Probability theory has long provided a rigorous framework for quantifying uncertainty, yet its extension to infinite sets introduces profound conceptual challenges and opportunities. Contemporary ...

Probability theory forms the mathematical backbone for quantifying uncertainty and random events, providing a rigorous language with which to describe both everyday phenomena and complex scientific ...

In this paper, we consider the function f p ( t )= 2p X 2 ( 2p t+p;p ) , where χ²(x; n) defined by X 2 ( x;p )= 2 −p/2 Γ( p/2 ) e −x/2 x p/2−1 , is the density function of a χ²-distribution with n ...

In [14], Sanov proved that if FN is the empirical cumulative distribution function (cdf) of a sample drawn from a population whose true cdf is F0 and Ω is a set of cdf's which satisfies certain ...