scholarly journals Note on the Background to the Subject: Theory of Risk, Fundamental Mathematics and Applications

1961 ◽  
Vol 1 (5) ◽  
pp. 256-264 ◽  
Author(s):  
Carl Philipson
Keyword(s):  
Finite Interval ◽  
General Theorem ◽  
Compound Poisson Process ◽  
Positive Real ◽  
Completely Monotonic ◽  
The Subject ◽  
Image Position ◽  
The Right ◽  
Necessary And Sufficient ◽  
Subject Theory

By a general theorem the necessary and sufficient condition for a function φ0 (σ) being completely monotonic for σ lying in the right semi-plane, i.e. that the nth derivative with respect to σ has the sign of (— 1)n, is that the function may be represented by the Laplace-Stieltjes integral , where U (v) is a non-decreasing function of v, independent of σ and bounded in every finite interval, σ a real or complex variable represented in the right semi-plane, s a real constant ≤ the real part of σ. By the notation φn (σ) we designate , which for U (v) being independent of σ, as assumed above, is equal to .Definition 1. A compound Poisson process (in the narrow sense) is a process for which the probability distribution of the number of changes in the random function Y (t), constituting the process, occurring while the parameter, which is represented on the positive real axis, is in the interval (o, t) for every value of t, is defined by the following relationthe function φn (σ) being defined by the integral given above and subject to the condition that φ0 (σ) tends to unity, when σ tends to s. The function U (v) in the integrand of φn (σ) is, then, a distribution function which defines the risk distribution, in this case said to be t-independent.

On the rate of convergence of probabilities of moderate deviations

1970 ◽  
Vol 11 (1) ◽  
pp. 91-94 ◽  
Author(s):  
V. K. Rohatgi
Keyword(s):  
Sufficient Conditions ◽  
Moderate Deviations ◽  
Independent Random Variables ◽  
Standard Normal Distribution ◽  
Positive Real ◽  
Moment Conditions ◽  
Standard Normal ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Asymptotic Forms

Let {Xn: n ≧ 1} be a sequence of independent random variables and write Letand let . Suppose that converges in law to the standard normal distribution (see [5, 280] for necessary and sufficient conditions). Let {xn} be a monotonic sequence of positive real numbers such that xn → ∞ as n → ∞. Then as n → ∞ for all ε > 0. [6] Rubin and Sethuraman call probabilities of the form probabilities of moderate deviations and obtain asymptotic forms for such probabilities under appropriate moment conditions.

A note on a condition satisfied by certain random walks

1977 ◽  
Vol 14 (04) ◽  
pp. 843-849 ◽  
Author(s):  
R. A. Doney
Keyword(s):  
Final Section ◽  
Domain Of Attraction ◽  
Partial Result ◽  
Stable Law ◽  
Left Hand ◽  
Common Distribution ◽  
Common Distribution Function ◽  
Image Position ◽  
The Right ◽  
Necessary And Sufficient

The problem considered is to elucidate under what circumstances the condition holds, where and Xi are independent and have common distribution function F. The main result is that if F has zero mean, and (*) holds with F belongs to the domain of attraction of a completely asymmetric stable law of parameter 1/γ. The cases are also treated. (The case cannot arise in these circumstances.) A partial result is also given for the case when and the right-hand tail is ‘asymptotically larger’ than the left-hand tail. For 0 < γ < 1, (*) is known to be a necessary and sufficient condition for the arc-sine theorem to hold for Nn , the number of positive terms in (S 1, S 2, …, Sn ). In the final section we point out that in the case γ = 1 a limit theorem of a rather peculiar type can hold for Nn.

Some Moment Problems in a Finite Interval

1968 ◽  
Vol 20 ◽  
pp. 960-966 ◽  
Author(s):  
D. Leviatan
Keyword(s):  
Finite Interval ◽  
Moment Problems ◽  
Image Position ◽  
Necessary And Sufficient

Let the sequence {λi} (i ≧ 0) satisfy the following conditions.1.2.3.We shall deal with the following moment problems: what are the conditions, necessary and sufficient, on a sequence {μn} (n ≧ O) in order that it should possess the representation1.1

scholarly journals A Generalized Model for the Risk Process and its Application to a Tentative Evaluation of Outstanding Liabilities

