ÎÁ ÓÑÒÎÉ×ÈÂÎÑÒÈ ÐÀÑÏÐÅÄÅËÅÍÈÉ U

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