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].

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].

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].

Waitingtimes between record observations

1967 ◽  
Vol 4 (01) ◽  
pp. 206-208 ◽  
Author(s):  
Marcel F. Neuts
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Asymptotic Normality ◽  
Central Limit ◽  
Law Of Large Numbers ◽  
Continuous Distribution ◽  
Random Variables ◽  
Large Numbers ◽  
Upper Record ◽  
Independent Identically Distributed

If Δ r denotes the waitingtime between the (r − 1)st and the rth upper record in a sequence of independent, identically distributed random variables with a continuous distribution, then it is shown that Δ r satisfies the weak law of large numbers and a central limit theorem. This theorem supplements those of Foster and Stuart and Rényi, who investigated the index Vr of the rth upper record. Qualitatively the theorems establish the intuitive fact that for higher records, the waitingtime between the last two records outweighs even the total waitingtime for previous records. This explains also why the asymptotic normality of logVr is very inadequate for approximation purposes—Barton and Mallows.

Waitingtimes between record observations

1967 ◽  
Vol 4 (1) ◽  
pp. 206-208 ◽  
Author(s):  
Marcel F. Neuts
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Asymptotic Normality ◽  
Central Limit ◽  
Law Of Large Numbers ◽  
Continuous Distribution ◽  
Random Variables ◽  
Large Numbers ◽  
Upper Record ◽  
Independent Identically Distributed

If Δr denotes the waitingtime between the (r − 1)st and the rth upper record in a sequence of independent, identically distributed random variables with a continuous distribution, then it is shown that Δr satisfies the weak law of large numbers and a central limit theorem.This theorem supplements those of Foster and Stuart and Rényi, who investigated the index Vr of the rth upper record.Qualitatively the theorems establish the intuitive fact that for higher records, the waitingtime between the last two records outweighs even the total waitingtime for previous records. This explains also why the asymptotic normality of logVr is very inadequate for approximation purposes—Barton and Mallows.

Limit distributions of the number of renewals and waiting times

1976 ◽  
Vol 13 (2) ◽  
pp. 301-312
Author(s):  
N. R. Mohan
Keyword(s):  
Distribution Function ◽  
Limit Distribution ◽  
Infinite Sequence ◽  
Waiting Times ◽  
Random Variables ◽  
Stable Laws ◽  
Domains Of Attraction ◽  
Limit Distributions

Let {Xn} be an infinite sequence of independent non-negative random variables. Let the distribution function of Xi, i = 1, 2, …, be either F1 or F2 where F1 and F2 are distinct. Set Sn = X1 + X2 + … + Xn and for t > 0 define and Zt = SN(t)+1 – t. The limit distributions of N(t), Yt and Zt as t → ∞ are obtained when F1 and F2 are in the domains of attraction of stable laws with exponents α1 and α2, respectively and Sn properly normalised has the composition of these two stable laws as its limit distribution.

Limit distributions of the number of renewals and waiting times

1976 ◽  
Vol 13 (02) ◽  
pp. 301-312
Author(s):  
N. R. Mohan
Keyword(s):  
Distribution Function ◽  
Limit Distribution ◽  
Infinite Sequence ◽  
Waiting Times ◽  
Random Variables ◽  
Stable Laws ◽  
Domains Of Attraction ◽  
Limit Distributions

Let {X n} be an infinite sequence of independent non-negative random variables. Let the distribution function of Xi , i = 1, 2, …, be either F 1 or F 2 where F 1 and F 2 are distinct. Set Sn = X 1 + X 2 + … + Xn and for t > 0 define and Zt = SN (t)+1 – t. The limit distributions of N(t), Yt and Zt as t → ∞ are obtained when F 1 and F 2 are in the domains of attraction of stable laws with exponents α 1 and α 2 , respectively and Sn properly normalised has the composition of these two stable laws as its limit distribution.

scholarly journals Exact and Limiting Probability Distributions of Some Smirnov Type Statistics

1965 ◽  
Vol 8 (1) ◽  
pp. 93-103 ◽  
Author(s):  
Miklós Csörgo
Keyword(s):  
Distribution Function ◽  
Random Sample ◽  
Empirical Distribution Function ◽  
Empirical Distribution ◽  
Probability Distributions ◽  
Continuous Distribution ◽  
Random Variables ◽  
Random Variable ◽  
Exact Distributions ◽  
Image Position

Let F(x) be the continuous distribution function of a random variable X and Fn(x) be the empirical distribution function determined by a random sample X1, …, Xn taken on X. Using the method of Birnbaum and Tingey [1] we are going to derive the exact distributions of the random variablesand and where the indicated sup' s are taken over all x' s such that -∞ < x < xb and xa ≤ x < + ∞ with F(xb) = b, F(xa) = a in the first two cases and over all x' s so that Fn(x) ≤ b and a ≤ Fn(x) in the last two cases.

scholarly journals A Limit Theorem for Random Products of Trimmed Sums of i.i.d. Random Variables

2011 ◽  
Vol 2011 ◽  
pp. 1-13
Author(s):  
Fa-mei Zheng
Keyword(s):  
Distribution Function ◽  
Limit Theorem ◽  
Wiener Process ◽  
Continuous Distribution ◽  
Random Variables ◽  
Standard Wiener Process ◽  
Trimmed Sums ◽  
Fixed Constant ◽  
Random Products ◽  
Trimmed Sum

Let be a sequence of independent and identically distributed positive random variables with a continuous distribution function , and has a medium tail. Denote and , where , , and is a fixed constant. Under some suitable conditions, we show that , as , where is the trimmed sum and is a standard Wiener process.

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.

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