorem ◽  
Doubly Stochastic Poisson Process ◽  
Necessary And Sufficient

In this note, necessary and sufficient conditions for the central limit theorem for the number of events in a doubly stochastic Poisson process are given.

Multiple comparisons and sums of dissociated random variables

1985 ◽  
Vol 17 (01) ◽  
pp. 147-162 ◽  
Author(s):  
A. D. Barbour ◽  
G. K. Eagleson
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Multiple Comparisons ◽  
The Central Limit Theorem ◽  
Normally Distributed

Sufficient conditions for a sum of dissociated random variables to be approximately normally distributed are derived. These results generalize the central limit theorem for U-statistics and provide conditions which can be verified in a number of applications. The method of proof is that due to Stein (1970).

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