ÎÁ ÓÑÒÎÉ×ÈÂÎÑÒÈ ÐÀÑÏÐÅÄÅËÅÍÈÉ U-ÑÒÀÒÈÑÒÈÊ ÂÒÎÐÎÃÎ ÏÎÐßÄÊÀ ∗ Î. ßíóøêÿâè÷åíå Institute of Mathematics and Informatics (Akademijos, 4, Vilnius LT - 08663, Lithuania) and Vilnius Pedagogical University (Studentu 39, Vilnius LT-08106, Lithuania), e-mail:olgjan@zebra.lt 1 Ââåäåíèå è ôîðìóëèðîâêà ðåçóëüòàòà Ïóñòü X, X1 ,...,Xn - íåçàâèñèìûå, îäèíàêîâî ðàñïðåäåëåííûå ñëó÷àéíûå âåëè÷èíû, ïðèíèìàþùèå çíà÷åíèÿ â èçìåðèìîì ïðîñòðàíñòâå (Θ, <). Ïóñòü h : Θ2 → R è g : Θ → R èçìåðèìûå ôóíêöèè, ïðèíèìàþùèå äåéñòâèòåëüíûå çíà÷åíèÿ. Ïóñòü h ñèììåòðè÷íà, òî åñòü h(x, y) = h(y, x) äëÿ âñåõ x, y ∈ Θ. Ïðåäïîëîæèì, ÷òî Eg(X) = 0, Eh(x, X) = 0, x ∈ Θ. Ðàññìîòðèì U -ñòàòèñòèêó T = T (X1 , ..., Xn ) = n−1 h(Xi , Xk ) + n−1 X X g(Xi ). (1) 1≤i≤n 1≤i<k≤n Ïóñòü Z, Z1 ,...,Zn - íåçàâèñèìûå, îäèíàêîâî ðàñïðåäåëåííûå ñëó÷àéíûå âåëè÷èíû, òàêæå ïðèíèìàþùèå çíà÷åíèÿ â èçìåðèìîì ïðîñòðàíñòâå (Θ, <), äëÿ êîòîðûõ âûïîëíåííû ñëåäóþùèå óñëîâèÿ E ∗ g(Z) = 0, E h(x, Z) = 0, E g 2 (Z) = Eg 2 (X), (2) The research was partially supported by the Lithuanian State Science and Stud- ies Foundation, grant No. T-15/07 1 E E h(x, Z)h(y, Z) = Eh(x, X)h(y, X), h(x, Z)g(Z) = Eh(x, X)g(X) x ∈ Θ. Îáîçíà÷èì L(Y ) ðàñïðåäåëåíèå ñëó÷àéíîé âåëè÷èíû Y .  íàñòîÿùåé ðàáîòå äîêàçûâàåòñÿ, ÷òî L(T (X1 , ..., Xn )) cáëèæàåòñÿ ñ L(T (Z1 , ..., Zn )), êîãäà n → ∞ ïðè âûïîëíåíèè óñëîâèé (2), (4) è îöåíèâàåòñÿ ñêîðîñòü ýòîãî ñáëèæåíèÿ. Òàêèì îáðàçîì, äîêàçûâàåòñÿ, ÷òî ïðè ñîáëþäåíèè ïåðå÷èñëåííûõ ìîìåíòíûõ óñëîâèé ðàñïðåäåëåíèÿ U -ñòàòèñòèêè âòîðîãî ïîðÿäêà îáëàäàþò ñâîéñòâîì óñòîé÷èâîñòè â òîì ñìûñëå, ÷òî îíè ñëàáî çàâèñÿò îò ðàñïðåäåëåíèÿ íà÷àëüíîé ñëó÷àéíîé âåëè÷èíû X . Ïîëó÷åííûé ðåçóëüòàò ìîæíî òàêæå ðàññìàòðèâàòü êàê ïðîìåæóòî÷íûé ïðè íàõîæäåíèè ñêîðîñòè ñõîäèìîñòè ñòàòèñòèêè T ê ñâîåìó ïðåäåëüíîìó ðàñïðåäåëåíèþ. Ââåäåì îáîçíà÷åíèå ∆n = ρ(L(T (X1 , ..., Xn )), L(T (Z1 , ..., Zn ))), ãäå ρ - êîëìîãîðîâñêàÿ (èëè ðàâíîìåðíàÿ) ìåòðèêà. Ïóñòü q1 , q2 , ...- ñîáñòâåííûå çíà÷åíèÿ îïåðàòîðà Ãèëüáåðòà-Øìèäòà Q, ñîîòâåòñòâóþùåãî ÿäðó h (ñì. ðàçäåë 2). Íå òåðÿÿ îáùíîñòè, ìû ìîæåì ïðåäïîëîæèòü, ÷òî |q1 | ≥ |q2 | ≥ ... .  