The rate of convergence of the distribution of the maximum of cumulative sums of independent nonidentically distributed random vectors to a limiting law

1973 ◽  
Vol 13 (2) ◽  
pp. 262-268
Author(s):  
V. Paulauskas ◽  
S. Steishunas
Keyword(s):  
Rate Of Convergence ◽  
Random Vectors ◽  
Distribution Of The Maximum ◽  
Cumulative Sums

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.

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