Achievable rates for multiple descriptions

Consider a sequence of independent identically distributed (i.i.d.) random variables
X_{l},X_{2}, \cdots, X_{n}
and a distortion measure
d(X_{i},X̂_{i})
on the estimates
X̂_{i}
of
X_{i}
. Two descriptions
i(X)\in \{1,2, \cdots ,2^{nR_{1}\}
and
j(X)\in \{1,2, \cdots,2^{nR_{2}\}
are given of the sequence
X=(X_{1}, X_{2}, \cdots ,X_{n})
. From these two descriptions, three estimates
(i(X)), X2(j(X))
, and
\hat{X}_{O}(i(X),j(X))
are formed, with resulting expected distortions
E \frac{1/n} \sum^{n}_{k=1} d(X_{k}, \hat{X}_{mk})=D_{m}, m=0,1,2.
We find that the distortion constraints
D_{0}, D_{1}, D_{2}
are achievable if there exists a probability mass distribution
p(x)p(\hat{x}_{1},\hat{x}_{2},\hat{x}_{0}|x)
with
Ed(X,\hat{x}_{m})\leq D_{m}
such that
R_{1}>I(X;\hat{X}_{1}),
R_{2}>I(X;\hat{X}_{2}),
where
I(\cdot)
denotes Shannon mutual information. These rates are shown to be optimal for deterministic distortion measures.