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.

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].

A functional equation technique for obtaining wiener process probabilities associated with theorems of Kolmogorov-Smirnov type

1959 ◽  
Vol 55 (4) ◽  
pp. 328-332 ◽  
Author(s):  
J. Kiefer ◽  
D. V. Lindley
Keyword(s):  
Functional Equation ◽  
Distribution Function ◽  
Wiener Process ◽  
Limit Distribution ◽  
Sample Distribution ◽  
Probabilistic Computation ◽  
Limit Distributions ◽  
Kolmogorov Smirnov

1. Introduction. Since the first proofs by Kolmogorov (13) and Smirnov ((14), (15)) of their well-known results on the limit distribution of the deviations of the sample distribution function, many alternative proofs of these results have been given. For example, we may cite the various approaches of Feller (4), Doob (3), Kac (8), Gnedenko and Korolyuk(7), and Anderson and Darling (1). The approaches of (3), (8) and (1) rest on a probabilistic computation regarding the Wiener process, and are justified by the paper of Donsker (2) (see also (11)). Of all these approaches, only those of (8) and (1) can be extended to obtain the limit distributions of the ‘k–sample’ generalizations of the Kolmogorov-Smirnov statistics suggested in (9), and the author ((9), (10)) and Gihman(6) carried out such proofs.

Maximum Waiting Times are Asymptotically Independent

1992 ◽  
Vol 1 (3) ◽  
pp. 251-264 ◽  
Author(s):  
Tamás F. Móri
Keyword(s):  
Waiting Time ◽  
Limit Distribution ◽  
Infinite Sequence ◽  
Waiting Times ◽  
Finite Alphabet ◽  
Cantelli Lemma ◽  
Converse Part

For every n consider a subset Hn of the patterns of length n over a fixed finite alphabet. The limit distribution of the waiting time until each element of Hn appears in an infinite sequence of independent, uniformly distributed random letters was determined in an earlier paper. This time we prove that these waiting times are getting independent as n → ∞. Our result is used for applying the converse part of the Borel–Cantelli lemma to problems connected with such waiting times, yielding thus improvements on some known theorems.

On a generalized finite-capacity storage model

1983 ◽  
Vol 20 (3) ◽  
pp. 663-674 ◽  
Author(s):  
Samuel W. Woolford
Keyword(s):  
Distribution Function ◽  
Markov Chain ◽  
State Space ◽  
Random Variables ◽  
Time Dependent ◽  
Finite Capacity ◽  
Random Input ◽  
Storage Model ◽  
Limit Distributions

This paper considers a finite-capacity storage model defined on a Markov chain {Xn; n = 0, 1, ·· ·}, having state space J ⊆ {1, 2, ·· ·}. If Xn = j, then there is a random ‘input' Vn(j) (a negative input implying a demand) of ‘type' j, having a distribution function Fj(·). We assume that {Vn(j)} is an i.i.d. sequence of random variables, taken to be independent of {Xn} and of {Vn (k)}, for k ≠ j. Here, the random variables Vn(j) represent instantaneous ‘inputs' of type j for our storage model. Within this framework, we establish certain limit distributions for the joint processes (Zn, Xn) and (Zn, Qn, Ln), where Zn (defined in (1.2)) is the level of storage at time n, Qn (defined in (1.3)) is the cumulative overflow at time n, and Ln (defined in (1.4)) is the cumulative demand lost due to shortage of supply up to time n. In addition, an expression for the time-dependent distribution of (Zn, Xn) is obtained.

On a generalized finite-capacity storage model

1984 ◽  
Vol 16 (1) ◽  
pp. 23-23
Author(s):  
Samuel W. Woolford
Keyword(s):  
Distribution Function ◽  
Markov Chain ◽  
State Space ◽  
Random Variables ◽  
Time Dependent ◽  
Finite Capacity ◽  
Random Input ◽  
Storage Model ◽  
Limit Distributions

This paper considers a finite-capacity storage model defined on a Markov chain {Xn; n = 0, 1, …}, having state space J ⊆ {1, 2, …}. If Xn, = j, then there is a random. ‘input’ Vn(j) (a negative input implying a demand) of ‘type’ j, having a distribution function Fj (·). We assume that {Vn (j)} is an i.i.d. sequence of random variables, taken to be independent of {Xn} and of {Vn(k)}, for k ≠ = j. Here, the random variables Vn(j) represent instantaneous ‘inputs’ of type j for our storage model. Within this framework, we establish certain limit distributions for the joint processes (zn, Xn) and (Zn, OnLn), where Zn, is the level of storage at time n, Qn is the cumulative overflow at time n, and Ln is the cumulative demand lost due to shortage of supply up to time n. In addition, an expression for the time-dependent distribution of (Zn, Xn) is obtained.

