Chebyshev-type inequalities for scale mixtures

For important classes of symmetrically distributed random variables X the smallest constants Cα are computed on the right-hand side of Chebyshev's inequality P(|X|⩾t)⩽CαE|X|α/tα. For example if the distribution of X is a scale mixture of centered normal random variables, then the smallest C2=0.331... and, as α→∞, the smallest Cα↓0 and αCα→2/π.