äàëüíåéøåì áóêâîé c ìû áóäåì îáîçíà÷àòü ïîëîæèòåëüíûå àáñîëþòíûå êîíñòàíòû, êîòîðûå ìîãóò ìåíÿòüñÿ îò ñòðî÷êè ê ñòðî÷êå è îò ôîðìóëû ê ôîðìóëå. Íèæå äîêàçàíà ñëåäóþùàÿ òåîðåìà. Îáîçíà÷èì γs = E |g(X)|s , γ¯s = E |g(Z)|s , βs = E |h(X, X1 )|s , β¯s = E |h(Z, Z1 )|s , (3) ãäå s > 0, β := max(β3 +β18/5 +n−3/2 γ3 +n−9/5 γ18/5 +1, β¯3 +β̄18/5 +n−3/2 γ¯3 +n−9/5 γ̄18/5 +1), è ïðåäïîëîæèì, ÷òî β < ∞, β2 > 0, β̄2 > 0. (4) Óñëîâèå β2 > 0 îçíà÷àåò, ÷òî êâàäðàòè÷íàÿ ÷àñòü ñòàòèñòèêè T íå ÿâëÿåòñÿ àññèìïòîòè÷åñêè ïðåíåáðåæèìîé, à, ñëåäîâàòåëüíî, ñòàòèñòèêà T íå ÿâëÿåòñÿ àññèìïòîòè÷åñêè íîðìàëüíîé. 2 Òåîðåìà. Ïðè âûïîëíåíèè óñëîâèé (2) è (4) ñïðàâåäëèâî íåðàâåíñòâî: ∆n ≤ cβ 1/6 |q1 |−1/2 n−1/12 . (5) Ïîðÿäîê îöåíêè ÿâëÿåòñÿ îïòèìàëüíûì, ò.ê. Â.Ñåíàòîâûì áûëî ïîêàçàíî â [5], ÷òî â ÖÏÒ â ìíîãîìåðíîì ýâêëèäîâîì ïðîñòðàíñòâå äëÿ øàðîâ, öåíòð êîòîðûõ íàõîäèòñÿ íå â íóëå, ñîîòâåòñòâóþùèå −k/12 ñêîðîñòè ñõîäèìîñòè îïðåäåëÿþòñÿ âûðàæåíèåì O( (q1n...qk )1/2 ), ïðè óñëîâèè, ÷òî |qk | > 0 è k ≤ 6.  íàøåì ñëó÷àå èìååòñÿ èíôîðìàöèÿ ëèøü î ïåðâîì ñîáñòâåííîì çíà÷åíèè, ïîýòîìó ïîðÿäîê îöåíêè áóäåò ðàâåí 1/12. Ïîëüçóÿñü ñëó÷àåì àâòîð õî÷åò ïîáëàãîäàðèòü ïðîôåññîðà Â. Áåíòêóñà, êîòîðûé ñôîðìóëèðîâàë ðàññìàòðèâàåìóþ çàäà÷ó, à òàêæå ïîäñêàçàë ãëàâíóþ èäåþ äîêàçàòåëüñòâà. Àâòîð òàêæå õî÷åò ïîáëàãîäàðèòü ïðîôåññîðà Â. Ñåíàòîâà çà êîíñóëüòàöèè ïî âîïðîñó ïîñòðîåíèÿ îöåíêè ñíèçó. 2 Ñïåöèàëüíîå ïðåäñòàâëåíèå U -ñòàòèñòèêè âòîðîãî ïîðÿäêà Áåíòêóñ è Ãåòöå ïðåäëîæèëè â [4] ñëåäóþùåå ïðåäñòàâëåíèå U ñòàòèñòèêè âòîðîãî ïîðÿäêà. Ðàññìîòðèì èçìåðèìîå ïðîñòðàíñòâî (Θ, <, µ) ñ ìåðîé µ = L(X), ÿâëÿþùåéñÿ ðàñïðåäåëåíèåì ñ.â. X . Îáîçíà÷èì L2 = L2 (Θ, <, µ) ãèëüáåðòîâî ïðîñòðàíñòâî èíòåãðèðóåìûõ â êâàäðàòå äåéñòâèòåëüíûõ ôóíêöèé. Îïåðàòîð Ãèëüáåðòà-Øìèäòà äëÿ ÿäðà h îïðåäåëÿåòñÿ ñëåäóþùèì îáðàçîì: Qf (x) = Z h(x, y)f (y)µ(dy), f ∈ L2 . Θ Îáîçíà÷èì {ej : j ≥ 1} ïîëíóþ îðòîíîðìèðîâàííóþ ñèñòåìó ñîáñòâåííûõ ôóíêöèé îïåðàòîðà Q. Ïðè ýòîì ìû ìîæåì âûáðàòü íóìåðàöèþ òàêèì îáðàçîì, ÷òîáû àáñîëþòíûå âåëè÷èíû ñîîòâåòñòâóþùèõ ñîáñòâåííûõ çíà÷åíèé óáûâàëè ñ ðîñòîì íîìåðà: |q1 | ≥ |q2 | ≥ .... Ñïðàâåäëèâî ðàâåíñòâî E h2 (X, X1 ) = X qj2 < ∞, h(x, y) = j≥1 X j≥1 3 qj ej (x)ej (y). (6) Ðàññìîòðèì ïîäïðîñòðàíñòâî L2 (g, h) ⊂ L2 , ïîðîæäàåìîå ôóíêöèåé g è ñîáñòâåííûìè ôóíêöèÿìè ej , ñîîòâåòñòâóþùèìè ñîáñòâåííûì çíà÷åíèÿì qj 6= 0, j = 1, 2, .... Ââîäÿ íîðìèðîâàííóþ ñîáñòâåííóþ ôóíêöèþ e0 , òàêóþ ÷òî Qe0 = 0, ìû ìîæåì ñ÷èòàòü, ÷òî e0 , e1 , ... îðòîíîðìèðîâàííûé áàçèñ ïîäïðîñòðàíñòâà L2 (g, h). Ïîýòîìó, g(X) = X aj ej (X), γ2 = E g 2 (X) = j≥0 X a2j , (7) j≥0 ãäå aj = E g(X)ej (X). Íåòðóäíî âèäåòü, ÷òî E ej (X) = 0, äëÿ âñåõ j . Ñëåäîâàòåëüíî, (ej (X))j≥0 ÿâëÿåòñÿ îðòîíîðìèðîâàííîé ñèñòåìîé ñîáñòâåííûõ ôóíêöèé, ìàòåìàòè÷åñêîå îæèäàíèå êîòîðûõ ðàâíî 0. Ìû áóäåì ïðåäïîëàãàòü, ÷òî âñå ñëó÷àéíûå âåëè÷èíû è âåêòîðû ÿâëÿþòñÿ íåçàâèñèìûìè, åñëè íå îãîâîðåíî ïðîòèâíîå. Îáîçíà÷èì ∞ ïðîñòðàíñòâî âñåõ äåéñòâèòåëüíûõ ïîñëåäîâàòåëüíîñòåé x = (x0 , x1 , x2 , ...), xj ∈ . Ãèëüáåðòîâî ïðîñòðàíñòâî l2 ⊂ ∞ ñîñòîèò èç x ∈ ∞ , òàêèõ, ÷òî R R R R |x|2 =def hx, xi, |x| < ∞, hx, yi = X xj yj . j≥0 Ðàññìîòðèì ñëó÷àéíûé âåêòîð X =def (e0 (X), e1 (X), ...), R êîòîðûé ïðèíèìàåò çíà÷åíèÿ â ∞ . Òàê êàê {ej (X)}j≥0 ÿâëÿåòñÿ ñèñòåìîé íå êîððåëèðîâàííûõ ñëó÷àéíûõ âåëè÷èí ñî ñðåäíèì 0 è äèñïåðñèåé, ðàâíîé 1, ñëó÷àéíûé âåêòîð X èìååò åäèíè÷íóþ êîâàðèàöèîííóþ ìàòðèöó è ñðåäíåå 0. Èñïîëüçóÿ (6) è (7), çàïèøåì: h(X, X1 ) = hQX, X1 i, g(X) = ha, Xi, R (8) R ãäå Qx = (0, q1 x1 , q2 x2 , ...), äëÿ x ∈ ∞ , è a = (aj )j≥0 ∈ ∞ . Ðàâåíñòâà (8) ïîçâîëÿþò ñ÷èòàòü, ÷òî Θ ÿâëÿåòñÿ ïðîñòðàíñòâîì ∞ , ñëó÷àéíûé âåêòîð X ÿâëÿåòñÿ ñëó÷àéíûì âåêòîðîì, ïðèíèìàþùèì çíà÷åíèÿ â ∞ , ñî ñðåäíèì 0 è åäèíè÷íîé êîâàðèàöèîííîé ìàòðèöåé è, êðîìå òîãî, R R h(X, X1 ) = hQX, X1 i, g(X) = ha, Xi.  ÷àñòíîñòè, íå òåðÿÿ îáùíîñòè, ìû ìîæåì ïðåäïîëîæèòü, ÷òî h(x, y) è g(x) ÿâëÿþòñÿ ëèíåéíûìè ôóíêöèÿìè êàæäîãî èç èõ àðãóìåíòîâ. 4 Ïóñòü G, Gi , 1 ≤ i ≤ n, - íåçàâèñèìûå îäèíàêîâî ðàñïðåäåëåííûå ñëó÷àéíûå âåêòîðû Gi = (G1,i , G2,i , ...) ñî çíà÷åíèÿìè â ∞ , ãäå G1,i , G2,i , ... - íåçàâèñèìûå îäèíàêîâî ñòàíäàðòíî íîðìàëüíî ðàñïðåäåëåííûå ñëó÷àéíûå âåëè÷èíû.  [4] íà ñòð. 461 ïîêàçàíî, ÷òî ñëó÷àéíûå âåëè÷èíû G ìîæíî âûáðàòü òàê ÷òîáû áûëè ñïðàâåäëèâû ðàâåíñòâà R E g(G) = 0, E E 3 E (9) g 2 (G) = Eg 2 (X), h(x, G) = 0, E h(x, G)h(y, G) = Eh(x, X)h(y, X), h(x, G)g(G) = Eh(x, X)g(X) x ∈ Θ. Ëåììà Äëÿ äîêàçàòåëüñòâà òåîðåìû íàì íóæíà ñëåäóþùàÿ ëåììà. Ëåììà. Ïóñòü η äåéñòâèòåëüíàÿ ñòàíäàðòíî íîðìàëüíî ðàñïðåäåëåííàÿ ñëó÷àéíàÿ âåëè÷èíà è q1 > 0. Òîãäà äëÿ ôóíêöèè ðàñïðåäåëåíèÿ H(x) = P{q1 η 2 < x} âûïîëíÿåòñÿ óñëîâèå Ëèïøèöà. −1/2 √ |H(x + ε) − H(x)| ≤ cq1 (10) ε. Ïëîòíîñòü ñëó÷àéíîé âåëè÷èíû η 2 çàïèñûâàåòñÿ ñëåäóþùèì îáðàçîì: Äîêàçàòåëüñòâî ëåììû. f (x) = √ 1 √ e−x/2 , 2Γ(1/2) x ïðè x > 0, è f (x) = 0, ïðè x ≤ 0. Îáîçíà÷èì H1 (x) ôóíêöèþ ðàñïðåäåëåíèÿ ñëó÷àéíîé âåëè÷èíû η 2 . Òîãäà Z x+ε H1 (x + ε) − H1 (x) = f (u)du. x 0 Àíàëèçèðóÿ ïåðâóþ ïðîèçâîäíóþ H1 ýòîé ôóíêöèè, ëåãêî âèäåòü, ÷òî ïîñëåäíÿÿ íå âîçðàñòàåò. Ïîýòîìó ìû ìîæåì çàïèñàòü |H1 (x + ε) − H1 (x)| ≤ c Z 0 ε √ e−u √ du ≤ c ε. u Ïåðåõîäÿ òåïåðü îò ôóíêöèè ðàñïðåäåëåíèÿ H1 ê ôóíêöèè ðàñïðåäåëåíèÿ H , ïîëó÷àåì óòâåðæäåíèå ëåììû. Äîêàæåì íàøó òåîðåìó, ò.å. ïîëó÷èì îöåíêó (5). 5 4 Äîêàçàòåëüñòâî òåîðåìû Ââåäåì îáîçíà÷åíèÿ ηi = Gi /qi , i = 1, .... Èñïîëüçóÿ íåçàâèñèìîñòü ηi , ηj , i 6= j è äîêàçàííóþ ëåììó, íåòðóäíî âèäåòü, ÷òî ôóíêöèÿ ðàñïðåäåëåíèÿ ñòàòèñòèêè T (G1 , ..., Gn ) óäîâëåòâîðÿåò óñëîâèþ Ëèïøèöà ñ ïîêàçàòåëåì 1/2. Äîêàæåì, ÷òî äëÿ ëþáîãî ε > 0 ñïðàâåäëèâî íåðàâåíñòâî: 0 c √ ρ(L(T (X1 , ..., Xn )), L(T (G1 , ..., Gn ))) ≤ q ε + ∆, |q1 | (11) ãäå ∆ = max |Eϕ(T (X1 , ..., Xn )) − Eϕ(T (G1 , ..., Gn ))|, ϕ è max áåðåòñÿ ïî âñåì áåñêîíå÷íî äèôôåðåíöèðóåìûì ôóíêöèÿì ϕ òàêèì, ÷òî |ϕ(k) (u)| ≤ c1 /εk , k = 1, 2, 3, è 0 ≤ ϕ(u) ≤ 1, ïðè÷åì, ëèáî ϕ(u) = 1, ïðè u ≤ x − , è ϕ(u) = 0, ïðè u ≥ x, (12) ëèáî ϕ(u) = 1, ïðè u ≤ x, è ϕ(u) = 0, ïðè u ≥ x + . (13) 0 Çäåñü è äàëåå c , c1 , c2 , ... - íåêîòîðûå àáñîëþòíûå êîíñòàíòû. Ïîëîæèì δ ∗ = P{T (X1 , ..., Xn ) ≤ x} − P{T (G1 , ..., Gn ) ≤ x}. Äîêàæåì (11) äëÿ ñëó÷àÿ δ ∗ ≥ 0. Ðàññìîòðèì ôóíêöèþ ϕ äëÿ ñëó÷àÿ, êîãäà âûïîëíÿåòñÿ óñëîâèå (13). Òîãäà ñïðàâåäëèâî ñëåäóþùåå íåðàâåíñòâî: δ ∗ = EI{T (X1 , ..., Xn ) ≤ x} − P{T (G1 , ..., Gn ) ≤ x} ≤ |Eϕ(T (X1 , ..., Xn )) − Eϕ(T (G1 , ..., Gn ))| + |Eϕ(T (G1 , ..., Gn )) − P{T (G1 , ..., Gn ) ≤ x}| ≤ ∆ + P{x ≤ T (G1 , ..., Gn ) ≤ x + ε}. Èñïîëüçóÿ óñëîâèå Ëèïøèöà, ïîëó÷àåì (11). Ñëó÷àé, êîãäà δ ∗ < 0, ðàññìàòðèâàåòñÿ àíàëîãè÷íî. Îöåíèì ∆∗ (ϕ) = |Eϕ(T (X1 , ..., Xn )) − Eϕ(T (G1 , ..., Gn ))|. Íåòðóäíî âèäåòü, ÷òî ∆∗ (ϕ) ≤ |Eϕ(T (X1 , ..., Xn )) − Eϕ(T (X1 , ..., Xn−1 , Gn ))|+ 6 |Eϕ(T (X1 , ..., Xn−1 , Gn )) − Eϕ(T (X1 , ..., Xn−2 , Gn−1 , Gn ))|+ (14) ... + |Eϕ(T (X1 , G2 , ..., Gn )) − Eϕ(T (G1 , ..., Gn ))| = ∆∗1,n + ... + ∆∗n,n . Äîêàæåì, ÷òî ρ(T (X1 , ..., Xn ), T (G1 , ..., Gn )) ≤ c2 |q1 |−1/2 β 1/6 n−1/12 . (15) Äëÿ ýòîãî èñïîëüçóåì èíäóêöèþ ïî n. Ïðåäïîëîæèì, ÷òî äëÿ âñåõ m ≤ n − 1, ñïðàâåäëèâî íåðàâåíñòâî ρ(L(T (X1 , ..., Xi−1 , Gi , ..., Gm )), L(T (G1 , ..., Gm ))) ≤ c2 |q1 |−1/2 β 1/6 m−1/12 (16) äëÿ âñåõ 2 ≤ i ≤ m è âñåõ ôóíêöèé h è g , äëÿ êîòîðûõ âûïîëíÿþòñÿ óñëîâèÿ (6), (7) äëÿ çàäàííûõ qi è ai . Ëåãêî âèäåòü, ÷òî íåðàâåíñòâî (16) ñïðàâåäëèâî äëÿ m = 2. Äåéñòâèòåëüíî, ïóñòü m = 2. Òàê êàê ρ ≤ 1, äîñòàòî÷íî ïîêàçàòü, ÷òî |q1 |−1/2 β 1/6 ≥ 1. Íåòðóäíî âèäåòü, ÷òî |q1 |−1/2 β 1/6 ≥ |q1 |−1/2 (E |h|3 )1/6 = (q1−2 (E |h|3 )2/3 )1/4 . Èñïîëüçóÿ (6), ïîëó÷àåì |q1 |−1/2 β 1/6 ≥ (q1−2 E |h|2 )1/4 = (q1−2 (q12 + q22 + ...))1/4 ≥ 1. Óòâåðæäåíèå äîêàçàíî. Äîêàæåì, ÷òî íåðàâåíñòâî (16) âûïîëíÿåòñÿ äëÿ m = n. Äåéñòâèòåëüíî, T (X1 , ..., Xn ) = 1 {h(X1 , X2 )+g(X1 )+...+h(X1 , Xn−1 )+g(Xn−1 )+h(X1 , Xn )+g(Xn )+ n h(X2 , X3 ) + ... + h(X2 , Xn )+ ... h(Xn−2 , Xn−1 ) + h(Xn−2 , Xn )+ h(Xn−1 , Xn )}. Îáîçíà÷èì w - ñóììó âñåõ ÷ëåíîâ, êîòîðûå íå ñîäåðæàò àðãóìåíòà Xn : w= 1 {h(X1 , X2 )+g(X1 )+...+h(X1 , Xn−1 )+g(Xn−1 )+...+h(Xn−2 , Xn−1 )} n 7 è ln - ñóììó âñåõ ÷ëåíîâ, êîòîðûå ñîäåðæàò àðãóìåíò Xn : ln = 1 {h(X1 , Xn ) + g(Xn ) + ... + h(Xn−2 , Xn ) + h(Xn−1 , Xn )}. n Çàìåíÿÿ Xn íà Gn , ïîëó÷àåì 1 T (X1 , ..., Xn−1 , Gn ) = w+ {h(X1 , Gn )+g(Gn )+...+h(Xn−2 , Gn )+h(Xn−1 , Gn )} = n w + ln∗ , ãäå ln∗ - ìû ïîëó÷èëè èç ln çàìåíÿÿ Xn íà Gn . Ðàçëîæèì â ðÿä Òåéëîðà 1 000 1 00 0 ϕ(x + y) = ϕ(x) + ϕ (x)y + ϕ (x)y 2 + Eϕ (x + τ y)(1 − τ )2 y 3 . 