scholarly journals Limit Theorems for the Inductive Mean on Metric Trees

2010 ◽  
Vol 47 (04) ◽  
pp. 1136-1149 ◽  
Author(s):  
Bojan Basrak
Keyword(s):  
Strong Consistency ◽  
Limit Theorems ◽  
Random Variables ◽  
Expected Value ◽  
Asymptotic Distributions ◽  
Weighted Interpolation ◽  
Underlying Distribution ◽  
Data Point ◽  
Definition Of

For random variables with values on binary metric trees, the definition of the expected value can be generalized to the notion of a barycenter. To estimate the barycenter from tree-valued data, the so-called inductive mean is constructed recursively using the weighted interpolation between the current mean and a new data point. We show the strong consistency of the inductive mean, but also that it, somewhat peculiarly, converges towards the true barycenter with different rates, and asymptotic distributions depending on the small variations of the underlying distribution.

scholarly journals Limit Theorems for the Inductive Mean on Metric Trees

2010 ◽  
Vol 47 (4) ◽  
pp. 1136-1149 ◽  
Author(s):  
Bojan Basrak
Keyword(s):  
Strong Consistency ◽  
Limit Theorems ◽  
Random Variables ◽  
Expected Value ◽  
Asymptotic Distributions ◽  
Weighted Interpolation ◽  
Underlying Distribution ◽  
Data Point ◽  
Definition Of

For random variables with values on binary metric trees, the definition of the expected value can be generalized to the notion of a barycenter. To estimate the barycenter from tree-valued data, the so-called inductive mean is constructed recursively using the weighted interpolation between the current mean and a new data point. We show the strong consistency of the inductive mean, but also that it, somewhat peculiarly, converges towards the true barycenter with different rates, and asymptotic distributions depending on the small variations of the underlying distribution.

A Poisson limit theorem for incomplete symmetric statistics

1983 ◽  
Vol 20 (01) ◽  
pp. 47-60 ◽  
Author(s):  
M. Berman ◽  
G. K. Eagleson
Keyword(s):  
Limit Theorem ◽  
Limit Theorems ◽  
Random Variables ◽  
Asymptotic Distributions ◽  
Poisson Limit ◽  
Symmetric Statistics ◽  
Limit Result ◽  
Independent Identically Distributed

Silverman and Brown (1978) have derived Poisson limit theorems for certain sequences of symmetric statistics, based on a sample of independent identically distributed random variables. In this paper an incomplete version of these statistics is considered and a Poisson limit result shown to hold. The powers of some tests based on the incomplete statistic are investigated and the main results of the paper are used to simplify the derivations of the asymptotic distributions of some statistics previously published in the literature.

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].

scholarly journals The Strong Laws of Large Numbers for Set-Valued Random Variables in Fuzzy Metric Space

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1192
Author(s):  
Li Guan ◽  
Juan Wei ◽  
Hui Min ◽  
Junfei Zhang
Keyword(s):  
Metric Space ◽  
Limit Theorems ◽  
Hausdorff Metric ◽  
Random Variables ◽  
Fuzzy Metric Space ◽  
Laws Of Large Numbers ◽  
Strong Laws ◽  
Fuzzy Metric ◽  
Large Numbers ◽  
Definition Of

In this paper, we firstly introduce the definition of the fuzzy metric of sets, and discuss the properties of fuzzy metric induced by the Hausdorff metric. Then we prove the limit theorems for set-valued random variables in fuzzy metric space; the convergence is about fuzzy metric induced by the Hausdorff metric. The work is an extension from the classical results for set-valued random variables to fuzzy metric space.

A Poisson limit theorem for incomplete symmetric statistics

1983 ◽  
Vol 20 (1) ◽  
pp. 47-60 ◽  
Author(s):  
M. Berman ◽  
G. K. Eagleson
Keyword(s):  
Limit Theorem ◽  
Limit Theorems ◽  
Random Variables ◽  
Asymptotic Distributions ◽  
Poisson Limit ◽  
Symmetric Statistics ◽  
Limit Result ◽  
Independent Identically Distributed

Silverman and Brown (1978) have derived Poisson limit theorems for certain sequences of symmetric statistics, based on a sample of independent identically distributed random variables. In this paper an incomplete version of these statistics is considered and a Poisson limit result shown to hold. The powers of some tests based on the incomplete statistic are investigated and the main results of the paper are used to simplify the derivations of the asymptotic distributions of some statistics previously published in the literature.

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 central limit theorem by remainder term for renewal processes

1992 ◽  
Vol 24 (2) ◽  
pp. 267-287 ◽  
Author(s):  
Allen L. Roginsky
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Remainder Term ◽  
Limit Theorems ◽  
Random Variables ◽  
Central Limit Theorems ◽  
Renewal Processes

Three different definitions of the renewal processes are considered. For each of them, a central limit theorem with a remainder term is proved. The random variables that form the renewal processes are independent but not necessarily identically distributed and do not have to be positive. The results obtained in this paper improve and extend the central limit theorems obtained by Ahmad (1981) and Niculescu and Omey (1985).

Sign in / Sign up

Export Citation Format

Share Document