scholarly journals Probabilistic Investigations on the Explosion of Solutions of the KAC Equation with Infinite Energy Initial Distribution

2008 ◽  
Vol 45 (01) ◽  
pp. 95-106 ◽  
Author(s):  
Eric Carlen ◽  
Ester Gabetta ◽  
Eugenio Regazzini
Keyword(s):  
Random Variables ◽  
Random Variable ◽  
Initial Distribution ◽  
Initial Energy ◽  
Infinite Variance ◽  
Borel Sets ◽  
Second Moment ◽  
Quantitative Estimates ◽  
Kac Equation ◽  
Kac’S Model

Gabetta and Regazzini (2006b) have shown that finiteness of the initial energy (second moment) is necessary and sufficient for the solution of the Kac's model Boltzmann equation to converge weakly (Cb-convergence) to a probability measure onR. Here, we complement this result by providing a detailed analysis of what does actually happen when the initial energy is infinite. In particular, we prove that such a solution converges vaguely (C0-convergence) to the zero measure (which is identically 0 on the Borel sets ofR). More precisely, we prove that the total mass of the limiting distribution splits into two equal masses (of value ½ each), and we provide quantitative estimates on the rate at which such a phenomenon takes place. The methods employed in the proofs also apply in the context of sums of weighted independent and identically distributed random variablesx̃1,x̃2, …, where these random variables have an infinite second moment and zero mean. Then, withTn:= ∑j=1ηnλj,nx̃j, with max1 ≤j≤ ηnλj,n→ 0 (asn→ +∞), and ∑j=1ηnλj,n2= 1,n= 1, 2, …, the classical central limit theorem suggests thatTshould in some sense converge to a ‘normal random variable of infinite variance’. Again, in this setting we prove quantitative estimates on the rate at which the mass splits into adherent masses to -∞ and +∞, or to ∞, that are analogous to those we have obtained for the Kac equation. Although the setting in this case is quite classical, we have not uncovered any previous results of a similar type.

scholarly journals Probabilistic Investigations on the Explosion of Solutions of the KAC Equation with Infinite Energy Initial Distribution

2008 ◽  
Vol 45 (1) ◽  
pp. 95-106 ◽  
Author(s):  
Eric Carlen ◽  
Ester Gabetta ◽  
Eugenio Regazzini
Keyword(s):  
Random Variables ◽  
Random Variable ◽  
Initial Distribution ◽  
Initial Energy ◽  
Infinite Variance ◽  
Borel Sets ◽  
Second Moment ◽  
Quantitative Estimates ◽  
Kac Equation ◽  
Kac’S Model

Gabetta and Regazzini (2006b) have shown that finiteness of the initial energy (second moment) is necessary and sufficient for the solution of the Kac's model Boltzmann equation to converge weakly (Cb-convergence) to a probability measure on R. Here, we complement this result by providing a detailed analysis of what does actually happen when the initial energy is infinite. In particular, we prove that such a solution converges vaguely (C0-convergence) to the zero measure (which is identically 0 on the Borel sets of R). More precisely, we prove that the total mass of the limiting distribution splits into two equal masses (of value ½ each), and we provide quantitative estimates on the rate at which such a phenomenon takes place. The methods employed in the proofs also apply in the context of sums of weighted independent and identically distributed random variables x̃1, x̃2, …, where these random variables have an infinite second moment and zero mean. Then, with Tn := ∑j=1ηnλj,nx̃j, with max1 ≤ j ≤ ηnλj,n → 0 (as n → +∞), and ∑j=1ηnλj,n2 = 1, n = 1, 2, …, the classical central limit theorem suggests that T should in some sense converge to a ‘normal random variable of infinite variance’. Again, in this setting we prove quantitative estimates on the rate at which the mass splits into adherent masses to -∞ and +∞, or to ∞, that are analogous to those we have obtained for the Kac equation. Although the setting in this case is quite classical, we have not uncovered any previous results of a similar type.

scholarly journals Random Variables and Product of Probability Spaces

2013 ◽  
Vol 21 (1) ◽  
pp. 33-39
Author(s):  
Hiroyuki Okazaki ◽  
Yasunari Shidama
Keyword(s):  
Random Variables ◽  
Random Variable ◽  
Borel Sets ◽  
The Real ◽  
Probability Spaces ◽  
Countably Infinite

Summary We have been working on the formalization of the probability and the randomness. In [15] and [16], we formalized some theorems concerning the real-valued random variables and the product of two probability spaces. In this article, we present the generalized formalization of [15] and [16]. First, we formalize the random variables of arbitrary set and prove the equivalence between random variable on Σ, Borel sets and a real-valued random variable on Σ. Next, we formalize the product of countably infinite probability spaces.