2 2 Çäåñü τ - ðàâíîìåðíî ðàñïðåäåëåííàÿ íà èíòåðâàëå [0, 1] è íå çàâèñÿùàÿ îò äðóãèõ ñëó÷àéíàÿ âåëè÷èíà. Çàïèøåì ðàçëîæåíèå äëÿ x = w è y = ln , ïîëó÷èì ∆∗1,n = |Eϕ(T (X1 , ..., Xn ))−Eϕ(T (X1 , ..., Xn−1 , Gn ))| = Eϕ(w)−Eϕ(w)+ 1 00 1 00 2 ∗ 2 Eϕ (w)ln − Eϕ (w)(ln ) + 2 2 1 000 1 000 3 2 ∗ ∗ 3 2 Eϕ (w + ln τ )ln (1 − τ ) − Eϕ (w + ln τ )(ln ) (1 − τ ) . 2 2 Ôèêñèðóåì ñëó÷àéíûå âåëè÷èíû X1 , ..., Xn−1 , τ . Îáîçíà÷èì EX1 ,...,Xn−1 ,τ ìàòåìàòè÷åñêîå îæèäàíèå ïî ñëó÷àéíûì âåëè÷èíàì X1 , ..., Xn−1 , τ , EXn - ìàòåìàòè÷åñêîå îæèäàíèå ïî ñëó÷àéíîé âåëè÷èíå Xn è òàê äàëåå. Ïåðåïèøåì âûðàæåíèå ñëåäóþùèì îáðàçîì: E 0 0 ϕ (w)ln − Eϕ (w)ln∗ + 0 0 ∆∗1,n = EX1 ,...,Xn−1 ,τ ϕ (w)EXn ln − EX1 ,...,Xn−1 ,τ ϕ (w)EXn ln∗ + 1 1 00 00 2 ∗ 2 EX1 ,...,Xn−1 ,τ ϕ (w)EXn ln − EX1 ,...,Xn−1 ,τ ϕ (w)EXn (ln ) + 2 2 1 000 1 000 3 2 ∗ ∗ 3 2 Eϕ (w + ln τ )ln (1 − τ ) − Eϕ (w + ln τ )(ln ) (1 − τ ) . 2 2 Èñïîëüçóÿ (9), ïîëó÷àåì 1 ∆∗1,n = 2 E 000 ϕ (w + ln τ )ln3 (1 − τ )2 − 8 1 000 ∗ ∗ 3 2 Eϕ (w + ln τ )(ln ) (1 − τ ) . (17) 2 000 Îáîçíà÷èì |Eϕ (w + ln τ )ln3 (1 − τ )2 | áóêâîé ρ è îöåíèì åãî çíà÷åíèå. Ïðåäïîëîæèì, ÷òî n − 1 - ÷åòíîå ÷èñëî.  ïðîòèâíîì ñëó÷àåìû P 0 ìîæåì ðàññìàòðèâàòü ñòàòèñòèêó T = (n−1)−1 1≤i<k≤n−1 h(Xi , Xk )+ P n−1 1≤i≤n−1 g(Xi ). Ëåãêî âèäåòü, ÷òî ïîðÿäîê ñêîðîñòè ñõîäèìîñòè ñòàòèñòèêè T ñîâïàäàåò ñ ïîðÿäêîì ñêîðîñòè ñõîäèìîñòè ñòàòèñòèêè 0 T . Çàïèøåì ln ñëåäóþùèì îáðàçîì: 0 00 ln = ln + ln , ãäå 0 ln = X X 1 1 00 (g(Xn ) + h(Xk , Xn )), ln = h(Xk , Xn ). n n 1≤k≤(n−1)/2 (n−1)/2<k≤n−1 Èñïîëüçóÿ íåðàâåíñòâî (a + b)3 ≤ 8(a3 + b3 ), ñïðàâåäëèâîå äëÿ a > 0, b > 0, ïîëó÷àåì 8c1 0 00 0 3 0 00 00 3 ( E I(x ≤ w+τ ln +τ ln ≤ x+ε)|ln | +E I(x ≤ w+τ ln +τ ln ≤ x+ε)|ln | ) = ε3 (18) 8c1 (ρ1 + ρ2 ). ε3 Îöåíèì òîëüêî ρ1 , ò.ê. îöåíêà ρ2 àíàëîãè÷íà. Ôèêñèðóåì ñëó÷àéíûå âåëè÷èíû X1 , ..., X(n−1)/2 , Xn , τ. Ïóñòü E∗ = EX(n+1)/2 ,...,Xn−1 , òîãäà ρ≤ 0 0 00 ρ1 = E |ln |3 E∗ I(x ≤ w + τ ln + τ ln ≤ x + ε) ≤ E |ln |3 sup P x ≤ w + τ ln ≤ x + ε 0 00 n o x (ìû èñïîëüçóåì óñëîâèå íåçàâèñèìîñòè) 0 n o = E |ln |3 sup P x ≤ T ∗ ≤ x + ε , x 00 ãäå T ∗ = T ∗ (X(n+1)/2 , ..., Xn−1 ). Ìû ïîëó÷èëè T ∗ èç w + τ ln , êîãäà çàôèêñèðîâàëè X1 , ..., X(n−1)/2 , Xn , τ. Ñîîòâåòñòâóþùàÿ ôóíêöèÿ h∗ = h îñòàåòñÿ áåç èçìåíåíèÿ, à g ∗ èìååò ñëåäóþùèé âèä g ∗ (Xj ) = g(Xj )+ X h(Xk , Xj )+h(Xn , Xj ), j = (n+1)/2, ..., n−1. 