Some limit theorems for large deviations of sums of independent random variables with a common distribution function from the domain of attraction of the normal law

2005 ◽  
Vol 127 (1) ◽  
pp. 1767-1783 ◽  
Author(s):  
L. V. Rozovsky
Keyword(s):  
Distribution Function ◽  
Large Deviations ◽  
Limit Theorems ◽  
Random Variables ◽  
Domain Of Attraction ◽  
Independent Random Variables ◽  
Common Distribution ◽  
Common Distribution Function ◽  
Normal Law

Probability maximizing approach to a secretary problem by random change-point of the distribution law of the observed process

1984 ◽  
Vol 21 (1) ◽  
pp. 98-107 ◽  
Author(s):  
Minoru Yoshida
Keyword(s):  
Distribution Function ◽  
Stopping Time ◽  
Random Variables ◽  
Secretary Problem ◽  
Independent Random Variables ◽  
Distribution Law ◽  
Common Distribution ◽  
Common Distribution Function ◽  
Random Moment ◽  
The Moment

Before some random moment θ, independent identically distributed random variables x1, · ··, xθ–1 with common distribution function μ (dx) appear consecutively. After the moment θ, independent random variables xθ, xθ+1, · ·· have another common distribution function f (x)μ (dx). Our information about θ can be constructed only by successively observed values of the x's.In this paper we find an optimal stopping policy by which we can maximize the probability that the quantity associated with the stopping time is the largest of all θ + m – 1 quantities for a given integer m.

On the asymptotic normality of Hill's estimator

1995 ◽  
Vol 118 (2) ◽  
pp. 375-382 ◽  
Author(s):  
Sándor Csörgő ◽  
László Viharos
Keyword(s):  
Distribution Function ◽  
Asymptotic Normality ◽  
Order Statistics ◽  
Random Variables ◽  
Fixed Number ◽  
Distribution Functions ◽  
Independent Random Variables ◽  
Hill’S Estimator ◽  
Common Distribution ◽  
Common Distribution Function

Let X, X1, X2, …, be independent random variables with a common distribution function F(x) = P {X ≤ x}, x∈ℝ, and for each n∈ℕ, let X1, n ≤ … ≤ Xn, n denote the order statistics pertaining to the sample X1, …, Xn. We assume that 1–F(x) = x−1/cl(x), 0 < x < ∞, where l is some function slowly varying at infinity and c > 0 is any fixed number. The class of all such distribution functions will be denoted by .

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.

Probability maximizing approach to a secretary problem by random change-point of the distribution law of the observed process

1984 ◽  
Vol 21 (01) ◽  
pp. 98-107
Author(s):  
Minoru Yoshida
Keyword(s):  
Distribution Function ◽  
Stopping Time ◽  
Random Variables ◽  
Secretary Problem ◽  
Independent Random Variables ◽  
Distribution Law ◽  
Common Distribution ◽  
Common Distribution Function ◽  
Random Moment ◽  
The Moment

Before some random moment θ, independent identically distributed random variables x 1, · ··, xθ– 1 with common distribution function μ (dx) appear consecutively. After the moment θ, independent random variables xθ, xθ +1, · ·· have another common distribution function f (x)μ (dx). Our information about θ can be constructed only by successively observed values of the x's. In this paper we find an optimal stopping policy by which we can maximize the probability that the quantity associated with the stopping time is the largest of all θ + m – 1 quantities for a given integer m.

Laws of the iterated logarithm for sums of the middle portion of the sample

1987 ◽  
Vol 101 (2) ◽  
pp. 301-312 ◽  
Author(s):  
Erich Haeusler ◽  
David M. Mason
Keyword(s):  
Order Statistics ◽  
Random Variables ◽  
Domain Of Attraction ◽  
Empirical Processes ◽  
Independent Random Variables ◽  
Middle Portion ◽  
Stable Law ◽  
Iterated Logarithm ◽  
Common Distribution ◽  
Common Distribution Function

AbstractLet X1, X2, …, be a sequence of independent random variables with common distribution function F in the domain of attraction of a stable law and, for each n ≥ 1, let X1, n ≤ … ≤ Xn, n denote the order statistics based on the first n of these random variables. It is shown that sums of the middle portion of the order statistics of the form , where (kn)n ≥ 1 is a sequence of non-negative integers such that kn → ∞ and kn/n → 0 as n → ∞ at an appropriate rate, can be normalized and centred so that the law of the iterated logarithm holds. The method of proof is based on the almost sure properties of weighted uniform empirical processes.

The central limit problem for trimmed sums

1987 ◽  
Vol 102 (2) ◽  
pp. 329-349 ◽  
Author(s):  
Philip S. Griffin ◽  
William E. Pruitt
Keyword(s):  
Distribution Function ◽  
Random Variables ◽  
Random Variable ◽  
Limit Problem ◽  
Absolute Value ◽  
Trimmed Sums ◽  
Common Distribution ◽  
Common Distribution Function ◽  
Central Limit Problem ◽  
Trimmed Sum

Let X, X1, X2,… be a sequence of non-degenerate i.i.d. random variables with common distribution function F. For 1 ≤ j ≤ n, let mn(j) be the number of Xi satisfying either |Xi| > |Xj|, 1 ≤ i ≤ n, or |Xi| = |Xj|, 1 ≤ i ≤ j, and let (r)Xn = Xj if mn(j) = r. Thus (r)Xn is the rth largest random variable in absolute value from amongst X1, …, Xn with ties being broken according to the order in which the random variables occur. Set (r)Sn = (r+1)Xn + … + (n)Xn and write Sn for (0)Sn. We will refer to (r)Sn as a trimmed sum.

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