On the locations of maxima and minima in a sequence of exchangeable random variables

2021 ◽  
Vol 105 (0) ◽  
pp. 35-50
Author(s):  
D. Ferger
Keyword(s):  
Infinite Sequence ◽  
Random Variables ◽  
Finite Sequence ◽  
Extreme Value ◽  
Extreme Value Statistics ◽  
Classical Approach ◽  
Linear Transformations ◽  
Exchangeable Random Variables

We show for a finite sequence of exchangeable random variables that the locations of the maximum and minimum are independent from every symmetric event. In particular they are uniformly distributed on the grid without the diagonal. Moreover, for an infinite sequence we show that the extrema and their locations are asymptotically independent. Here, in contrast to the classical approach we do not use affine-linear transformations. Moreover it is shown how the new transformations can be used in extreme value statistics.

A note on Spitzer's lemma and its application to the maximum of partial sums of dependent random variables

1976 ◽  
Vol 13 (2) ◽  
pp. 361-364 ◽  
Author(s):  
M. E. Solari ◽  
J. E. A. Dunnage
Keyword(s):  
Infinite Sequence ◽  
Random Variables ◽  
Finite Sequence ◽  
Partial Sums ◽  
Dependent Random Variables ◽  
Finite Segment ◽  
Exchangeable Random Variables ◽  
Partial Sum

We give an expression for the expectation of max (0, S1, …, Sn) where Sk is the kth partial sum of a finite sequence of exchangeable random variables X1, …, Xn. When the Xk are also independent, the formula we give has already been obtained by Spitzer; and when the sequence is a finite segment of an infinite sequence of exchangeable random variables, it is a consequence of a theorem of Hewitt.

A note on Spitzer's lemma and its application to the maximum of partial sums of dependent random variables

1976 ◽  
Vol 13 (02) ◽  
pp. 361-364
Author(s):  
M. E. Solari ◽  
J. E. A. Dunnage
Keyword(s):  
Infinite Sequence ◽  
Random Variables ◽  
Finite Sequence ◽  
Partial Sums ◽  
Dependent Random Variables ◽  
Finite Segment ◽  
Exchangeable Random Variables ◽  
Partial Sum

We give an expression for the expectation of max (0, S 1, …, Sn ) where Sk is the kth partial sum of a finite sequence of exchangeable random variables X 1, …, Xn . When the Xk are also independent, the formula we give has already been obtained by Spitzer; and when the sequence is a finite segment of an infinite sequence of exchangeable random variables, it is a consequence of a theorem of Hewitt.

scholarly journals Joint Distributions of Counts of Strings in Finite Bernoulli Sequences

2012 ◽  
Vol 49 (3) ◽  
pp. 758-772 ◽  
Author(s):  
Fred W. Huffer ◽  
Jayaram Sethuraman
Keyword(s):  
Poisson Process ◽  
Random Vector ◽  
Joint Distribution ◽  
Infinite Sequence ◽  
Random Variables ◽  
Finite Sequence ◽  
Factorial Moments ◽  
Bernoulli Random Variables ◽  
Joint Distributions ◽  
Bernoulli Sequence

An infinite sequence (Y1, Y2,…) of independent Bernoulli random variables with P(Yi = 1) = a / (a + b + i - 1), i = 1, 2,…, where a > 0 and b ≥ 0, will be called a Bern(a, b) sequence. Consider the counts Z1, Z2, Z3,… of occurrences of patterns or strings of the form {11}, {101}, {1001},…, respectively, in this sequence. The joint distribution of the counts Z1, Z2,… in the infinite Bern(a, b) sequence has been studied extensively. The counts from the initial finite sequence (Y1, Y2,…, Yn) have been studied by Holst (2007), (2008b), who obtained the joint factorial moments for Bern(a, 0) and the factorial moments of Z1, the count of the string {1, 1}, for a general Bern(a, b). We consider stopping the Bernoulli sequence at a random time and describe the joint distribution of counts, which extends Holst's results. We show that the joint distribution of counts from a class of randomly stopped Bernoulli sequences possesses the mixture of independent Poissons property: there is a random vector conditioned on which the counts are independent Poissons. To obtain these results, we extend the conditional marked Poisson process technique introduced in Huffer, Sethuraman and Sethuraman (2009). Our results avoid previous combinatorial and induction methods which generally only yield factorial moments.

scholarly journals Joint Distributions of Counts of Strings in Finite Bernoulli Sequences

2012 ◽  
Vol 49 (03) ◽  
pp. 758-772 ◽  
Author(s):  
Fred W. Huffer ◽  
Jayaram Sethuraman
Keyword(s):  
Poisson Process ◽  
Random Vector ◽  
Joint Distribution ◽  
Infinite Sequence ◽  
Random Variables ◽  
Finite Sequence ◽  
Factorial Moments ◽  
Bernoulli Random Variables ◽  
Joint Distributions ◽  
Bernoulli Sequence

