5441337178

Denne the function

H(y) := inf {t | v (((, oo)) < y] , which is the inverse taił function, and observe that

{(u, x) | u > v((x,oo))} D {(u,a:) | H(u) < #}, as u > u ((H(u), oo)) > u ((x, oo)). This leads to

-f [ — ¹{t.>^((_a:,oo))}('Łi,a:)d^(xc) < - [    / -l(H(_u)<,)(«,l)dj/(l)

Jo Jo ^u    Jo Jo ^u

= - f v([H(u),oo))—.

Jo    ^u

Since u < u ([H(u), oo)), we finally get the desired estimation.    □

Now, for a certain class of functions, we establish the multidimensional ver-sion of ineąuality (5). For the simplicity, we formulate this result for the Gaus-sian measure.

Lemma 2. Let g: Cⁿ —> R+ be a bounded function satisfying

1)    ^((eⁱ012i,...,eⁱ0"z_n)) = g(z) for any z e Cⁿ and 0i,... ,0„ G R,

2)    for any w, z G Cⁿ the condition |iyfc| < \zk\, k = 1,... ,n implies g(w) < g{z). Then

Ent^ g < g(z) (^- - du_n(z).

r /Ula \

(6)

Proof. One piece of notation: for a fixed vector r = (r*i,..., r_n) G (R+)ⁿ we de-note r^k = (ri,..., rk-i, rk+1, • • •, r_n) G (R+)^n_1, and then define the functions

9k (®) =0(ri_ł...,rfc_i,ar_łr_fc+i,...,r_n), k = l,...,n.

Notice that for a function h: C —> R+ obeying the property 1) we get

[ h(z)dvi(z) = -5- [    [ h{re^l0)e ^r2/²rdrdO = [ h(r)dp(r),

Jc    27t Jo Jo    J₀

where p denotes the probability measure on R+ with the density at r given by re~^r '². Therefore

L^g{z) ■ⁿ)^di/n{z)=L>»^9{r)    ■ⁿ) ^d^^n(r)

dp®

⁼LML/^{x)&-¹)^Mx

Applying Lemma 1 for the function gjj and the measure p we obtain the estimation

f 9{z)(~--n]di'_n(z)> f ^Ent^9j’d/i®"(r

j C"    V ¹    /    d(R₊)»^

> Ent_M®» g = Ent„_n g,

where the last ineąuality follows from subadditivity of entropy (for example see [Led, Proposition 5.6]).    □

4