1≤k≤(n−1)/2 9 Èñïîëüçóÿ íåðàâåíñòâî (16), ïîëó÷àåì 0 2c2 4β ∗ 2 1/12 ε+ q , |q1 | |q1 | n − 1 c ∗ √ sup P(x ≤ T ≤ x + ε) ≤ q x (19) ∗ ãäå β ∗ = β3 + β18/5 + 23/2 (n − 1)−3/2 γ3∗ + 29/5 (n − 1)−9/5 γ18/5 + 1, γs∗ = ∗ s E∗ |g (X)| . Èñïîëüçóÿ òåîðåìó 20 èç [5], p.89, ïîëó÷àåì 0 n18/5 E|ln |18/5 = E|g(Xn ) + h(Xk , Xn )|18/5 ≤ X 1≤k≤(n−1)/2 c4 (γ18/5 + ((n − 1)/2)9/5 β18/5 ) ≤ c4 βn9/5 . Àíàëîãè÷íûì îáðàçîì ïîëó÷àåì 0 n3 E|ln |3 = E|g(Xn ) + X h(Xk , Xn )|3 ≤ 1≤k≤(n−1)/2 c3 (γ3 + ((n − 1)/2)3/2 β3 ). Íåòðóäíî âèäåòü, ÷òî 0 (20) |l |3 ≤ c3 βn−3/2 . E n Îöåíèì E 0 (|ln |3 β ∗ 1/6 ). Èñïîëüçóÿ íåðàâåíñòâî Ãåëüäåðà, ïîëó÷àåì E 0 0 (|ln |3 β ∗ 1/6 ) ≤ (E |ln |18/5 )5/6 (E β ∗ )1/6 = 0 (E |ln |18/5 )5/6 (E (β3 +β18/5 +23/2 (n−1)−3/2 γ3∗ +29/5 (n−1)−9/5 |g ∗ (X)|18/5 ))1/6 . Ñïðàâåäëèâû ñëåäóþùèå îöåíêè E |g ∗ (X)|18/5 ≤ c4 (γ18/5 +((n+1)/2)9/5 β18/5 ), E |g ∗ (X)|3 ≤ c4 (γ3 +((n+1)/2)3/2 β3 ). Cëåäîâàòåëüíî, E 0 1/6 0 (|ln |3 β ∗ 1/6 ) ≤ 2c4 β 1/6 (E |ln |18/5 )5/6 ≤ 2c4 βn−3/2 . Îáúåäèíÿÿ (19), (20), (21) è ó÷èòûâàÿ, ÷òî β > 1, ïîëó÷àåì β ρ1 ≤ q |q1 |n3/2 √ 0 c c3 ε + 6c2 c4 β 1/6 (n − 1)−1/12 ). 10 (21) Àíàëîãè÷íûå îöåíêè ñïðàâåäëèâû äëÿ âòîðîãî ñëàãàåìîãî â (17). Òàêèì îáðàçîì, √ c1 β 0 (c c3 ε + 6c2 c4 β 1/6 (n − 1)−1/12 ). ∆∗1,n ≤ 16 q |q1 |ε3 n3/2 Ïóñòü ε = δn−1/6 β 1/3 , òîãäà ïîëó÷àåì √ c0 c δ 6c2 c4 −1/2 1/6 −13/12 3 ∗ ∆1,n ≤ 16c1 + |q1 | β (n) . δ3 δ3 Âñïîìèíàÿ (14) è (11), ïîëó÷àåì 0 −1/2 ρ(L(T (X1 , ..., Xn )), L(T (G1 , ..., Gn ))) ≤ c |q1 | 0√ c0 c √ ε+16c1 c0 c √ 3 δ3 δ 6c2 c4 −1/2 1/6 −1/12 + 3 |q1 | β (n) = δ 6c2 c4 −1 1/6 −1/12 |q1 | β (n) . δ3 Âûáèðàÿ δ è c2 òàê, ÷òîáû ñîîòíîøåíèÿ 0√ c2 ≥ 2c δ, 16c1 (c3 + 12c4 ) ≤ δ 3 , c δ + 16c1 3 δ3 δ √ + èìåëè ìåñòî, ïîëó÷àåì ρ(L(T (X1 , ..., Xn )), L(T (G1 , ..., Gn ))) ≤ c2 |q1 |−1/2 β 1/6 n−1/12 . Èñïîëüçóÿ òåïåðü íåðàâåíñòâî òðåóãîëüíèêà ∆n ≤ ρ(L(T (X1 , ..., Xn )), L(T (G1 , ..., Gn )))+ρ(L(T (G1 , ..., Gn )), L(T (Z1 , ..., Zn ))), è îöåíèâàÿ àíàëîãè÷íî ïðåäûäóùåìó âòîðîå ñëàãàåìîå, ïîëó÷àåì óòâåðæäåíèå òåîðåìû. Òåîðåìà äîêàçàíà. Ëèòåðàòóðà 1. Êîðîëþê Â.