On the law of the iterated logarithm for inter-record times

1970 ◽  
Vol 7 (02) ◽  
pp. 432-439 ◽  
Author(s):  
William E. Strawderman ◽  
Paul T. Holmes
Keyword(s):  
Distribution Function ◽  
Probability Space ◽  
Continuous Distribution ◽  
Random Variables ◽  
Continuous Distribution Function ◽  
Record Times ◽  
Iterated Logarithm ◽  
The Law ◽  
Image Position ◽  
Independent Identically Distributed

Let X 1, X2, X 3 , ··· be independent, identically distributed random variables on a probability space (Ω, F, P); and with a continuous distribution function. Let the sequence of indices {Vr } be defined as Also define The following theorem is due to Renyi [5].

On the law of the iterated logarithm for inter-record times

1970 ◽  
Vol 7 (2) ◽  
pp. 432-439 ◽  
Author(s):  
William E. Strawderman ◽  
Paul T. Holmes
Keyword(s):  
Distribution Function ◽  
Probability Space ◽  
Continuous Distribution ◽  
Random Variables ◽  
Continuous Distribution Function ◽  
Record Times ◽  
Iterated Logarithm ◽  
The Law ◽  
Image Position ◽  
Independent Identically Distributed

Let X1, X2, X3, ··· be independent, identically distributed random variables on a probability space (Ω, F, P); and with a continuous distribution function. Let the sequence of indices {Vr} be defined as Also define The following theorem is due to Renyi [5].

Records from a power model of independent observations

1995 ◽  
Vol 32 (4) ◽  
pp. 982-990 ◽  
Author(s):  
Ishay Weissman
Keyword(s):  
Distribution Function ◽  
Wide Spectrum ◽  
Continuous Distribution ◽  
Random Variables ◽  
Continuous Distribution Function ◽  
Power Model ◽  
Record Times ◽  
Independent Sequence ◽  
Independent Observations

Records from are analyzed, where {Yj} is an independent sequence of random variables. Each Yj has a continuous distribution function Fj = Fλj for some distribution F and some λ j > 0. We study records, record times and related quantities for this sequence. Depending on the sequence of powers , a wide spectrum of behaviour is exhibited.

A note on the waiting times between record observations

1969 ◽  
Vol 6 (03) ◽  
pp. 711-714 ◽  
Author(s):  
Paul T. Holmes ◽  
William E. Strawderman
Keyword(s):  
Distribution Function ◽  
Infinite Sequence ◽  
Waiting Times ◽  
Continuous Distribution ◽  
Random Variables ◽  
Expected Value ◽  
Underlying Distribution ◽  
Upper Record ◽  
Image Position ◽  
Independent Identically Distributed

Let X 1, X 2, X 3,··· be independent, identically distributed random variables with a continuous distribution function and let the sequence of indices {Vr } be defined as follows: and for r ≧ 1, V r is the trial on which the rth (upper) record observation occurs. {V r} will be an infinite sequence of random variables since the underlying distribution function of the X's is continuous. It is well known that the expected value of V r. is infinite for every r (see, for example, Feller [1], page 15). Also define and for r > 1 δr is the number of trials between the (r - l)th and the rth record. The distributions of the random variables Vr and δ r do not depend on the distribution of the original random variables. It can be shown (see Neuts [2], page 206 or Tata 1[4], page 26) that The following theorem is due to Neuts [2].

The distributions of interrecord fillings

2016 ◽  
Vol 26 (4) ◽  
Author(s):  
Oleg P. Orlov ◽  
Nikolay Yu. Pasynkov
Keyword(s):  
Distribution Function ◽  
Continuous Distribution ◽  
Random Variables ◽  
Record Values ◽  
Continuous Distribution Function ◽  
Record Times ◽  
Initial Sequence

AbstractIn a sequence of independent positive random variables with the same continuous distribution function a monotonic subsequence of record values is chosen. A corresponding sequence of record times divides the initial sequence into interrecord intervals. Let

Records from a power model of independent observations

1995 ◽  
Vol 32 (04) ◽  
pp. 982-990
Author(s):  
Ishay Weissman
Keyword(s):  
Distribution Function ◽  
Wide Spectrum ◽  
Continuous Distribution ◽  
Random Variables ◽  
Continuous Distribution Function ◽  
Power Model ◽  
Record Times ◽  
Independent Sequence ◽  
Independent Observations

Records from are analyzed, where {Yj } is an independent sequence of random variables. Each Yj has a continuous distribution function Fj = Fλj for some distribution F and some λ j > 0. We study records, record times and related quantities for this sequence. Depending on the sequence of powers , a wide spectrum of behaviour is exhibited.