1965 ◽  
Vol 3 (3) ◽  
pp. 215-238 ◽  
Author(s):  
Carl Philipson
Keyword(s):  
Poisson Process ◽  
Compound Poisson Process ◽  
Random Variable ◽  
Weak Sense ◽  
Risk Process ◽  
Distribution Functions ◽  
Wide Sense ◽  
Risk Distribution ◽  
Image Position ◽  
The Right

A compound Poisson process, in this context abbreviated to cPp, is defined by a probability distribution of the number m of events in the interval (o, τ) of the original scale of the process parameter, assumed to be one-dimensional, in the following form.where du shall be inserted for t, λτ being the intensity function of a Poisson process with the expected number t of events in the interval (O, τ) and U(ν, τ) is the distribution function of ν for every fixed value of τ, here called the risk distribution. If the inverse of is substituted for τ, in the right membrum of (1), the function obtained is a function of t.If the risk distribution is defined by the general form U(ν, τ) the process defined by (1) is called a cPp in the wide sense (i.w.s.). In the sequel two particular cases for U(ν, τ) shall be considered, namely when it has the form of distribution functions, which define a primary process being stationary (in the weak sense) or non-stationary, and when it is equal to U1(ν) independently of τ. The process defined by (1) is in these cases called a stationary or non-stationary (s. or n.s.)cPp and a cPpin the narrow sense (i.n.s.) respectively. If a process is non-elementary i.e. the size of one change in the random function constituting the process is a random variable, the distribution of this variable conditioned by the hypothesis that such a change has occurred at τ is here called the change distribution and denoted by V(x, τ), or, if it is independent of τ, by V1(x). In an elementary process the size of one change is a constant, so that, in this case, the change distribution reduces to the unity distribution E(x — k), where E(ξ) is equal to I, o, if ξ is non-negative, negative respectively, and k is a given constant.

Sufficiency theorems for local minimizers of the multiple integrals of the calculus of variations

2001 ◽  
Vol 131 (1) ◽  
pp. 155-184 ◽  
Author(s):  
Ali Taheri
Keyword(s):  
Field Theory ◽  
Calculus Of Variations ◽  
Sufficient Conditions ◽  
Local Minimizer ◽  
Local Minimizers ◽  
Indirect Approach ◽  
The Subject ◽  
Multiple Integrals ◽  
Image Position ◽  
The Right

Let Ω ⊂ Rn be a bounded domain and let f : Ω × RN × RN×n → R. Consider the functional over the class of Sobolev functions W1,q(Ω;RN) (1 ≤ q ≤ ∞) for which the integral on the right is well defined. In this paper we establish sufficient conditions on a given function u0 and f to ensure that u0 provides an Lr local minimizer for I where 1 ≤ r ≤ ∞. The case r = ∞ is somewhat known and there is a considerable literature on the subject treating the case min(n, N) = 1, mostly based on the field theory of the calculus of variations. The main contribution here is to present a set of sufficient conditions for the case 1 ≤ r < ∞. Our proof is based on an indirect approach and is largely motivated by an argument of Hestenes relying on the concept of ‘directional convergence’.

Random walk and electric currents in networks

1959 ◽  
Vol 55 (2) ◽  
pp. 181-194 ◽  
Author(s):  
C. St J. A. Nash-Williams
Keyword(s):  
Random Walk ◽  
Electrical Network ◽  
Positive Real ◽  
Geometrical Characterization ◽  
Locally Finite ◽  
Biased Random Walk ◽  
Geometrical Character ◽  
Complementary Set ◽  
Image Position ◽  
Necessary And Sufficient

ABSTRACTLet G be a locally finite connected graph and c be a positive real-valued function defined on its edges. Let D(ξ) denote the sum of the values of c on the edges incident with a vertex ξ. A particle starts at some vertex α and performs an infinite random walkin which (i) the ξj are vertices of G, (ii), λj. is an edge joining ξj–1 to ξj (j = 1, 2, 3, …), (iii) if λ is any edge incident with ξj, thenLet υ be a set of vertices of G such that the complementary set of vertices is finite and includes α. A geometrical characterization is given of the probability (τ, say) that the particle will visit some element of υ without first returning to α. An essentially equivalent problem is obtained by regarding G as an electrical network and c(λ) as the conductance of an edge λ; the current flowing through the network from α to υ when an external agency maintains α at potential I and all elements of υ at potential 0 is found to be τD(α).A necessary and sufficient condition (of a geometrical character) for the particle to be certain to return to α. is obtained; and, as an application, a new proof is given of a conjecture of Gillis (3) regarding centrally biased random walk on an n–dimensional lattice.

