このレポートはIan Stewart ’GALOIS THEORY THIRD EDITION’ のChapter9 の解説である.ここでは正規拡大と分解体の概念を定義し,両者が同値であることを証明する.体はC の部分体のみを扱う. 始めにGalois の紹介をする. Galois は群論,Galois 理論の業績で不朽の名を残し...
Proof of Helly-Bray Theorem, Continuity Theorem and Cram´er -Wold Theorem Theorem(Helly-Bray)Let g(x) be bounded,continuous function and assume thatfFn(x)g is a sequence of uniformly bounded,non decreasing distribution functions which converges to F(x) at all ...
Proof of Useful Propositions as to Stochastic Convergence Proposition(a)LetfYYng be a sequence of n 1 random vectors withYYn d ! Y.SupposefXXng is a sequence of n 1 random vectors such that(XXn YYn) p ! 0Then XXn converges in distribution toYY. Proof Denote th...
Result as to Uniformly Integrable Sequence Result Suppose that there exists r >1 and M < 1 such that EjXtXt kj r < M for all t , k and that P 1 j = 1 jhj j < 1 . Define Yt = P 1 j = 1 hj Xt j . Then fYtYt kg is uniformaly integrable for any integer k. This imp...