A note on the waiting times between record observations

1969 ◽  
Vol 6 (3) ◽  
pp. 711-714 ◽  
Author(s):  
Paul T. Holmes ◽  
William E. Strawderman
Keyword(s):  
Distribution Function ◽  
Infinite Sequence ◽  
Waiting Times ◽  
Continuous Distribution ◽  
Random Variables ◽  
Expected Value ◽  
Underlying Distribution ◽  
Upper Record ◽  
Image Position ◽  
Independent Identically Distributed

Let X1,X2,X3,··· be independent, identically distributed random variables with a continuous distribution function and let the sequence of indices {Vr} be defined as follows: and for r ≧ 1, Vr is the trial on which the rth (upper) record observation occurs. {Vr} will be an infinite sequence of random variables since the underlying distribution function of the X's is continuous. It is well known that the expected value of Vr. is infinite for every r (see, for example, Feller [1], page 15). Also define and for r > 1 δr is the number of trials between the (r - l)th and the rth record. The distributions of the random variables Vr and δr do not depend on the distribution of the original random variables. It can be shown (see Neuts [2], page 206 or Tata 1[4], page 26) that The following theorem is due to Neuts [2].

A clustering law for some discrete order statistics

2003 ◽  
Vol 40 (01) ◽  
pp. 226-241 ◽  
Author(s):  
Sunder Sethuraman
Keyword(s):  
Distribution Function ◽  
Positive Integer ◽  
Order Statistics ◽  
Law Of Large Numbers ◽  
Asymptotic Properties ◽  
Random Variables ◽  
Large Numbers ◽  
The Law ◽  
Sample Maximum ◽  
Independent Identically Distributed

Let X 1, X 2, …, X n be a sequence of independent, identically distributed positive integer random variables with distribution function F. Anderson (1970) proved a variant of the law of large numbers by showing that the sample maximum moves asymptotically on two values if and only if F satisfies a ‘clustering’ condition, In this article, we generalize Anderson's result and show that it is robust by proving that, for any r ≥ 0, the sample maximum and other extremes asymptotically cluster on r + 2 values if and only if Together with previous work which considered other asymptotic properties of these sample extremes, a more detailed asymptotic clustering structure for discrete order statistics is presented.

scholarly journals On the number and sum of near-record observations

2005 ◽  
Vol 37 (03) ◽  
pp. 765-780 ◽  
Author(s):  
N. Balakrishnan ◽  
A.G. Pakes ◽  
A. Stepanov
Keyword(s):  
Distribution Function ◽  
Asymptotic Properties ◽  
Continuous Distribution ◽  
Random Variables ◽  
Continuous Distribution Function ◽  
Record Time ◽  
Record Value ◽  
Independent And Identically Distributed

Let X 1,X 2,… be a sequence of independent and identically distributed random variables with some continuous distribution function F. Let L(n) and X(n) denote the nth record time and the nth record value, respectively. We refer to the variables X i as near-nth-record observations if X i ∈(X(n)-a,X(n)], with a>0, and L(n)<i<L(n+1). In this work we study asymptotic properties of the number of near-record observations. We also discuss sums of near-record observations.

On the converse to the iterated logarithm law

1968 ◽  
Vol 5 (01) ◽  
pp. 210-215 ◽  
Author(s):  
C. C. Heyde
Keyword(s):  
Random Variables ◽  
Iterated Logarithm ◽  
Logarithm Law ◽  
The Law ◽  
Image Position ◽  
Iterated Logarithm Law ◽  
Independent And Identically Distributed

Let Xi, i = 1, 2, 3,… be a sequence of independent and identically distributed random variables with law ℓ(X) and write. if EX = 0 and EX2 = σ2 < ∞, the law of the iterated logarithm (Hartman and Wintner [1]) tells us that

A clustering law for some discrete order statistics

2003 ◽  
Vol 40 (1) ◽  
pp. 226-241 ◽  
Author(s):  
Sunder Sethuraman
Keyword(s):  
Distribution Function ◽  
Positive Integer ◽  
Order Statistics ◽  
Law Of Large Numbers ◽  
Asymptotic Properties ◽  
Random Variables ◽  
Large Numbers ◽  
The Law ◽  
Sample Maximum ◽  
Independent Identically Distributed

Let X1, X2, …, Xn be a sequence of independent, identically distributed positive integer random variables with distribution function F. Anderson (1970) proved a variant of the law of large numbers by showing that the sample maximum moves asymptotically on two values if and only if F satisfies a ‘clustering’ condition, In this article, we generalize Anderson's result and show that it is robust by proving that, for any r ≥ 0, the sample maximum and other extremes asymptotically cluster on r + 2 values if and only if Together with previous work which considered other asymptotic properties of these sample extremes, a more detailed asymptotic clustering structure for discrete order statistics is presented.

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