We study the large deviation principle for stochastic processes of the form $\{{\sum }_{k=1}^{\infty }{x}_k\left(t\right){\xi }_k:t\in T\}$, where ${\left\{{\xi }_k\right\}}_{k=1}^{\infty }$ is a sequence of i.i.d.r.v.'s with mean zero and $x_k\left(t\right)\in ℝ$. We present necessary and sufficient conditions for the large deviation principle for these stochastic processes in several situations. Our approach is based in showing the large deviation principle of the finite dimensional distributions and an exponential asymptotic equicontinuity condition. In order to get the exponential asymptotic equicontinuity condition,...

We study the large deviation principle for stochastic processes of the form $\{{\sum }_{k=1}^{\infty }{x}_k\left(t\right){\xi }_k:t\in T\}$, where ${\left\{{\xi }_k\right\}}_{k=1}^{\infty }$ is a sequence of i.i.d.r.v.’s with mean zero and $x_k\left(t\right)\in ℝ$. We present necessary and sufficient conditions for the large deviation principle for these stochastic processes in several situations. Our approach is based in showing the large deviation principle of the finite dimensional distributions and an exponential asymptotic equicontinuity condition. In order to get the exponential asymptotic equicontinuity condition,...

The least absolute shrinkage and selection operator (LASSO) is a popular technique for simultaneous estimation and model selection. There have been a lot of studies on the large sample asymptotic distributional properties of the LASSO estimator, but it is also well-known that the asymptotic results can give a wrong picture of the LASSO estimator's actual finite-sample behaviour. The finite sample distribution of the LASSO estimator has been previously studied for the special case of orthogonal models....

We study deformations of the free convolution arising via invertible transformations of probability measures on the real line T:μ ↦ Tμ. We define new associative convolutions of measures by $\mu {⊞}_T\nu =T^{-1}\left(T\mu ⊞T\nu \right)$. We discuss infinite divisibility with respect to these convolutions, and we establish a Lévy-Khintchine formula. We conclude the paper by proving that for any such deformation of free probability all probability measures μ have the Nica-Speicher property, that is, one can find their convolution power ${\mu }^{{⊞}_{T}s}$ for...

In this paper, we introduce a new stochastic order between continuous non-negative random variables called the PLR (proportional likelihood ratio) order, which is closely related to the usual likelihood ratio order. The PLR order can be used to characterize random variables whose logarithms have log-concave (log-convex) densities. Many income random variables satisfy this property and they are said to have the IPLR (increasing proportional likelihood ratio) property (DPLR property). As an application,...

The space of distribution functions endowed with the metric introduced in [5] is separable.

We study deformations of the classical convolution. For every invertible transformation T:μ ↦ Tμ, we are able to define a new associative convolution of measures by $\mu {*}_T\nu =T^{-1}(T\mu *T\nu )$. We deal with the $V_a$-deformation of the classical convolution. We prove the analogue of the classical Lévy-Khintchine formula. We also show the central limit measure, which turns out to be the standard Gaussian measure. Moreover, we calculate the Poisson measure in the $V_a$-deformed classical convolution and give the orthogonal polynomials...