Ken-ichi Yoshihara

Conditional empirical processes defined by φ - mixing sequences

For a random vector (X, Y) in R 2 let m (y/x)=P(Y≤y|X=x) be the conditional distribution function. It is assumed that (X 1 , Y 1 ), (X 2 , Y 2 ), ... form a strictly stationary Φ-mixing sequence of random vectors with the same distribution as (X, Y). Considering an estimate mn (•|x) of m(•|x) it is shown that (nan) 1/2 {mn(•|x 0 )-m (•|x 0 )} converges to a certain Gaussian process depending on x 0 , where (an) is a sequence of bandwiths

Density estimation for samples satisfying a certain absolute regularity condition

Abstract Let {ξ i } be an absolutely regular sequence of identically distributed random variables having common density function f(x) . Let H k (x,y) ( k =1, 2,…) be a sequence of Borel-measurable functions and f n ( x )= n −1 ( H n ( x , ξ 1 )+…+ H n ( x , ξ n )) the empirical density function. In this paper, the asymptotic property of the probability P (sup x | f n ( x )− f ( x )|> e ) ( n →∞) is studied.

Limiting behavior of one-sample rank-order statistics for absolutely regular processes

SummaryLetξi be a strictly stationary, absolutely regular process defined on a probability space (Ω,A,P), i.e.,ξis satisfy the condition

Note on multidimensional empirical processes for [phi]-mixing random vectors

\beta (n) = E\{ \mathop {\sup }\limits_{A \in \mathcal{M}_n^\infty } |P(A|M_{ - \infty }^0 ) - P(A)|\} \downarrow 0

A weak convergence theorem for functionals of sums of independent random variables

whereξab (a≦b) denote theσ-algebra of events generated byσa,...σb.(It is known thatσi is absolutely regular if {σi} isφ-mixing, i.e.

Almost Sure Invariance Principles for V-and U-Statistics Based on Weakly Dependent Random Variables

\phi (n) = \mathop {\sup }\limits_{B \in \mathcal{M}_{ - \infty }^0 ,A \in \mathcal{M}_n^\infty } |P(A \cap B) - P(A)P(B)|/P(B) \downarrow 0.