Collected Works of Jaroslav Hájek: With Commentary (Wiley by M. Hušková, R. Beran, V. Dupac

By M. Hušková, R. Beran, V. Dupac

"H?jek was once definitely a statistician of large energy who, in his rather brief lifestyles, contributed basic effects over quite a lot of topics..." V. Barnett, collage of Nottingham.H?jek's writings in statistics are usually not in basic terms seminal yet shape a robust unified physique of conception. this can be really the case along with his reports of non-parametric statistics. His booklet "The concept of Rank Test", with ?id?k, was once defined by means of W. Hoeffding as nearly the final word at the topic. H?jek's paintings nonetheless has nice value this present day, for instance his learn has proved hugely suitable to contemporary investigations on bootstrap diagnostics. a lot of H?jek's paintings is scattered during the literature and a few of it rather inaccessible, present in basic terms within the unique Czech model. This publication presents a worthwhile unified textual content of the collective works of H?jek with extra essays via the world over well known members. unquestionably this booklet could be crucial examining to trendy researchers in nonparametric facts.

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The following theorem is just as important as 28 and n simp1er - the most remarkable part being without doubt assertion (a) on uniform integrability. -00 1301 THEOREM. Let (Xn)n~o be a supermartinga1e relative to the family (F ) <0' Suppose that the family (X n ) is bounded in Ll n n(30. v. Q, n-+- oo lE [X nl 1i m < +00. (a) The family (X n ) is then uniformly integrable. v. s. 3) Proof. 1). 4) lim n-+- oo lE [X nl - lE [XKl Then 0 ~ lE [X n] - lE [X K] ~ E for all n constant large enough for the integral ~ ~ E.

THEOREM. (a) Suppose that the sequence (X-) is uniformly integrable n (which is the case in particular if the sequence (X n) is uniformly integrable or positive). , in other words £ closes X on the right. v. X00 closing X on the right, the sequence (X~) is uniformly integrable, hence £ also closes X on the right and £ ~ E [XJFooJ ~ Proof. 21. Let A E Fn . e. £ closes X. ). We saw in no. v. E [X:IF n] are uniformly integrable and so therefore also are the X~. V GENERALITIES AND THE DISCRETE CASE 24 We shall see in a moment (the martingale case, no.

V. lE [Xn+mIF n] are bounded above by Xn which is integrable, Yn is integrable if and only if limm lE[lE[Xn+mIFJ] >-00, a condition which is independent of n and means simply that limm lE [X m] > -00. If this condition holds, then (taking limits in Ll) In other words, the process Y is a martingale and the process Z a positive supermartingale. On the other hand = X- Y whence it follows that limn :IE [Zn] = O. Consequently Z is a potential. If H is a submartingale bounded above by X, then whence Hn ~ Yn as m ~ 00.

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