The problem of recovering the multivariate probability density function f from the moments of its Radon transform Rf is studied. The approximation of the Radon transform Rf itself is obtained from the moments of f. Under the mild conditions on f the uniform rates of convergence for the proposed...

Let {X,Xn,n≥1} be a sequence of i.i.d. random variables with zero mean and finite variance. Set Sn=∑k=1nXk, EX2=σ20, λα(ϵ)=∑n=1∞P(|Sn|≥n1/2+αϵ), 0α1. In this paper, we discuss the rate of the approximation of σ1/αcα by ϵ1/αλα(ϵ) under suitable conditions, and extend the...

Let {X,Xn,n≥1} be a sequence of i.i.d. random variables with mean zero, and set Sn=∑k=1nXk, TX(t)=EX2I(|X|t). Heyde (1975) proved precise asymptotics for ∑n=1∞P(|Sn|≥nϵ) as ϵ↘0. In this paper, we obtain a convergence rate in a theorem of Heyde (1975) under a second moment...