  We define embedding of an $n$-dimensional normed space into $L_{-p},\ 0<p<n$
by extending analytically with respect to $p$ the corresponding property of the
classical $L_p$-spaces. The well-known connection between embeddings into $L_p$
and positive definite functions is extended to the case of negative $p$ by
showing that a normed space embeds in $L_{-p}$ if and only if $\|x\|^{-p}$ is a
positive definite distribution. Using this criterion, we generalize the recent
solutions to the 1938 Schoenberg's problems by proving that the spaces
$\ell_q^n,\ 2<q\le \infty$ embed in $L_{-p}$ if and only if $p\in [n-3,n).$ We
show that the technique of embedding in $L_{-p}$ can be applied to stable
processes in some situations where standard methods do not work. As an example,
we prove inequalities of correlation type for the expectations of norms of
stable vectors. In particular, for every $p\in [n-3,n),$ $\Bbb
E(\max_{i=1,...,n} |X_i|^{-p}) \ge \Bbb E(\max_{i=1,...,n} |Y_i|^{-p}),$ where
$X_1,...,X_n$ and $Y_1,...,Y_n$ are jointly $q$-stable symmetric random
variables, $0<q\le 2,$ so that, for some $k\in \Bbb N,\ 1\le k <n,$ the vectors
$(X_1,...,X_k)$ and $(X_{k+1},...,X_n)$ have the same distributions as
$(Y_1,...,Y_k)$ and $(Y_{k+1},...,Y_n),$ respectively, but $Y_i$ and $Y_j$ are
independent for every choice of $1\le i\le k,\ k+1\le j\le n.$
