變敍的項的共同分佈的極限

<正> 合X為一隨機变數,其分佈律為F(x).今對X作n次相互獨立的觀察,便得n個值,把這n個值依其大小順序排列為

ON THE LIMIT OF THE COMMON DISTRIBUTION OF TERMS OF VARIATIONAL SERIES

Let X be a random variable, whose distribution law is F(x). Arranging the results of n independent trials on X, we obtain the corresponding variational series: ξ~1≤ξ_2≤…≤ξ_n.(1) In this paper the limit of the common distribution of two middle terms and that of a side term of constant rank and a middle term are investigated. Two general theorems are obtained:Theorem 1. Suppose that as n→∞, k_1/n→λ_1,k_2/n→λ_2, 0<λ_1<λ_2<1. In order that for given sequences of constants a_n, (> 0), b_n and c_n (> 0), d_n, we have on the continuity points of Φ (x, y), it is necessary and sufficent that where and The increasing functions u(x) and v(y) are uniquely determined from Φ (x,y) by the following relations:Theorem 2. Let k_1 < k_2, where k_1 is fixed and k_2/n→λ_2 (n→∞) and 0 < λ_2<1. In order that for given sequences of constants A_n (> 0), B_n and C_n (> 0), D_n, we have, as n →∞ on the continuity points of Ψ (x,y), it is necessary and sufficient that: U_n(x) =n F(A_n x+B_n) →U(x) (n→∞), V_n(y)=F(C_n y +D_n)-λ_(k_2)/τ_(k_2)→V(y) (n→∞), where λ_(k_2) and τ_(k_2) as in Theorem 1. The increasing functions U(x) and V(x) are uniquely determined from Ψ (x,y) by the following relations: