Tauberian Conditions Under Which Convergence Follows from Statistical Summability by Weighted Means

Let $$(p_n)$$ be a sequence of nonnegative numbers such that $$p_0>0$$ and $$ P_n:=\sum _{k=0}^{n}p_k\rightarrow \infty \,\,\,\,\text {as}\,\,\,\,n\rightarrow \infty . $$ Let $$(s_n)$$ be a sequence of real and complex numbers. The weighted mean of $$(s_n)$$ is defined by $$ t_n:=\frac{1}{P_n}\sum _{k=0}^{n}p_k s_k\,\,\,\,\text {for}\,\,\,\,n =0,1,2,\ldots $$ We obtain some sufficient conditions, under which the existence of the limit $$\lim s_n=\mu $$ follows from that of st- $$\lim t_n=\mu $$ , where $$\mu $$ is a finite number. If $$(s_n)$$ is a sequence of real numbers, then these Tauberian conditions are one-sided. If $$(s_n)$$ is a sequence of complex numbers, these Tauberian conditions are two-sided. These Tauberian conditions are satisfied if $$(s_n)$$ satisfies the one-sided condition of Landau type relative to $$(P_n)$$ in the case of real sequences or if $$(s_n)$$ satisfies the two-sided condition of Hardy type relative to $$(P_n)$$ in the case of complex numbers.