Right Bol Quasi-Fields

1969 ◽  
Vol 21 ◽  
pp. 1409-1420
Author(s):  
Michael J. Kallaher
Keyword(s):  
Division Ring ◽  
Near Field ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Image Position ◽  
The Right ◽  
Necessary And Sufficient ◽  
Bol Loops

We shall consider quasi-fields which satisfy the multiplicative Identity1.1(1.1) will be called the right Bol law and a quasi-field satisfying it will be called a right Bol quasi-held. Moufang quasi-fields, i.e., those satisfying the Moufang identity1.2were studied in (5). Quasi-fields satisfying the left Bol identity1.3were studied by Burn (3) and the author (6). Such quasi-fields are called Bol quasi-fields.Our investigation will parallel the investigations in (5; 6). In § 2 we derive necessary and sufficient conditions for a right Bol quasi-field to be an alternative division ring and also criteria for it to be a near-field. With this information we derive in §§ 3 and 4 new characterizations of Moufang planes similar to those in (5; 6).Loops satisfying (1.1) have been studied by Robinson (10). He calls such loops Bol loops.

scholarly journals Monotonicity of a ratio involving trigamma and tetragamma functions

2020 ◽  
Author(s):  
Feng Qi
Keyword(s):  
Sufficient Conditions ◽  
Laplace Transforms ◽  
Necessary And Sufficient Conditions ◽  
Exponential Functions ◽  
Monotonic Functions ◽  
Completely Monotonic ◽  
Completely Monotonic Functions ◽  
The Right ◽  
Necessary And Sufficient ◽  
Logarithmic Concavity

In the paper, by convolution theorem of the Laplace transforms, Bernstein's theorem for completely monotonic functions, and logarithmic concavity of a function involving exponential functions, the author(1) finds necessary and sufficient conditions for a ratio involving trigamma and tetragamma functions to be monotonic on the right real semi-axis;(2) and presents alternative proofs of necessary and sufficient conditions for a function and its negativity involving trigamma and tetragamma functions to be completely monotonic on the positive semi-axis.These results generalizes known conclusions recently obtained by the author.

A note on a condition satisfied by certain random walks

1977 ◽  
Vol 14 (4) ◽  
pp. 843-849 ◽  
Author(s):  
R. A. Doney
Keyword(s):  
Final Section ◽  
Domain Of Attraction ◽  
Partial Result ◽  
Stable Law ◽  
Left Hand ◽  
Common Distribution ◽  
Common Distribution Function ◽  
Image Position ◽  
The Right ◽  
Necessary And Sufficient

The problem considered is to elucidate under what circumstances the condition holds, where and Xi are independent and have common distribution function F. The main result is that if F has zero mean, and (*) holds with F belongs to the domain of attraction of a completely asymmetric stable law of parameter 1/γ. The cases are also treated. (The case cannot arise in these circumstances.) A partial result is also given for the case when and the right-hand tail is ‘asymptotically larger’ than the left-hand tail. For 0 < γ < 1, (*) is known to be a necessary and sufficient condition for the arc-sine theorem to hold for Nn, the number of positive terms in (S1, S2, …, Sn). In the final section we point out that in the case γ = 1 a limit theorem of a rather peculiar type can hold for Nn.

scholarly journals Möbius convolutions and the Riemann hypothesis

2005 ◽  
Vol 2005 (22) ◽  
pp. 3599-3608 ◽  
Author(s):  
Luis Báez-Duarte
Keyword(s):  
Real Axis ◽  
Riemann Hypothesis ◽  
Entire Functions ◽  
General Theorem ◽  
Positive Real Axis ◽  
Positive Real ◽  
Necessary And Sufficient Criteria ◽  
Necessary And Sufficient

The well-known necessary and sufficient criteria for the Riemann hypothesis of M. Riesz and of Hardy and Littlewood, based on the order of certain entire functions on the positive real axis, are here embedded in a general theorem for a class of entire functions, which in turn is seen to be a consequence of a rather transparent convolution criterion. Some properties of the convolutions involved sharpen what is hitherto known for the Riesz function.

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