On the variance of the maximum of partial sums of n-exchangeable random variables with applications

1980 ◽  
Vol 17 (2) ◽  
pp. 432-439
Author(s):  
A. A. Anis ◽  
M. Gharib
Keyword(s):  
General Formula ◽  
Random Variables ◽  
Partial Sums ◽  
Direct Relevance ◽  
Exchangeable Random Variables ◽  
Hurst Effect

A general formula is obtained for the variance of the maximum of partial sums of n-exchangeable random variables, derived from a result of Spitzer's. The formula is applied in particular to obtain the variance of the maximum of adjusted rescaled partial sums of normal summands. This is of direct relevance to the Hurst effect.

On the variance of the maximum of partial sums of n-exchangeable random variables with applications

1980 ◽  
Vol 17 (02) ◽  
pp. 432-439
Author(s):  
A. A. Anis ◽  
M. Gharib
Keyword(s):  
General Formula ◽  
Random Variables ◽  
Partial Sums ◽  
Direct Relevance ◽  
Exchangeable Random Variables ◽  
Hurst Effect

A general formula is obtained for the variance of the maximum of partial sums of n-exchangeable random variables, derived from a result of Spitzer's. The formula is applied in particular to obtain the variance of the maximum of adjusted rescaled partial sums of normal summands. This is of direct relevance to the Hurst effect.

scholarly journals ON THE FOURTH MOMENT OF THE MAXIMUM OF PARTIAL SUMS OF $n$-EXCHANGEABLE RANDOM VARIABLES WITH APPLICATIONS

1997 ◽  
Vol 27 (3) ◽  
pp. 247-256
Author(s):  
M. GHARIB
Keyword(s):  
Time Series ◽  
Stochastic Processes ◽  
General Formula ◽  
Random Variables ◽  
Partial Sums ◽  
Fourth Moment ◽  
Direct Relevance ◽  
Exchangeable Random Variables ◽  
Reservoir Design

A general formula is obtained for the fourth moment of the maximum of partial sums of $n$-exchangeable random variables derived from a result of Spitzer. The formula is applied in particular to obtain the fourth moment of the maximum of adjusted partial sums of normal summands. This is of direct relevance to reservoir design and the analysis of the structure of stochastic processes and time series.

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.

On the expected range and expected adjusted range of partial sums of exchangeable random variables

1973 ◽  
Vol 10 (3) ◽  
pp. 671-677 ◽  
Author(s):  
D. C. Boes ◽  
J.D. Salas-La Cruz
Keyword(s):  
Storage Capacity ◽  
Random Variables ◽  
Partial Sums ◽  
Sums Of Random Variables ◽  
Exchangeable Random Variables ◽  
Expected Values

Studies of storage capacity of reservoirs, under the assumption of infinite storage, lead to the problem of finding the distribution of the range or adjusted range of partial sums of random variables.In this paper, formulas for the expected values of the range and adjusted range of partial sums of exchangeable random variables are presented. Such formulas are based on an elegant result given in Spitzer (1956). Some consequences of the aforementioned formulas are discussed.

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 expected range and expected adjusted range of partial sums of exchangeable random variables

1973 ◽  
Vol 10 (03) ◽  
pp. 671-677 ◽  
Author(s):  
D. C. Boes ◽  
J.D. Salas-La Cruz
Keyword(s):  
Storage Capacity ◽  
Random Variables ◽  
Partial Sums ◽  
Sums Of Random Variables ◽  
Exchangeable Random Variables ◽  
Expected Values

Studies of storage capacity of reservoirs, under the assumption of infinite storage, lead to the problem of finding the distribution of the range or adjusted range of partial sums of random variables. In this paper, formulas for the expected values of the range and adjusted range of partial sums of exchangeable random variables are presented. Such formulas are based on an elegant result given in Spitzer (1956). Some consequences of the aforementioned formulas are discussed.

On the ordered partial sums of real random variables

1977 ◽  
Vol 14 (1) ◽  
pp. 75-88 ◽  
Author(s):  
Lajos Takács
Keyword(s):  
Random Variables ◽  
Partial Sums ◽  
Markov Sequence ◽  
Stieltjes Transforms ◽  
Independent And Identically Distributed

In 1952 Pollaczek discovered a remarkable formula for the Laplace-Stieltjes transforms of the distributions of the ordered partial sums for a sequence of independent and identically distributed real random variables. In this paper Pollaczek's result is proved in a simple way and is extended for a semi-Markov sequence of real random variables.

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