Asymptotic behaviour of partial densities and their derivatives

D. M. Čibisov

Teoriâ veroâtnostej i ee primeneniâ, Tome 25 (1980) no. 1, pp. 3-17

Résumé

If $p(u_1,\dots,u_k)$ be a $k$-variate density then we call partial densities the functions of a part of variables $u_1,\dots,u_k$ when remaining variables are fixed. Let $(Y_{i0},Y_{1i},\dots,Y_{pi})$ for $i=1,\dots,n$ be i.i.d. random $(p+1)$-vectors, $$ T_{nl}=n^{-1/2}\sum_{i=1}^nY_{li},\qquad l=0,1,\dots,p. $$ Denote by $p_n(u_0,u_1,\dots,u_p)$ the density of $(T_{n0},T_{n1},\dots,T_{np})$, let for $\nu=0,1,\dots$ $$ p_n^{(\nu)}(u_0,u_1,\dots,u_p)=(\partial/\partial u_0)^\nu p_n(u_0,u_1,\dots,u_p). $$ Given $p_1$, $0\le p_1$, let $\mathbf u_1=(u_0,\dots,u_{p_1})$, $\mathbf u_2=(u_{p_1+1},\dots,u_{p})$, $$ q_{n,\nu}(\mathbf u_2)=\sup[|p_n^{\nu}(\mathbf u_1,\mathbf u_2)|;\mathbf u_i\in R^{p_1+1}]. $$ It is proved under certain conditions that $q_{n,\nu}(\mathbf u_2)$ in some respects behaves like a density function. Namely, for any $\nu=0,1,\dots$ $$ \int_{R^{p-p_1}}q_{n,\nu}(\mathbf u_2)\,d\mathbf u_2\le C\infty. $$ 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 $(Y_{01},\dots,Y_{p1})$ satisfies the Cramér's condition (C) the above stated results hold for appropriately smoothed version of $T_{n0},\dots,T_{np}$.