On a Problem of Infinite Divisibility

Let $f(t)$ be a characteristic function. The question on infinite divisibility of $g_{2k}(t)=f^{(2k)}(t)/f^{(2k)}(0)$ is considered. There are given the condition for that function not to be infinite divisible. Some examples of infinite divisibility of $g_{2k}(t)$ are given.

The External World

The Yogācāra school of Mahāyāna Buddhism denies the existence of external objects, holding that only mental entities are ultimately real. This chapter examines the arguments developed by Yogācāra philosophers for that thesis, as well as objections raised by Buddhist realists. It begins with examination of Buddhist arguments against physicalism, which were principally aimed at the Cārvāka school of Indian materialism. It then discusses the route to idealism by way of the representationalist theory of sense perception that was supported by a time-lag argument. Idealism as such was subsequently supported by appeal to parsimony, as well as by considerations to do with infinite divisibility, and arguably by the claim that physical objects and cognitions are never grasped separately.

Characteristic Functions

This chapter begins with a look at convolutions and the distribution of sums of random variables. It briefly surveys complex number theory before defining the characteristic function and studying its properties, with a range of examples. The concept of infinite divisibility is introduced. The important inversion theorem is treated and finally consideration is given to characteristic functions in conditional distributions.

Gamma variance model: Fractional Fourier Transform (FRFT)

The paper examines the Fractional Fourier Transform (FRFT) based technique as a tool for obtaining the probability density function and its derivatives; and mainly for fitting stochastic model with the fundamental probabilistic relationships of infinite divisibility. The probability density functions are computed, and the distributional proprieties are reviewed for Variance-Gamma (VG) model. The VG model has been increasingly used as an alternative to the Classical Lognormal Model (CLM) in modelling asset prices. The VG model was estimated by the FRFT. The data comes from the SPY ETF historical prices. The Kolmogorov-Smirnov (KS) goodness-of-fit shows that the VG model fits better the cumulative distribution of the sample data than the CLM. The best VG model comes from the FRFT estimation.

The Implantation Argument: Simulation Theory is Proof that God Exists

I introduce the implantation argument, a new argument for the existence of God. Spatiotemporal extensions believed to exist outside of the mind, composing an external physical reality, cannot be composed of either atomlessness (infinite divisibility, atomless gunk), or of Democritean atoms (extended simples), and therefore the inner experience of an external reality containing spatiotemporal extensions believed to exist outside of the mind does not represent the external reality (inner mind does not represent external, mind-independent, reality), the mind is a mere cinematic-like mindscreen (a mindscreen simulation), implanted into the mind by a creator-God. It will be shown that only a creator-God can be the implanting creator of the mindscreen simulation (the creator of reality), and other simulation theories, such as Bostrom’s famous account, that do not involve a creator-God as the mindscreen simulation creator, involve a reification fallacy.

On the Accuracy of the Generalized Gamma Approximation to Generalized Negative Binomial Random Sums

We investigate the proximity in terms of zeta-structured metrics of generalized negative binomial random sums to generalized gamma distribution with the corresponding parameters, extending thus the zeta-structured estimates of the rate of convergence in the Rényi theorem. In particular, we derive upper bounds for the Kantorovich and the Kolmogorov metrics in the law of large numbers for negative binomial random sums of i.i.d. random variables with nonzero first moments and finite second moments. Our method is based on the representation of the generalized negative binomial distribution with the shape and exponent power parameters no greater than one as a mixed geometric law and the infinite divisibility of the negative binomial distribution.

On the Notion of Reproducibility and Its Full Implementation to Natural Exponential Families

Let F=Fθ:θ∈Θ⊂R be a family of probability distributions indexed by a parameter θ and let X1,⋯,Xn be i.i.d. r.v.’s with L(X1)=Fθ∈F. Then, F is said to be reproducible if for all θ∈Θ and n∈N, there exists a sequence (αn)n≥1 and a mapping gn:Θ→Θ,θ⟼gn(θ) such that L(αn∑i=1nXi)=Fgn(θ)∈F. In this paper, we prove that a natural exponential family F is reproducible iff it possesses a variance function which is a power function of its mean. Such a result generalizes that of Bar-Lev and Enis (1986, The Annals of Statistics) who proved a similar but partial statement under the assumption that F is steep as and under rather restricted constraints on the forms of αn and gn(θ). We show that such restrictions are not required. In addition, we examine various aspects of reproducibility, both theoretically and practically, and discuss the relationship between reproducibility, convolution and infinite divisibility. We suggest new avenues for characterizing other classes of families of distributions with respect to their reproducibility and convolution properties .