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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 $T_{n0},…,T_{np}$.

Full text: PDF file (762 kB)

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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