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.

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.

Some Inequalities of Partial Sums for Negatively Dependent Random Variables

2012 ◽  
Vol 195-196 ◽  
pp. 694-700
Author(s):  
Hai Wu Huang ◽  
Qun Ying Wu ◽  
Guang Ming Deng
Keyword(s):  
Random Variables ◽  
Independent Random Variables ◽  
Partial Sums ◽  
Exponential Inequalities ◽  
Dependent Random Variables ◽  
Probability Inequalities ◽  
Associated Random Variables ◽  
Negatively Dependent Random Variables

The main purpose of this paper is to investigate some properties of partial sums for negatively dependent random variables. By using some special numerical functions, and we get some probability inequalities and exponential inequalities of partial sums, which generalize the corresponding results for independent random variables and associated random variables. At last, exponential inequalities and Bernsteins inequality for negatively dependent random variables are presented.

Moment bounds for non-stationary dependent sequences

1994 ◽  
Vol 31 (3) ◽  
pp. 731-742 ◽  
Author(s):  
Tae Yoon Kim
Keyword(s):  
Random Variables ◽  
Partial Sums ◽  
Unified Approach ◽  
Dependent Random Variables ◽  
Combinatorial Argument ◽  
Stationarity Condition ◽  
Weakly Dependent Random Variables ◽  
Mixing Sequences ◽  
Moment Bounds ◽  
Weakly Dependent

We provide a unified approach for establishing even-moment bounds for partial sums for a class of weakly dependent random variables satisfying a stationarity condition. As applications, we discuss moment bounds for various types of mixing sequences. To obtain even-moment bounds, we use a ‘combinatorial argument' developed by Cox and Kim (1990).

Moments for first-passage and last-exit times, the minimum, and related quantities for random walks with positive drift

1986 ◽  
Vol 18 (04) ◽  
pp. 865-879 ◽  
Author(s):  
Svante Janson
Keyword(s):  
Random Walks ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Necessary And Sufficient Conditions ◽  
Partial Sums ◽  
Exit Times ◽  
First Passage ◽  
Partial Sum ◽  
Positive Expectation ◽  
Necessary And Sufficient

Consider the sequence of partial sums of a sequence of i.i.d. random variables with positive expectation. We study various random quantities defined by the sequence of partial sums, e.g. the time at which the first or last crossing of a given level occurs, the value of the partial sum immediately before or after the crossing, the minimum of all partial sums. Necessary and sufficient conditions are given for the existence of moments of these quantities.

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.

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