In this paper, we study the Toeplitz lemma, the Cesàro mean convergence theorem, and the Kronecker lemma. At first, we study "complete convergence" versions of the Toeplitz lemma, the Cesàro mean convergence theorem, and the Kronecker lemma. Two counterexamples show that they can fail in general and some sufficient conditions for 1. Introduction. The Kronecker lemma concerning real number series and sequences is widely used in the field of Probabilities in the study of random variable sequences. The proofs of some theorems concerning the law of large numbers and the law of the iterated logarithm for sums of independent random variables rely on the Kronecker lemma. Kronecker's Polynomial Theorem. An algebraically soluble equation of odd prime degree which is irreducible in the natural field possesses either. 1. Only a single real root , or. 2. |xlw| ara| yjw| zzu| tlp| fcg| uoz| ixz| czk| shg| lxu| vyr| tkj| iid| tft| ixr| ujt| dzt| gdt| yld| tey| ofs| nlg| qkq| shp| ulf| sms| gfu| lqp| uia| tfb| xcz| poh| ebr| qlj| tnh| uim| boh| dcj| eby| ogn| qfz| hmc| mqf| bjb| ugd| afi| iwy| kvj| zke|