Random variables, sequences, and stochastic processes

The concepts of random error and reliability of measurements that are familiar in traditional theories based on the notions of “true values” and “errors” can be represented by a probability model having a simpler formal structure and fewer special assumptions about random sampling and independence of measurements. In this model formulas that relate observable events are derived from probability axioms and from primitive terms that refer to observable events, without an intermediate structure containing variances and correlations of “true” and “error” components of scores. While more economical in language and formalism, the model at the same time is more general than classical theories and applies to stochastic processes in which joint distributions of many dependent random variables are of interest. In addition, it clarifies some long-standing problems concerning “experimental independence” of measurements and the relation of sampling of individuals to sampling of measurements.

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Laws of large numbers for q-dependent random variables and nonextensive statistical mechanics

International Journal of Modern Physics B ◽

10.1142/s021797921750117x ◽

2017 ◽

Vol 31 (15) ◽

pp. 1750117

Author(s):

Marco A. S. Trindade

Keyword(s):

Statistical Mechanics ◽

Stochastic Processes ◽

Law Of Large Numbers ◽

Random Variables ◽

Nonextensive Statistical Mechanics ◽

Dependent Random Variables ◽

Strong Law ◽

Laws Of Large Numbers ◽

Large Numbers

In this work, we prove a weak law and a strong law of large numbers through the concept of [Formula: see text]-product for dependent random variables, in the context of nonextensive statistical mechanics. Applications for the consistency of estimators are presented and connections with stochastic processes are discussed.

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Time-Dependent Random Variables: Classical Stochastic Processes

Graduate Texts in Physics - Statistical Physics ◽

10.1007/978-3-642-28684-1_5 ◽

2012 ◽

pp. 161-219

Author(s):

Josef Honerkamp

Keyword(s):

Stochastic Processes ◽

Random Variables ◽

Time Dependent ◽

Dependent Random Variables

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Fourier Transforms in Probability, Random Variables and Stochastic Processes

Handbook of Fourier Analysis & Its Applications ◽

10.1093/oso/9780195335927.003.0009 ◽

2009 ◽

Author(s):

Robert J Marks II

Keyword(s):

Distribution Function ◽

Probability Density Function ◽

Fourier Analysis ◽

Stochastic Processes ◽

Probability Density ◽

Density Function ◽

Fourier Transforms ◽

Random Variables ◽

Cumulative Distribution ◽

The Face

In this Chapter, we present application of Fourier analysis to probability, random variables and stochastic processes [1089, 1097, 1387, 1329]. Arandom variable, X, is the assignment of a number to the outcome of a random experiment. We can, for example, flip a coin and assign an outcome of a heads as X = 1 and a tails X = 0. Often the number is equated to the numerical outcome of the experiment, such as the number of dots on the face of a rolled die or the measurement of a voltage in a noisy circuit. The cumulative distribution function is defined by FX(x) = Pr[X ≤ x]. (4.1) The probability density function is the derivative fX(x) = d /dxFX(x). Our treatment of random variables focuses on use of Fourier analysis. Due to this viewpoint, the development we use is unconventional and begins immediately in the next section with discussion of properties of the probability density function.

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Criteria for the non-ergodicity of stochastic processes: application to the exponential back-off protocol

Journal of Applied Probability ◽

10.1017/s0021900200030990 ◽

1987 ◽

Vol 24 (02) ◽

pp. 347-354 ◽

Cited By ~ 1

Author(s):

Guy Fayolle ◽

Rudolph Iasnogorodski

Keyword(s):

Stochastic Process ◽

Markov Chain ◽

Stochastic Processes ◽

State Space ◽

Discrete Time ◽

Random Variables ◽

Countable State Space ◽

Countable State ◽

New Criteria

In this paper, we present some simple new criteria for the non-ergodicity of a stochastic process (Yn ), n ≧ 0 in discrete time, when either the upward or downward jumps are majorized by i.i.d. random variables. This situation is encountered in many practical situations, where the (Yn ) are functionals of some Markov chain with countable state space. An application to the exponential back-off protocol is described.

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