Let µ be an even Borel probability measure on R. For every N > n consider N independent random vectors X1, . . . , XN in R n , with independent coordinates having distribution µ. We establish a sharp threshold for the product measure µn of the random polytope KN := conv X1, . . . , XN in R n under the assumption that the Legendre transform Λ * µ of the logarithmic moment generating function of µ satisfies the conditionAn application is a sharp threshold for the case of the product measure ν n p = ν ⊗n p , p 1 with density (2γp) −n exp(− x p p ), where • p is the ℓ n p -norm and γp = Γ(1 + 1/p).
Let $$\mu $$ μ be a log-concave probability measure on $${\mathbb R}^n$$ R n and for any $$N>n$$ N > n consider the random polytope $$K_N=\textrm{conv}\{X_1,\ldots ,X_N\}$$ K N = conv { X 1 , … , X N } , where $$X_1,X_2,\ldots $$ X 1 , X 2 , … are independent random points in $${\mathbb R}^n$$ R n distributed according to $$\mu $$ μ . We study the question if there exists a threshold for the expected measure of $$K_N$$ K N . Our approach is based on the Cramer transform $$\Lambda _{\mu }^{*}$$ Λ μ ∗ of $$\mu $$ μ . We examine the existence of moments of all orders for $$\Lambda _{\mu }^{*}$$ Λ μ ∗ and establish, under some conditions, a sharp threshold for the expectation $${\mathbb {E}}_{\mu ^N}[\mu (K_N)]$$ E μ N [ μ ( K N ) ] of the measure of $$K_N$$ K N : it is close to 0 if $$\ln N\ll {\mathbb {E}}_{\mu }(\Lambda _{\mu }^{*})$$ ln N ≪ E μ ( Λ μ ∗ ) and close to 1 if $$\ln N\gg {\mathbb {E}}_{\mu }(\Lambda _{\mu }^{*})$$ ln N ≫ E μ ( Λ μ ∗ ) . The main condition is that the parameter $$\beta (\mu )=\textrm{Var}_{\mu }(\Lambda _{\mu }^{*})/({\mathbb {E}}_{\mu }(\Lambda _{\mu }^{*}))^2$$ β ( μ ) = Var μ ( Λ μ ∗ ) / ( E μ ( Λ μ ∗ ) ) 2 should be small.
Let µ be a log-concave probability measure on R n and for any N > n consider the random polytope KN = conv{X1, . . . , XN }, where X1, X2, . . . are independent random points in R n distributed according to µ. We study the question if there exists a threshold for the expected measure of KN . Our approach is based on the Cramer transform Λ * µ of µ. We examine the existence of moments of all orders for Λ * µ and establish, under some conditions, a sharp threshold for the expectation E µ N [µ(KN )] of the measure of KN : it is close to 0 if ln N ≪ Eµ(Λ * µ ) and close to 1 if ln N ≫ Eµ(Λ * µ ). The main condition is that the parameter β(µ) = Varµ(Λ * µ )/(Eµ(Λ * µ )) 2 should be small.
Given a probability measure µ on R n , Tukey's half-space depth is defined for anywhere Lµ is the isotropic constant of µ and c1, c2 > 0 are absolute constants. The proofs combine large deviations techniques with a number of facts from the theory of Lq-centroid bodies of log-concave probability measures. The same ideas lead to general estimates for the expected measure of random polytopes whose vertices have a log-concave distribution.
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