On Some Limit Theorems Involving the Empirical Distribution Function

1967 ◽  
Vol 10 (5) ◽  
pp. 739-741
Author(s):  
Miklós Csörgo
Keyword(s):  
Distribution Function ◽  
Limit Theorems ◽  
Empirical Distribution Function ◽  
Empirical Distribution ◽  
Continuous Distribution ◽  
Random Variables ◽  
Continuous Distribution Function ◽  
Independent Random Variables ◽  
Mutually Independent

Let X1 …, Xn be mutually independent random variables with a common continuous distribution function F (t). Let Fn(t) be the corresponding empirical distribution function, that isFn(t) = (number of Xi ≤ t, 1 ≤ i ≤ n)/n.Using a theorem of Manija [4], we proved among others the following statement in [1].

A New Proof of Some Results of Rényi and the Asymptotic Distribution of the Range of his Kolmogorov-Smirnov Type Random Variables

1967 ◽  
Vol 19 ◽  
pp. 550-558 ◽  
Author(s):  
Miklós Csörgö
Keyword(s):  
Distribution Function ◽  
Asymptotic Distribution ◽  
Empirical Distribution Function ◽  
Empirical Distribution ◽  
Continuous Distribution ◽  
Random Variables ◽  
Continuous Distribution Function ◽  
Independent Random Variables ◽  
Kolmogorov Smirnov ◽  
Mutually Independent

Let X1 X2, … , Xn be mutually independent random variables with a common continuous distribution function F(t). Let Fn(t) be the corresponding empirical distribution function, that is Fn(t) = (number of Xi ⩽ t, 1 ⩽ i ⩽ n)/n.

scholarly journals On a probability problem connected with Railway traffic

1991 ◽  
Vol 4 (1) ◽  
pp. 1-27 ◽  
Author(s):  
Lajos Takács
Keyword(s):  
Distribution Function ◽  
Asymptotic Behavior ◽  
Empirical Distribution ◽  
Continuous Distribution ◽  
Random Variables ◽  
Distribution Functions ◽  
Continuous Distribution Function ◽  
Independent Random Variables ◽  
Railway Traffic ◽  
Empirical Distribution Functions

Let Fn(x) and Gn(x) be the empirical distribution functions of two independent samples, each of size n, in the case where the elements of the samples are independent random variables, each having the same continuous distribution function V(x) over the interval (0,1). Define a statistic θn by θn/n=∫01[Fn(x)−Gn(x)]dV(x)−min0≤x≤1[Fn(x)−Gn(x)]. In this paper the limits of E{(θn/2n)r}(r=0,1,2,…) and P{θn/2n≤x} are determined for n→∞. The problem of finding the asymptotic behavior of the moments and the distribution of θn as n→∞ has arisen in a study of the fluctuations of the inventory of locomotives in a randomly chosen railway depot.

On the comparison of a theoretical and an empirical distribution function

1971 ◽  
Vol 8 (2) ◽  
pp. 321-330 ◽  
Author(s):  
Lajos Takács
Keyword(s):  
Distribution Function ◽  
Empirical Distribution Function ◽  
Empirical Distribution ◽  
Random Variables ◽  
Independent Random Variables ◽  
Common Distribution ◽  
Common Distribution Function ◽  
Mutually Independent

Let ξ1, ξ2, ···, ξm be mutually independent random variables having a common distribution function P{ξr≦x} = F(x)(r = 1, 2, ···, m). Let Fm(x) be the empirical distribution function of the sample (ξ1, ξ2, ···, ξm), that is, Fm(x) is defined as the number of variables ≦x divided by m.

On the comparison of a theoretical and an empirical distribution function

1971 ◽  
Vol 8 (02) ◽  
pp. 321-330 ◽  
Author(s):  
Lajos Takács
Keyword(s):  
Distribution Function ◽  
Empirical Distribution Function ◽  
Empirical Distribution ◽  
Random Variables ◽  
Independent Random Variables ◽  
Common Distribution ◽  
Common Distribution Function ◽  
Mutually Independent

Let ξ 1 , ξ2, ···, ξm be mutually independent random variables having a common distribution function P {ξ r ≦x} = F(x)(r = 1, 2, ···, m). Let Fm (x) be the empirical distribution function of the sample (ξ 1, ξ 2 , ···, ξm), that is, Fm (x) is defined as the number of variables ≦x divided by m.

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.

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.

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].

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&gt;0, and L(n)&lt;i&lt;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.

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