 Multiplicative entropy power inequality

ID: multiplicative-entropy-power-inequality

For independent positive random variables

Y_{1}, Y_{2}

μ_{i} = E lo g_{2} Y_{i}

Z_{i} = ln Y_{i}

gives

2^{2 h (Y_{1} Y_{2})} \geq 2^{2 μ_{2}} 2^{2 h (Y_{1})} + 2^{2 μ_{1}} 2^{2 h (Y_{2})} .

Indeed

h (Z_{i}) = h (Y_{i}) - μ_{i}

, while

h (Y_{1} Y_{2}) = h (Z_{1} + Z_{2}) + μ_{1} + μ_{2}

, provided the sum's differential entropy is defined. Positivity alone does not ensure these expressions exist:

Y = e^{T}

with

T

having Cauchy distribution has undefined

E lo g_{2} Y

// Target: algebra.bigb

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