A Conditional-Probability Approach to the Development of Spatial Pattern

1979 ◽  
Vol 11 (3) ◽  
pp. 313-326 ◽  
Author(s):  
J Odland
Keyword(s):  
Spatial Patterns ◽  
Transition Probabilities ◽  
Random Variables ◽  
Random Variable ◽  
Initial Distribution ◽  
American City ◽  
Spatial Lag ◽  
City Space ◽  
Lag Structure ◽  
Selection Of

A stochastic model for the development of spatial patterns is introduced and used to investigate the process of housing deterioration in an American city. Space is treated as a sequence of discrete locations and a spatial-lag structure is incorporated in the model by defining multivalued random variables whose values indicate conditions at a central location and at a series of spatial lags. The possible combinations of these values define the states of a Markov process, and a description of this process can be obtained by estimating probabilities for the transitions from state to state. Qualitative inferences about the effects of a process on existing spatial patterns are obtained by comparing an initial distribution, for the multivalued random variables, with the limiting distribution implied by the process description. Application of the model involves selection of an appropriate random variable as well as estimation of a set of transition probabilities. Results for Indianapolis in 1977 indicate that the probability of housing deterioration is strongly associated with the presence of deteriorated structures in nearby locations.

scholarly journals Statistical maps: A categorical approach

2007 ◽  
Vol 57 (1) ◽  
Author(s):  
Roman Frič
Keyword(s):  
Probability Theory ◽  
Random Variables ◽  
Random Variable ◽  
Measurable Space ◽  
Probability Measures ◽  
Fuzzy Probability ◽  
Categorical Approach ◽  
Borel Sets ◽  
Fuzzy Probability Theory ◽  
Fuzzy Random

AbstractIn probability theory, each random variable f can be viewed as channel through which the probability p of the original probability space is transported to the distribution p f, a probability measure on the real Borel sets. In the realm of fuzzy probability theory, fuzzy probability measures (equivalently states) are transported via statistical maps (equivalently, fuzzy random variables, operational random variables, Markov kernels, observables). We deal with categorical aspects of the transportation of (fuzzy) probability measures on one measurable space into probability measures on another measurable spaces. A key role is played by D-posets (equivalently effect algebras) of fuzzy sets.

scholarly journals Introduction to Stochastic Finance: Random Variables and Arbitrage Theory

2018 ◽  
Vol 26 (1) ◽  
pp. 1-9
Author(s):  
Peter Jaeger
Keyword(s):  
Interest Rate ◽  
Probability Measure ◽  
Random Variables ◽  
Random Variable ◽  
Market Model ◽  
Borel Sets ◽  
Risk Neutral ◽  
Arbitrage Opportunity ◽  
The Interest Rate ◽  
Definition Of

Summary Using the Mizar system [1], [5], we start to show, that the Call-Option, the Put-Option and the Straddle (more generally defined as in the literature) are random variables ([4], p. 15), see (Def. 1) and (Def. 2). Next we construct and prove the simple random variables ([2], p. 14) in (Def. 8). In the third section, we introduce the definition of arbitrage opportunity, see (Def. 12). Next we show, that this definition can be characterized in a different way (Lemma 1.3. in [4], p. 5), see (17). In our formalization for Lemma 1.3 we make the assumption that ϕ is a sequence of real numbers (there are only finitely many valued of interest, the values of ϕ in Rd). For the definition of almost sure with probability 1 see p. 6 in [2]. Last we introduce the risk-neutral probability (Definition 1.4, p. 6 in [4]), here see (Def. 16). We give an example in real world: Suppose you have some assets like bonds (riskless assets). Then we can fix our price for these bonds with x for today and x · (1 + r) for tomorrow, r is the interest rate. So we simply assume, that in every possible market evolution of tomorrow we have a determinated value. Then every probability measure of Ωfut1 is a risk-neutral measure, see (21). This example shows the existence of some risk-neutral measure. If you find more than one of them, you can determine – with an additional conidition to the probability measures – whether a market model is arbitrage free or not (see Theorem 1.6. in [4], p. 6.) A short graph for (21): Suppose we have a portfolio with many (in this example infinitely many) assets. For asset d we have the price π(d) for today, and the price π(d) (1 + r) for tomorrow with some interest rate r > 0. Let G be a sequence of random variables on Ωfut1, Borel sets. So you have many functions fk : {1, 2, 3, 4}→ R with G(k) = fk and fk is a random variable of Ωfut1, Borel sets. For every fk we have fk(w) = π(k)·(1+r) for w {1, 2, 3, 4}. $$\matrix{ {Today} & {Tomorrow} \cr {{\rm{only}}\,{\rm{one}}\,{\rm{scenario}}} & {\left\{ {\matrix{ {w_{21} = \left\{ {1,2} \right\}} \hfill \cr {w_{22} = \left\{ {3,4} \right\}} \hfill \cr } } \right.} \cr {{\rm{for}}\,{\rm{all}}\,d \in N\,{\rm{holds}}\,\pi \left( d \right)} & {\left\{ {\matrix{ {f_d \left( w \right) = G\left( d \right)\left( w \right) = \pi \left( d \right) \cdot \left( {1 + r} \right),} \hfill \cr {w \in w_{21} \,or\,w \in w_{22} ,} \hfill \cr {r > 0\,{\rm{is}}\,{\rm{the}}\,{\rm{interest}}\,{\rm{rate}}.} \hfill \cr } } \right.} \cr }$$ Here, every probability measure of Ωfut1 is a risk-neutral measure.

