We study the number of real roots a Kostlan random polynomial degree d in one variable. More generally, we consider counting measure set such polynomials. compute large asymptotics central moments these variables. As consequence, obtain strong Law Large Numbers and Central Limit Theorem. In particular, almost surely equidistribute as diverges. Moreover, fluctuations their converge distribution ...

The paper proves the Strong Law of Large Numbers for integral functionals random fields with unboundedly increasing covariances. case functional data and domain asymptotics is studied. Conditions to guarantee that holds true are provided. considered scenarios include wide classes non stationary fields. discussion about application weak long-range dependent numerical examples given.

The Loeb space construction in nonstandard analysis is applied to the theory of processes to reveal basic phenomena which cannot be treated using classical methods. An asymptotic interpretation of results established here shows that for a triangular array (or a sequence) of random variables, asymptotic uncorrelatedness or asymptotic pairwise independence is necessary and sufficient for the vali...