An infinite sequence (Y 1, Y 2,…) of independent Bernoulli random variables with P(Y i = 1) = a / (a + b + i - 1), i = 1, 2,…, where a > 0 and b ≥ 0, will be called a Bern(a, b) sequence. Consider the counts Z 1, Z 2, Z 3,… of occurrences of patterns or strings of the form {11}, {101}, {1001},…, respectively, in this sequence. The joint distribution of the counts Z 1, Z 2,… in the infinite Bern(a, b) sequence has been studied extensively. The counts from the initial finite sequence (Y 1, Y 2,…, Y n ) have been studied by Holst (2007), (2008b), who obtained the joint factorial moments for Bern(a, 0) and the factorial moments of Z 1, the count of the string {1, 1}, for a general Bern(a, b). We consider stopping the Bernoulli sequence at a random time and describe the joint distribution of counts, which extends Holst's results. We show that the joint distribution of counts from a class of randomly stopped Bernoulli sequences possesses the mixture of independent Poissons property: there is a random vector conditioned on which the counts are independent Poissons. To obtain these results, we extend the conditional marked Poisson process technique introduced in Huffer, Sethuraman and Sethuraman (2009). Our results avoid previous combinatorial and induction methods which generally only yield factorial moments.

scholarly journals Extreme value statistics of correlated random variables: A pedagogical review

2020 ◽  
Vol 840 ◽  
pp. 1-32 ◽  
Author(s):  
Satya N. Majumdar ◽  
Arnab Pal ◽  
Grégory Schehr
Keyword(s):  
Random Variables ◽  
Extreme Value ◽  
Extreme Value Statistics

scholarly journals GLOBAL FLUCTUATIONS IN PHYSICAL SYSTEMS: A SUBTLE INTERPLAY BETWEEN SUM AND EXTREME VALUE STATISTICS

2008 ◽  
Vol 22 (20) ◽  
pp. 3311-3368 ◽  
Author(s):  
MAXIME CLUSEL ◽  
ERIC BERTIN
Keyword(s):  
Extreme Values ◽  
Random Variables ◽  
Extreme Value ◽  
Physical Models ◽  
Extreme Value Statistics ◽  
Specific Class ◽  
Qualitative Properties ◽  
Rigorous Results ◽  
General Mapping ◽  
Global Fluctuations

Fluctuations of global additive quantities, like total energy or magnetization for instance, can in principle be described by statistics of sums of (possibly correlated) random variables. Yet, it turns out that extreme values (the largest value among a set of random variables) may also play a role in the statistics of global quantities, in a direct or indirect way. This review discusses different connections that may appear between problems of sums and of extreme values of random variables, and emphasizes physical situations in which such connections are relevant. Along this line of thought, standard convergence theorems for sums and extreme values of independent and identically distributed random variables are recalled, and some rigorous results as well as more heuristic reasonings are presented for correlated or non-identically distributed random variables. More specifically, the role of extreme values within sums of broadly distributed variables is addressed, and a general mapping between extreme values and sums is presented, allowing us to identify a class of correlated random variables whose sum follows (generalized) extreme value distributions. Possible applications of this specific class of random variables are illustrated on the example of two simple physical models. A few extensions to other related classes of random variables sharing similar qualitative properties are also briefly discussed, in connection with the so-called BHP distribution.

scholarly journals Random matrices with exchangeable entries

2020 ◽  
Vol 32 (07) ◽  
pp. 2050022
Author(s):  
Werner Kirsch ◽  
Thomas Kriecherbauer
Keyword(s):  
Asymptotic Behavior ◽  
Random Matrices ◽  
Infinite Sequence ◽  
Random Variables ◽  
Band Matrices ◽  
Exchangeable Random Variables ◽  
Classic Result ◽  
Symmetric Band ◽  
Probability Spaces ◽  
Operator Norms

We consider ensembles of real symmetric band matrices with entries drawn from an infinite sequence of exchangeable random variables, as far as the symmetry of the matrices permits. In general, the entries of the upper triangular parts of these matrices are correlated and no smallness or sparseness of these correlations is assumed. It is shown that the eigenvalue distribution measures still converge to a semicircle but with random scaling. We also investigate the asymptotic behavior of the corresponding [Formula: see text]-operator norms. The key to our analysis is a generalization of a classic result by de Finetti that allows to represent the underlying probability spaces as averages of Wigner band ensembles with entries that are not necessarily centered. Some of our results appear to be new even for such Wigner band matrices.

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