Results 261 to 270 of about 168,667 (308)

Sums of Independent Random Variables

1997
In Example 3 of Chapter 1 we calculated the probability that exactly k heads appear in n flips of a fair coin. In view of the construction of that example and the definition of independence given in the preceding chapter, we see that what we calculated is the distribution of the sum of n independent random variables, each of which has the Bernoulli ...
Bert Fristedt, Lawrence Gray
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Sums of Independent Random Variables

1991
Sums of independent random variables already appeared in the preceding chapters in some concrete situations (Gaussian and Rademacher averages, representation of stable random variables). On the intuitive basis of central limit theorems which approximate normalized sums of independent random variables by smooth limiting distributions (Gaussian, stable),
Michel Ledoux, Michel Talagrand
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Sums of Independent Random Variables

2000
Many of the important uses of Probability Theory flow from the study of sums of independent random variables. A simple example is from Statistics: if we perform an experiment repeatedly and independently, then the “average value” is given by \( \bar x = \frac{1} {n}\sum\nolimits_{j = 1}^n {X_j } \) where X j represents the outcome of the jth experiment.
Jean Jacod, Philip Protter
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Sums of Independent Random Variables

1999
The theory of decoupling aims at reducing the level of dependence in certain problems by means of inequalities that compare the original sequence to one involving independent random variables. It is therefore important to have information on results dealing with functionals of independent random variables.
Víctor H. de la Peña, Evarist Giné
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On Sums of Random Variables and Independence

The American Statistician, 1986
Abstract Suppose that the density of the sum of two random variables X and Y is given by the convolution of the two marginal densities. Although this condition is stronger than uncorrelatedness of X and Y, it does not imply stochastic independence, as is shown by three examples. A situation for which this fact may be relevant occurs in the construction
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Estimates of the Moments of Sums of Independent Random Variables

Theory of Probability & Its Applications, 1985
Let \(X_ i\), \(i\geq 1\), be independent random variables with zero mean, \(S_ n=X_ 1+...+X_ n\), \(A_{t,n}=E(| X_ 1|^ t+...+| X_ n|^ t)\), \(B_ n=A^{1/2}_{2,n}\). An exact (but complicated) upper bound for \(E| S_ n|^ t\) is given in terms of \(A_{t,n}\) and \(B_ n\). The optimality of other earlier proved (and more simple) upper bounds for \(E| S_ n|
Pinelis, I. F., Utev, S. A.
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Sums of Independent Random Variables

2010
The classical large-sample theory is about the sum of independent random variables. Even though large-sample techniques have expanded well beyond the classical theory, the foundation set up by the latter remains the best way to understand and further explore elements of large-sample theory.
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Sums of Independent Random Variables

1977
Two properties play a basic role in the study of independent r.v.’s: the Borel zero-one law and the multiplication theorem for expectations. Two general a.s. limit problems for sums of independent r.v.’s have been investigated: the a.s. convergence problem and the a.s. stability problem.
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Sums of Independent Random Variables

1986
This chapter collects a few facts, of the central limit theorem type, about distributions of sums of independent random variables.
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Sums of Independent Random Variables

1978
Of paramount concern in probability theory is the behavior of sums {Snn ≥1} of independent random variables {Xii ≥ 1}. The case where the {Xi} are i.i.d. is of especial interest and frequently lends itself to more incisive results. The sequence of sums {Snn ≥ 1} of i.i.d.
Yuan Shih Chow, Henry Teicher
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