scholarly journals Almost Sure Invariance Principles for Partial Sums of Mixing $B$-Valued Random Variables

1980 ◽  
Vol 8 (6) ◽  
pp. 1003-1036 ◽  
Author(s):  
J. Kuelbs ◽  
Walter Philipp
Keyword(s):  
Random Variables ◽  
Partial Sums ◽  
Invariance Principles

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 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.

scholarly journals First-crossing and ballot-type results for some nonstationary sequences

2007 ◽  
Vol 39 (02) ◽  
pp. 492-509 ◽  
Author(s):  
Claude Lefèvre
Keyword(s):  
Time Distribution ◽  
Numerical Evaluation ◽  
Random Variables ◽  
Parametric Family ◽  
Partial Sums ◽  
Variable Parameters ◽  
Type Formula ◽  
Polynomial Structure ◽  
Crossing Time ◽  
First Crossing Time

In this paper we consider the problem of first-crossing from above for a partial sums process m+S t , t ≥ 1, with the diagonal line when the random variables X t , t ≥ 1, are independent but satisfying nonstationary laws. Specifically, the distributions of all the X t s belong to a common parametric family of arithmetic distributions, and this family of laws is assumed to be stable by convolution. The key result is that the first-crossing time distribution and the associated ballot-type formula rely on an underlying polynomial structure, called the generalized Abel-Gontcharoff structure. In practice, this property advantageously provides simple and efficient recursions for the numerical evaluation of the probabilities of interest. Several applications are then presented, for constant and variable parameters.

The expected number of visits of a sequence of normal partial sums to each side of a boundary

1978 ◽  
Vol 15 (01) ◽  
pp. 192-198
Author(s):  
Moshe Pollar
Keyword(s):  
Brownian Motion ◽  
Random Variables ◽  
Partial Sums ◽  
Expected Number ◽  
Number Of Visits

Let be the sequence of partial sums of independent N(μ, 1) random variables. The boundary in the {(n, Sn )} plane which minimizes the expected number of times n that Sn will be below a boundary when μ = θ > 0 subject to a given expected number of visits that Sn will be above the same boundary when μ = 0 is shown to be linear and is also characterized. Similar results are shown for Brownian motion.

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