An Elementary Derivation of the Distribution of the Maximum of a Certain Random Walk

1974 ◽  
Vol 5 (2) ◽  
pp. 237-239
Author(s):  
B.D. Bunday ◽  
R.E. Scraton
Keyword(s):  
Random Walk ◽  
Elementary Derivation ◽  
Distribution Of The Maximum

The exponential rate of convergence of the distribution of the maximum of a random walk. Part II

1976 ◽  
Vol 13 (04) ◽  
pp. 733-740
Author(s):  
N. Veraverbeke ◽  
J. L. Teugels
Keyword(s):  
Random Walk ◽  
Rate Of Convergence ◽  
Real Line ◽  
Exponential Convergence ◽  
Exponential Rate ◽  
The Real ◽  
Distribution Of The Maximum ◽  
Exponential Rate Of Convergence ◽  
Successive Maximum

Let Gn (x) be the distribution of the nth successive maximum of a random walk on the real line. Under conditions typical for complete exponential convergence, the decay of Gn (x) – limn→∞ Gn (x) is asymptotically equal to H(x) γn n–3/2 as n → ∞where γ < 1 and H(x) a function solely depending on x. For the case of drift to + ∞, G ∞(x) = 0 and the result is new; for drift to – ∞we give a new proof, simplifying and correcting an earlier version in [9].

The exponential rate of convergence of the distribution of the maximum of a random walk. Part II

1976 ◽  
Vol 13 (4) ◽  
pp. 733-740 ◽  
Author(s):  
N. Veraverbeke ◽  
J. L. Teugels
Keyword(s):  
Random Walk ◽  
Rate Of Convergence ◽  
Real Line ◽  
Exponential Convergence ◽  
Exponential Rate ◽  
The Real ◽  
Distribution Of The Maximum ◽  
Exponential Rate Of Convergence ◽  
Successive Maximum

Let Gn (x) be the distribution of the nth successive maximum of a random walk on the real line. Under conditions typical for complete exponential convergence, the decay of Gn (x) – limn→∞ Gn(x) is asymptotically equal to H(x) γn n–3/2 as n → ∞where γ < 1 and H(x) a function solely depending on x. For the case of drift to + ∞, G∞(x) = 0 and the result is new; for drift to – ∞we give a new proof, simplifying and correcting an earlier version in [9].

The exponential rate of convergence of the distribution of the maximum of a random walk

1975 ◽  
Vol 12 (02) ◽  
pp. 279-288 ◽  
Author(s):  
N. Veraverbeke ◽  
J. L. Teugels
Keyword(s):  
Distribution Function ◽  
Random Walk ◽  
Rate Of Convergence ◽  
Exponential Convergence ◽  
Random Variables ◽  
Limiting Distribution ◽  
Partial Sums ◽  
Exponential Rate ◽  
Distribution Of The Maximum ◽  
Exponential Rate Of Convergence

Let Gn (x) be the distribution function of the maximum of the successive partial sums of independent and identically distributed random variables and G(x) its limiting distribution function. Under conditions, typical for complete exponential convergence, the decay of Gn (x) — G(x) is asymptotically equal to c.H(x)n −3/2 γn as n → ∞ where c and γ are known constants and H(x) is a function solely depending on x.

The maximum and time to absorption of a left-continuous random walk

1976 ◽  
Vol 13 (3) ◽  
pp. 444-454 ◽  
Author(s):  
P. J. Green
Keyword(s):  
Random Walk ◽  
Limit Theorem ◽  
Joint Distribution ◽  
Distribution Of The Maximum ◽  
Conditional Limit Theorem ◽  
Continuous Random Walk ◽  
Conditional Limit

For a left-continuous random walk, absorbing at 0, the joint distribution of the maximum and time to absorption is derived. A description of the tails of the distributions and a conditional limit theorem are obtained for the cases where absorption is certain.

The maximum of a random walk whose mean path has a maximum

1985 ◽  
Vol 17 (1) ◽  
pp. 85-99 ◽  
Author(s):  
H. E. Daniels ◽  
T. H. R. Skyrme
Keyword(s):  
Integral Equation ◽  
Random Walk ◽  
Diffusion Approximation ◽  
Conditional Expectation ◽  
Joint Distribution ◽  
Marginal Distribution ◽  
Brownian Bridge ◽  
Distribution Of The Maximum

This paper discusses the joint distribution of the maximum and the time at which it is attained, of a random walk whose mean path is a curvilinear trend which itself has a maximum. A typical example of such a problem is the distribution of the maximum number of infectives present during the course of an epidemic. Another example where the random walk is constrained to terminate at 0 after a given time is provided by the distribution of the strength and breaking extension of a bundle of fibres.A diffusion approximation to the joint distribution is obtained for the general case of a Brownian bridge. In the commonest class of cases which includes the two examples mentioned, a certain integral equation has to be solved. Its solution enables the marginal distribution of the time to reach the maximum to be tabulated, and the marginal distribution of the maximum confirms the results previously obtained by Daniels (1974) and Barbour (1975). Of particular interest is the conditional expectation of the maximum for a given time of attainment which behaves asymmetrically.

The exponential rate of convergence of the distribution of the maximum of a random walk

1975 ◽  
Vol 12 (2) ◽  
pp. 279-288 ◽  
Author(s):  
N. Veraverbeke ◽  
J. L. Teugels
Keyword(s):  
Distribution Function ◽  
Random Walk ◽  
Rate Of Convergence ◽  
Exponential Convergence ◽  
Random Variables ◽  
Limiting Distribution ◽  
Partial Sums ◽  
Exponential Rate ◽  
Distribution Of The Maximum ◽  
Exponential Rate Of Convergence

Let Gn(x) be the distribution function of the maximum of the successive partial sums of independent and identically distributed random variables and G(x) its limiting distribution function. Under conditions, typical for complete exponential convergence, the decay of Gn(x) — G(x) is asymptotically equal to c.H(x)n−3/2γn as n → ∞ where c and γ are known constants and H(x) is a function solely depending on x.

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