On a generalized storage model with moment assumptions

1981 ◽  
Vol 18 (02) ◽  
pp. 473-481 ◽  
Author(s):  
Prem S. Puri ◽  
Samuel W. Woolford
Keyword(s):  
Distribution Function ◽  
Renewal Process ◽  
Random Variables ◽  
Positive Zero ◽  
Markov Renewal Process ◽  
Random Input ◽  
Storage Model ◽  
Limit Distributions ◽  
Second Moments

This paper considers a semi-infinite storage model, of the type studied by Senturia and Puri [13] and Balagopal [2], defined on a Markov renewal process, {(Xn, Tn ), n = 0, 1, ·· ·}, with 0 = T 0 < T 1 < · ··, almost surely, where Xn takes values in the set {1, 2, ·· ·}. If at Tn, Xn = j, then there is a random ‘input' Vn (j) (a negative input implying a demand) of ‘type' j, having distribution function Fj (·). We assume that {Vn (j)} is an i.i.d. sequence of random variables, taken to be independent of {(Xn, Tn )} and of {Vn (k)}, for k ≠ j, and that Vn (j) has first and second moments. Here the random variables Vn (j) represent instantaneous ‘inputs' (a negative value implying a demand) of type j for our storage model. Under these assumptions, we establish certain limit distributions for the joint process (Z(t), L(t)), where Z(t) (defined in (2)) is the level of storage at time t and L(t) (defined in (3)) is the demand lost due to shortage of supply during [0, t]. Different limit distributions are obtained for the cases when the ‘average stationary input' ρ, as defined in (5), is positive, zero or negative.

On a generalized finite-capacity storage model

1983 ◽  
Vol 20 (03) ◽  
pp. 663-674
Author(s):  
Samuel W. Woolford
Keyword(s):  
Distribution Function ◽  
Markov Chain ◽  
State Space ◽  
Random Variables ◽  
Time Dependent ◽  
Finite Capacity ◽  
Random Input ◽  
Storage Model ◽  
Limit Distributions

This paper considers a finite-capacity storage model defined on a Markov chain {Xn ; n = 0, 1, ·· ·}, having state space J ⊆ {1, 2, ·· ·}. If Xn = j, then there is a random ‘input' Vn (j) (a negative input implying a demand) of ‘type' j, having a distribution function Fj (·). We assume that {Vn (j)} is an i.i.d. sequence of random variables, taken to be independent of {Xn } and of {Vn (k)}, for k ≠ j. Here, the random variables Vn (j) represent instantaneous ‘inputs' of type j for our storage model. Within this framework, we establish certain limit distributions for the joint processes (Zn, Xn ) and (Zn, Qn, Ln ), where Zn (defined in (1.2)) is the level of storage at time n, Qn (defined in (1.3)) is the cumulative overflow at time n, and Ln (defined in (1.4)) is the cumulative demand lost due to shortage of supply up to time n. In addition, an expression for the time-dependent distribution of (Zn, Xn ) is obtained.

scholarly journals On the Discrepancy of Random Walks on the Circle

2019 ◽  
Vol 14 (2) ◽  
pp. 73-86
Author(s):  
Alina Bazarova ◽  
István Berkes ◽  
Marko Raseta
Keyword(s):  
Distribution Function ◽  
Random Walks ◽  
Limit Distribution ◽  
Empirical Distribution Function ◽  
Empirical Distribution ◽  
Random Variables ◽  
Weak Limit ◽  
Absolutely Continuous ◽  
Skorohod Space ◽  
Star Discrepancy

AbstractLet X1,X2,... be i.i.d. absolutely continuous random variables, let {S_k} = \sum\nolimits_{j = 1}^k {{X_j}} (mod 1) and let D*N denote the star discrepancy of the sequence (Sk)1≤k≤N. We determine the limit distribution of \sqrt N D_N^* and the weak limit of the sequence \sqrt N \left( {{F_N}(t) - t} \right) in the Skorohod space D[0, 1], where FN (t) denotes the empirical distribution function of the sequence (Sk)1≤k≤N.

Sign in / Sign up

Export Citation Format

Share Document