One Form of the Strong Law of Large Numbers
著者
KINOSHITA Takuma
journal or
publication title
鹿児島大学教育学部研究紀要. 自然科学編
=Bulletin of the Faculty of Education,
Kagoshima University. Natural science
volume
13
page range
1-1
ONE FORM OF THE STRONG LAW OF IJARGE NUMBERS
By TAKUMA KINOSHITA
Kagoshima Uniyersity
§ I. 1億億0duction There are several limit theorems in the theory of probability
which are collectively known as the law of large numbers. In this paper we present
one form of the strong law of large numbers. The following theorem, which is one form of the strong law of large numbers, can be found in §47 of m.
THEOREM. If雄i is a sequence of independent functions with鼻nite variances,
such that ffnd4-0, n-., 2, ･･･････ and ∑n=-1響く-･ then the
sequence〈÷∑胡con-verges to 0 almost everywllere･
The conclusion of this theorem remains true if the assumption lf"d4-0･ n-1･ 2･
･･･････ is replaced by assu-ptions that臣(X) l≦c a･ e･ and ∑,7三I ∫-fil d･u is conver一
gent. We use the results and notation of m
§2. One fom of the Strong Law of Large Numbers. In this present section, We
establish a theorem concernmg One form of the strong law of large numbers which
described in §1. We shall arst state and prove two lemmas.
LEMMA 1. If ifni is a sequence of independent functions and C is a positive constant sueん
that lifn(坤≦c a･ e･ n-.･ 2･････ ,hen ∑n語eonyergesa･ e･ if and on,y.fboththe series
∑n=,I吾dp and ∑n=.去02 (fn) are convergent･
PROOF. This follows by applying §46･ D(1) to the sequence igni de魚ned by
gn-四囲
〟
LEMMA 2･ If (ynl is sequence of real numbers such that the series ∑n--.÷yn is
convergent･ then li詰∑i-i･. yi-0･
PROOF. Confer §47.C of m.
THEOREM. If ifni is a sequence of independent funelions, C is a positive eonslant such that
は-fn(X)∼ ≦c a･ e･ n-.･ 2･･･････` and.fEl･≡I Ifn dp and 2In:.籍are Convergent, then ,he
sequenee宮詣-i. fi) Converges ,0 0 a,most everywhere･
pROOF･ It follows from lemma 1 that ∑n讐l fun <- a･e･････From lemma 2･ the
sequ-ence (÷∑謝〉 converges to o a･ e･, since ∑書く- a･
e-REFERENCES
〔l〕 Paul R. Halmos, Measure Theory, D. Van Nostrand, New York 1950.