Ñ., Áîðîâñêèõ Þ.Â.,Ñêîðîñòü ñõîäèìîñòè âûðîæäåííîãî ôóíêöèîíàëà Ôîí-Ìèçåñà. Òåîð. âåð. è åå ïðèì. 33 (1988), .1, 136146. 2. Êðàìåð Ã., Ìàòåìàòè÷åñêèå ìåòîäû ñòàòèñòèêè, Ì.,(1975). 11 3. Íàãàåâ Ñ.Â., ×åáîòàðåâ Â.È., Îöåíêè ñêîðîñòè ñõîäèìîñòè â öåíòðàëüíîé ïðåäåëüíîé òåîðåìå â ïðîñòðàíñòâå l2 . Ìàò. àíàëèç è ñìåæíûå âîïðîñû ìàòåìàòèêè, Íîâîñèáèðñê: Íàóêà, (1978), ñòp. 153-182. 4. Bentkus V., Gotze F., Optimal bounds in non-gaussian limit theorems for U-statistics, The Annals of Probability, Vol. 27, No. 1 (1999) pp. 454-521. 5. Senatov V.V., Qualitative effects in the estimates of the convergence rate in the central limit theorem in multidimensional spaces. Proceedings of the Steclov Institute of Math.(1996), ISSN Pending, v.215, N 4. 6. Yanushkevichiene O., On the rate of convergence of second-degree random polynomials, Journal of math. scien., (1998), v. 92, N 3, pp. 3955-3959. 7. Yanushkevichiene O., Optimal rates of convergence of second degree polynomials in several metrics, Journal of math. scien., (2004), v. 122, N 4, pp. 3449-3451. Àííîòàöèÿ Let X, X1 ,...,Xn be independent identically distributed ran- dom variables taking values in a measurable space h(x, y) and guments g(x) be x, y ∈ Θ (Θ, <). Let real valued measurable functions of the arand be symmetric. We consider U- h(x, y) statistics of type T (X1 , ..., Xn ) = n−1 h(Xi , Xk ) + n−1 X X g(Xi ). 1≤i≤n 1≤i<k≤n Let qi (i ≥ 1) be eigenvalues of the Hilbert-Schmidt operator associated with the kernel h(x, y), and q1 be the largest in absolute value eigenvalue. We prove that cβ 1/6 , |q1 |n1/12 ∆n = ρ(T (X1 , ..., Xn ), T (G1 , ..., Gn )) ≤ p where Gi , 1 ≤ i ≤ n be i.i.d. Gaussian random vectors, ρ is β := E |h(X, X1 )|3 + E |h(X, X1 )|18/5 + n−3/2 E |g(X)|3 + n−9/5 E |g(X)|18/5 + 1 < ∞. a Kolmogorov (or uniform) distance and 12