Order statistics of partial sums of mutually independent random variables

Herein is exposed a simplified analytic proof of formulas for the characteristic functions of ordered partial sums of mutually independent identically distributed random variables. This problem which we had raised and solved in 1952 by another method, has since been treated by several authors (see Wendel [6]), and recently by de Smit [4], who made use of a kind of Wiener-Hopf decomposition†. On the contrary our present as well as our previous proof essentially uses the explicit solution of a certain singular integral equation in a complex domain.

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On asymptotic independence of the partial sums of positive and negative parts of independent random variables

Advances in Applied Probability ◽

10.1017/s0001867800038015 ◽

1971 ◽

Vol 3 (02) ◽

pp. 404-425

Author(s):

Howard G. Tucker

Keyword(s):

Stable Distribution ◽

Characteristic Exponent ◽

Random Variables ◽

Domain Of Attraction ◽

Distribution Functions ◽

Independent Random Variables ◽

Asymptotic Independence ◽

Partial Sums ◽

Common Distribution Function ◽

Independent Identically Distributed

The aim of this study is an investigation of the joint limiting distribution of the sequence of partial sums of the positive parts and negative parts of a sequence of independent identically distributed random variables. In particular, let {Xn} be a sequence of independent identically distributed random variables with common distribution functionF, assumeFis in the domain of attraction of a stable distribution with characteristic exponent α, 0 < α ≦ 2, and let {Bn} be normalizing coefficients forF. Let us denoteXn+=XnI[Xn> 0]andXn−= −XnI[Xn<0], so thatXn=Xn+-Xn−. LetF+andF−denote the distribution functions ofX1+andX1−respectively, and letSn(+)=X1++ · · · +Xn+,Sn(-)=X1−+ · · · +Xn−. The problem considered here is to find under what conditions there exist sequences of real numbers {an} and {bn} such that the joint distribution of (Bn-1Sn(+)+an,Bn-1Sn(-)+bn) converges to that of two independent random variables (U, V). As might be expected, different results are obtained when α < 2 and when α = 2. When α < 2, there always exist such sequences so that the above is true, and in this case bothUandVare stable with characteristic exponent a, or one has such a stable distribution and the other is constant. When α = 2, and if 0 < ∫x2dF(x) < ∞, then there always exist such sequences such that the above convergence takes place; bothUandVare normal (possibly one is a constant), and if neither is a constant, thenUandVarenotindependent. If α = 2 and ∫x2dF(x) = ∞, then at least one ofF+,F−is in the domain of partial attraction of the normal distribution, and a modified statement on the independence ofUandVholds. Various specialized results are obtained for α = 2.

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On asymptotic independence of the partial sums of positive and negative parts of independent random variables

Advances in Applied Probability ◽

10.2307/1426178 ◽

1971 ◽

Vol 3 (2) ◽

pp. 404-425 ◽

Cited By ~ 4

Author(s):

Howard G. Tucker

Keyword(s):

Stable Distribution ◽

Characteristic Exponent ◽

Random Variables ◽

Domain Of Attraction ◽

Distribution Functions ◽

Independent Random Variables ◽

Asymptotic Independence ◽

Partial Sums ◽

Common Distribution Function ◽

Independent Identically Distributed

The aim of this study is an investigation of the joint limiting distribution of the sequence of partial sums of the positive parts and negative parts of a sequence of independent identically distributed random variables. In particular, let {Xn} be a sequence of independent identically distributed random variables with common distribution function F, assume F is in the domain of attraction of a stable distribution with characteristic exponent α, 0 < α ≦ 2, and let {Bn} be normalizing coefficients for F. Let us denote Xn+ = XnI[Xn > 0] and Xn− = − XnI[Xn<0], so that Xn = Xn+ - Xn−. Let F+ and F− denote the distribution functions of X1+ and X1− respectively, and let Sn(+) = X1+ + · · · + Xn+, Sn(-) = X1− + · · · + Xn−. The problem considered here is to find under what conditions there exist sequences of real numbers {an} and {bn} such that the joint distribution of (Bn-1Sn(+) + an, Bn-1Sn(-) + bn) converges to that of two independent random variables (U, V). As might be expected, different results are obtained when α < 2 and when α = 2. When α < 2, there always exist such sequences so that the above is true, and in this case both U and V are stable with characteristic exponent a, or one has such a stable distribution and the other is constant. When α = 2, and if 0 < ∫ x2dF(x) < ∞, then there always exist such sequences such that the above convergence takes place; both U and V are normal (possibly one is a constant), and if neither is a constant, then U and V are not independent. If α = 2 and ∫ x2dF(x) = ∞, then at least one of F+, F− is in the domain of partial attraction of the normal distribution, and a modified statement on the independence of U and V holds. Various specialized results are obtained for α = 2.

