DSpace at VNU: On the convergence of the product of independent random operators

J M ath K yoto Univ (JMKYAZ) 27-2 (1987) 381-385 On the convergence of the product of independent random variables By Hiroshi SATO Introduction Let {X } be a sequence of integrable random variables on a probability space P ), a b e the o -algebra generated by {X ; < k < n }, denote th e mathematical expectation by E [ ] and the mathematical expectation o n a se t A E a by E [ ; A ] {X,} is upper sem i-bounded if there exists a positive constant K such that k (0, - n k E [X ,; X k > K ] < + oo If there exists a positive constant K such that X k —1, a s , a n d E[X ]= 0, k e N T h en in Paragraph we shall show the equivalence of the almost sure convergence of E x , and the L' -conver gence of n (1 + X ) (Theorem 2) Note that if {x } is a real sequence, then the convergence _ of E x„ does not imply the convergence of n (1 + x„) (for example x = ( —1)k k ) Conversely the convergence of ri (1 + x,) does not imply the convergence of ri (1+ k k k k k (for example x = ( —1)k k + (2k) ) A s an application in Paragraph we shall give necessary and sufficient conditions for the equivalence (mutual absolute continuity) of two infinite product measures based on the convergence of marginal densities (Theorem 3) Xk ) k - Sum of semi-bounded independent random variables In this paragraph we prove the following theorem Theorem L e t { X ,} b e a sequence o f upper sem i-bounded independent random v ariables such that E[X,] > 0, k e N T hen all of the following statements are equivalent Communicated by Professor S Watanabe, Jan 26, 1986 382 Hiroshi Saw (A) E X k converges in I ) (B) sup ER ± X l] < + Œ k k =1 n (c ) E Xk converges almost surely (D ) E Xk and E X i converge almost surely P ro o f ( A ) ( B ) a n d (D )(C ) a re triv ia l (B )(C ) is proved by the Doob's X k is a 4-matringale (W Stout [3], Theorem 2-7-2) theorem since S = k=1 (C) ( D ) Since {X,J is upper semi-bounded, there exists a positive constant K such that n E E[X ; X ,>K ]< + oo (1) k Define Yk = { X k, if X l