On the generation of a smooth Gaussian random variable to 5 standard deviations

2003 ◽  
Author(s):  
G.S. Muller ◽  
C.K. Pauw
Keyword(s):  
Random Variable ◽  
Gaussian Random Variable ◽  
Standard Deviations

scholarly journals ASYMPTOTIC CRAMÉR TYPE DECOMPOSITION FOR WIENER AND WIGNER INTEGRALS

2013 ◽  
Vol 16 (01) ◽  
pp. 1350005
Author(s):  
SOLESNE BOURGUIN ◽  
JEAN-CHRISTOPHE BRETON
Keyword(s):  
Random Variables ◽  
Random Variable ◽  
Gaussian Random Variable ◽  
Independent Random Variables ◽  
Asymptotic Decomposition ◽  
Type Decomposition

We investigate generalizations of the Cramér theorem. This theorem asserts that a Gaussian random variable can be decomposed into the sum of independent random variables if and only if they are Gaussian. We prove asymptotic counterparts of such decomposition results for multiple Wiener integrals and prove that similar results are true for the (asymptotic) decomposition of the semicircular distribution into free multiple Wigner integrals.

WICK CALCULUS FOR THE SQUARE OF A GAUSSIAN RANDOM VARIABLE WITH APPLICATION TO YOUNG AND HYPERCONTRACTIVE INEQUALITIES

2012 ◽  
Vol 15 (03) ◽  
pp. 1250018 ◽  
Author(s):  
ALBERTO LANCONELLI ◽  
LUIGI SPORTELLI
Keyword(s):  
Gaussian Measure ◽  
Type Inequality ◽  
Random Variable ◽  
Gaussian Random Variable ◽  
Wick Product ◽  
Probabilistic Interpretation ◽  
Chi Square ◽  
Wick Calculus

We investigate a probabilistic interpretation of the Wick product associated to the chi-square distribution in the spirit of the results obtained in Ref. 7 for the Gaussian measure. Our main theorem points out a profound difference from the previously studied Gaussian7 and Poissonian12 cases. As an application, we obtain a Young-type inequality for the Wick product associated to the chi-square distribution which contains as a particular case a known Nelson-type hypercontractivity theorem.

scholarly journals Nondecreasing Lower Bound on the Poisson Cumulative Distribution Function for z Standard Deviations Above the Mean

2013 ◽  
Vol 50 (4) ◽  
pp. 909-917
Author(s):  
M. Bondareva
Keyword(s):  
Distribution Function ◽  
Lower Bound ◽  
Cumulative Distribution Function ◽  
Random Variable ◽  
Step Function ◽  
Cumulative Distribution ◽  
Control Parameters ◽  
Poisson Random Variable ◽  
The Mean ◽  
Standard Deviations

In this paper we discuss a nondecreasing lower bound for the Poisson cumulative distribution function (CDF) at z standard deviations above the mean λ, where z and λ are parameters. This is important because the normal distribution as an approximation for the Poisson CDF may overestimate or underestimate its value. A sharp nondecreasing lower bound in the form of a step function is constructed. As a corollary of the bound's properties, for a given percent α and parameter λ, the minimal z is obtained such that, for any Poisson random variable with the mean greater or equal to λ, its αth percentile is at most z standard deviations above its mean. For Poisson distributed control parameters, the corollary allows simple policies measuring performance in terms of standard deviations from a benchmark.

Nondecreasing Lower Bound on the Poisson Cumulative Distribution Function for z Standard Deviations Above the Mean

2013 ◽  
Vol 50 (04) ◽  
pp. 909-917
Author(s):  
M. Bondareva
Keyword(s):  
Distribution Function ◽  
Lower Bound ◽  
Cumulative Distribution Function ◽  
Random Variable ◽  
Step Function ◽  
Cumulative Distribution ◽  
Control Parameters ◽  
Poisson Random Variable ◽  
The Mean ◽  
Standard Deviations

In this paper we discuss a nondecreasing lower bound for the Poisson cumulative distribution function (CDF) at z standard deviations above the mean λ, where z and λ are parameters. This is important because the normal distribution as an approximation for the Poisson CDF may overestimate or underestimate its value. A sharp nondecreasing lower bound in the form of a step function is constructed. As a corollary of the bound's properties, for a given percent α and parameter λ, the minimal z is obtained such that, for any Poisson random variable with the mean greater or equal to λ, its αth percentile is at most z standard deviations above its mean. For Poisson distributed control parameters, the corollary allows simple policies measuring performance in terms of standard deviations from a benchmark.

scholarly journals Multiplexing: B

2021 ◽  
pp. 59-69
Author(s):  
Jean Walrand
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Characteristic Function ◽  
Central Limit ◽  
Random Variables ◽  
Random Variable ◽  
Gaussian Random Variable ◽  
Wifi Networks ◽  
The Central Limit Theorem ◽  
Communication Links

AbstractChapter 10.1007/978-3-030-49995-2_3 used the Central Limit Theorem to determine the number of users that can safely share a common cable or link. We saw that this result is also fundamental to calculate confidence intervals. In this section, we prove this theorem. A key tool is the characteristic function that provides a simple way to study sums of independent random variables.Section 4.1 introduces the characteristic function and calculates it for a Gaussian random variable. Section 4.2 uses that function to prove the Central Limit Theorem. Section 4.3 uses the characteristic function to calculate the moments of a Gaussian random variable. The sum of squares of Gaussian random variables is a common model of noise in communication links. Section 4.4 proves a remarkable property of such a sum. Section 4.5 shows how to use characteristic functions to approximate binomial and geometric random variables. The error function arises in the calculation of the probability of errors in transmission systems and also in decisions based on random observations. Section 4.6 derives useful approximations of that function. Section 4.7 concludes the chapter with a discussion of an adaptive multiple access protocol similar to one used in WiFi networks.

scholarly journals The Golden Quantizer: The Complex Gaussian Random Variable Case

2018 ◽  
Vol 7 (3) ◽  
pp. 312-315
Author(s):  
Peter Larsson ◽  
Lars K. Rasmussen ◽  
Mikael Skoglund
Keyword(s):  
Random Variable ◽  
Gaussian Random Variable ◽  
Complex Gaussian Random Variable
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