scholarly journals Stochastic Comparisons of Some Distances between Random Variables

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 981
Author(s):  
Patricia Ortega-Jiménez ◽  
Miguel A. Sordo ◽  
Alfonso Suárez-Llorens
Keyword(s):  
Financial Risk ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Stochastic Ordering ◽  
Random Variable ◽  
Distribution Functions ◽  
Stochastic Comparisons ◽  
Stochastic Orderings ◽  
Absolute Value ◽  
The Difference

The aim of this paper is twofold. First, we show that the expectation of the absolute value of the difference between two copies, not necessarily independent, of a random variable is a measure of its variability in the sense of Bickel and Lehmann (1979). Moreover, if the two copies are negatively dependent through stochastic ordering, this measure is subadditive. The second purpose of this paper is to provide sufficient conditions for comparing several distances between pairs of random variables (with possibly different distribution functions) in terms of various stochastic orderings. Applications in actuarial and financial risk management are given.

scholarly journals Fully degenerate Bell polynomials associated with degenerate Poisson random variables

2021 ◽  
Vol 19 (1) ◽  
pp. 284-296
Author(s):  
Hye Kyung Kim
Keyword(s):  
Random Variables ◽  
Random Variable ◽  
Combinatorial Identities ◽  
Bell Polynomials ◽  
Poisson Random Variable ◽  
Central Moments ◽  
Special Polynomials ◽  
Special Numbers And Polynomials ◽  
New Type ◽  
Two Parameters

Abstract Many mathematicians have studied degenerate versions of quite a few special polynomials and numbers since Carlitz’s work (Utilitas Math. 15 (1979), 51–88). Recently, Kim et al. studied the degenerate gamma random variables, discrete degenerate random variables and two-variable degenerate Bell polynomials associated with Poisson degenerate central moments, etc. This paper is divided into two parts. In the first part, we introduce a new type of degenerate Bell polynomials associated with degenerate Poisson random variables with parameter α > 0 \alpha \hspace{-0.15em}\gt \hspace{-0.15em}0 , called the fully degenerate Bell polynomials. We derive some combinatorial identities for the fully degenerate Bell polynomials related to the n n th moment of the degenerate Poisson random variable, special numbers and polynomials. In the second part, we consider the fully degenerate Bell polynomials associated with degenerate Poisson random variables with two parameters α > 0 \alpha \gt 0 and β > 0 \beta \hspace{-0.15em}\gt \hspace{-0.15em}0 , called the two-variable fully degenerate Bell polynomials. We show their connection with the degenerate Poisson central moments, special numbers and polynomials.

A Note on the Accuracy of Normal Approximation of Random Quantities

2021 ◽  
Vol 73 (1) ◽  
pp. 62-67
Author(s):  
Ibrahim A. Ahmad ◽  
A. R. Mugdadi
Keyword(s):  
Sample Size ◽  
Order Statistic ◽  
Normal Approximation ◽  
Order Approximation ◽  
Random Variables ◽  
Random Variable ◽  
Limiting Distribution ◽  
Exact Order ◽  
Fixed Sample ◽  
Independent Identically Distributed

For a sequence of independent, identically distributed random variable (iid rv's) [Formula: see text] and a sequence of integer-valued random variables [Formula: see text], define the random quantiles as [Formula: see text], where [Formula: see text] denote the largest integer less than or equal to [Formula: see text], and [Formula: see text] the [Formula: see text]th order statistic in a sample [Formula: see text] and [Formula: see text]. In this note, the limiting distribution and its exact order approximation are obtained for [Formula: see text]. The limiting distribution result we obtain extends the work of several including Wretman[Formula: see text]. The exact order of normal approximation generalizes the fixed sample size results of Reiss[Formula: see text]. AMS 2000 subject classification: 60F12; 60F05; 62G30.

INFORMATION AND UNCERTAINTY IN A QUEUING SYSTEM

2007 ◽  
Vol 21 (3) ◽  
pp. 361-380 ◽  
Author(s):  
Refael Hassin
Keyword(s):  
Service Quality ◽  
Random Variables ◽  
Random Variable ◽  
Queuing System ◽  
Service Rate ◽  
Single Server ◽  
Profit Function

This article deals with the effect of information and uncertainty on profits in an unobservable single-server queuing system. We consider scenarios in which the service rate, the service quality, or the waiting conditions are random variables that are known to the server but not to the customers. We ask whether the server is motivated to reveal these parameters. We investigate the structure of the profit function and its sensitivity to the variance of the random variable. We consider and compare variations of the model according to whether the server can modify the service price after observing the realization of the random variable.

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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