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 Teor. Veroyatnost. i Primenen., 1980, Volume 25, Issue 1, Pages 3–17 (Mi tvp943)

Asymptotic behaviour of partial densities and their derivatives

D. M. Čibisov

Moscow

Abstract: If $p(u_1,…,u_k)$ be a $k$-variate density then we call partial densities the functions of a part of variables $u_1,…,u_k$ when remaining variables are fixed.
Let $(Y_{i0},Y_{1i},…,Y_{pi})$ for $i=1,…,n$ be i.i.d. random $(p+1)$-vectors,
$$T_{nl}=n^{-1/2}\sum_{i=1}^nY_{li},\qquad l=0,1,…,p.$$
Denote by $p_n(u_0,u_1,…,u_p)$ the density of $(T_{n0},T_{n1},…,T_{np})$, let for $\nu=0,1,…$
$$p_n^{(\nu)}(u_0,u_1,…,u_p)=(\partial/\partial u_0)^\nu p_n(u_0,u_1,…,u_p).$$
Given $p_1$, $0\le p_1<p$, let $\mathbf u_1=(u_0,…,u_{p_1})$, $\mathbf u_2=(u_{p_1+1},…,u_{p})$,
$$q_{n,\nu}(\mathbf u_2)=\sup[|p_n^{\nu}(\mathbf u_1,\mathbf u_2)|;\mathbf u_i\in R^{p_1+1}].$$

It is proved under certain conditions that $q_{n,\nu}(\mathbf u_2)$ in some respects behaves like a density function. Namely, for any $\nu=0,1,…$
$$\int_{R^{p-p_1}}q_{n,\nu}(\mathbf u_2) d\mathbf u_2\le C<\infty.$$
Moreover, for an arbitrary $l_1$, $p_1+1\le l_1\le p$, consider the function
$$Q_{n,\nu}(z)=\int_{u_{l_1}>z}q_{n,\nu}(\mathbf u_2) d\mathbf u_2.$$
We obtain an upper bound for $Q_{n,\nu}(z)$ similar to that for $\mathbf P\{T_{n,l_1}>z\}$.
If the distribution of $(Y_{01},…,Y_{p1})$ satisfies the Cramér's condition (C) the above stated results hold for appropriately smoothed version of $T_{n0},…,T_{np}$.

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English version:
Theory of Probability and its Applications, 1980, 25:1, 1–15

Bibliographic databases:

Received: 29.09.1978

Citation: D. M. Čibisov, “Asymptotic behaviour of partial densities and their derivatives”, Teor. Veroyatnost. i Primenen., 25:1 (1980), 3–17; Theory Probab. Appl., 25:1 (1980), 1–15

Citation in format AMSBIB
\Bibitem{Chi80} \by D.~M.~{\v C}ibisov \paper Asymptotic behaviour of partial densities and their derivatives \jour Teor. Veroyatnost. i Primenen. \yr 1980 \vol 25 \issue 1 \pages 3--17 \mathnet{http://mi.mathnet.ru/tvp943} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=560053} \zmath{https://zbmath.org/?q=an:0455.60025|0423.60026} \transl \jour Theory Probab. Appl. \yr 1980 \vol 25 \issue 1 \pages 1--15 \crossref{https://doi.org/10.1137/1125001} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1980LG24200001} 

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