On stochastic monomial densities and their entropy functions (Q2755165)

This article deals with different variations of the stationarity property for a sequence \(X=\{X_{\alpha}, \alpha\in Z^{+}\}\) of random variables with monomial probability density \(f_{\alpha}(x)=(\alpha+1)x^{\alpha}/(b^{\alpha+1}-a^{\alpha+1}),\;a\leq x\leq b\), together with their \(\Lambda\)-entropy functions \(\Lambda[f_{\alpha}]=-f_{\alpha}(x)\ln f_{\alpha}(x)\). If a random variable \(Y_{\alpha}\) has density in the form \(g_{\alpha}(x)=g(f_{\alpha}(x))\), then \(G_{\beta}\{g_{\alpha}(x)\}=\int_{a}^{b}g_{\alpha}(x)f_{\beta}(x) dx\) is called generalized moment. A sequence \(Y=\{Y_{\alpha},\alpha\in Z^{+}\}\) of real-valued random variables is said to be nonuniformly stationary if \(G_{\beta}\{g_{\alpha}(x)\}=G_{\beta}\{g_{\alpha-1}(x)\}, \forall \alpha,\beta, \beta\leq \alpha\). If \(G_{\beta}\{g_{\alpha}(x)\}\geq (\leq)\) \(G_{\beta}\{g_{\alpha-1}(x)\}\), \(\forall \alpha,\beta\), \(\beta\leq \alpha\), then \(Y\) is said to be nonuniformly sub- (super-)stationary. The author proves, in particular, that the sequence \(X\) is nonuniformly sub-stationary when \(b\gg a>0\), and \(X\) is nonuniformly stationary when \(b\to\infty\). Let the random variable \(W_{\alpha}\) have density \(\varphi_{\alpha}(x)=-{1\over\alpha}\int_{a}^{x}f_{\alpha}(y)\ln f_{\alpha}(y) dy\). Then the sequence \(W=\{W_{\alpha},\alpha\in Z^{+}\}\) is nonuniformly super-stationary when \(b\gg a>0\), and nonuniformly stationary when \(b\to\infty\).