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On a Problem of Fluctuations of Sums of Independent Random Variables

Journal of Applied Probability ◽

10.1017/s0021900200047537 ◽

1975 ◽

Vol 12 (S1) ◽

pp. 29-37

Author(s):

Lajos Takács

Keyword(s):

Limit Distribution ◽

Random Variables ◽

Independent Random Variables ◽

Partial Sums ◽

Discrete Random Variables ◽

Mutually Independent ◽

Independent And Identically Distributed

The author determines the distribution and the limit distribution of the number of partial sums greater than k (k = 0, 1, 2, …) for n mutually independent and identically distributed discrete random variables taking on the integers 1, 0, − 1, − 2, ….

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Approximate Solution Of A Singular Integral Equation With A Multiplicative Cauchy Kernel In The Half-Plane

Computational Methods in Applied Mathematics ◽

10.2478/cmam-2008-0010 ◽

2008 ◽

Vol 8 (2) ◽

pp. 143-154 ◽

Cited By ~ 1

Author(s):

P. KARCZMAREK

Keyword(s):

Integral Equation ◽

Approximate Solution ◽

Singular Integral Equation ◽

Half Plane ◽

Singular Integral ◽

Trigonometric Polynomials ◽

Cauchy Kernel

AbstractIn this paper, Jacobi and trigonometric polynomials are used to con-struct the approximate solution of a singular integral equation with multiplicative Cauchy kernel in the half-plane.

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On the distribution of the number of cyclically occurring random events

Journal of Applied Probability ◽

10.2307/3212338 ◽

1972 ◽

Vol 9 (3) ◽

pp. 681-683

Author(s):

Leon Podkaminer

Keyword(s):

Random Variables ◽

Independent Random Variables ◽

Time Intervals ◽

Time Period ◽

Random Events ◽

Mutually Independent

The probabilities of the occurrence of n events in a certain time period are calculated under the assumptions that the time intervals between the neighbouring events are mutually independent random variables, satisfying some analytic conditions.

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A semianalytical and finite-element solution to the unbonded contact between a frictionless layer and an FGM-coated half-plane

Mathematics and Mechanics of Solids ◽

10.1177/1081286517744600 ◽

2017 ◽

Vol 24 (2) ◽

pp. 448-464 ◽

Cited By ~ 7

Author(s):

Jie Yan ◽

Changwen Mi ◽

Zhixin Liu

Keyword(s):

Integral Equation ◽

Finite Element ◽

Singular Integral Equation ◽

Contact Problem ◽

Singular Integral ◽

Material Properties ◽

Elastic Layer ◽

Coated Substrate ◽

Receding Contact ◽

Contact Size

In this work, we examine the receding contact between a homogeneous elastic layer and a half-plane substrate reinforced by a functionally graded coating. The material properties of the coating are allowed to vary exponentially along its thickness. A distributed traction load applied over a finite segment of the layer surface presses the layer and the coated substrate against each other. It is further assumed that the receding contact between the layer and the coated substrate is frictionless. In the absence of body forces, Fourier integral transforms are used to convert the governing equations and boundary conditions of the plane receding contact problem into a singular integral equation with the contact pressure and contact size as unknowns. Gauss–Chebyshev quadrature is subsequently employed to discretize both the singular integral equation and the force equilibrium condition at the contact interface. An iterative algorithm based on the method of steepest descent has been proposed to numerically solve the system of algebraic equations, which is linear for the contact pressure but nonlinear for the contact size. Extensive case studies are performed with respect to the coating inhomogeneity parameter, geometric parameters, material properties, and the extent of the indentation load. As a result of the indentation, the elastic layer remains in contact with the coated substrate over only a finite interval. Exterior to this region, the layer and the coated substrate lose contact. Nonetheless, the receding contact size is always larger than that of the indentation traction. To validate the theoretical solution, we have also developed a finite-element model to solve the same receding contact problem. Numerical results of finite-element modeling and theoretical development are compared in detail for a number of parametric studies and are found to agree very well with each other.

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