On the asymptotic behaviour of the constants in the generalized Khintchine inequality (Q2896645)

Let \(\{ y_k(t)\}_{k=1}^n\), \(0<t<1\), be a system of mean zero independent real-valued random variables from \(L_p(0,1)\). Here the interval \((0,1)\) with the Lebesgue measure is considered as a probability space. Denote \(\beta_r=\sup\limits_{1\leq k\leq n}\| y_k\| _r\), \(r=\max (p,2)\), where \(\| \cdot \| _r\) is the \(L_r\)-norm. Then for any real numbers \(a_1,\ldots ,a_n\), NEWLINE\[NEWLINE A_p\left( \sum\limits_{k=1}^n a_k^2\right)^{1/2} \leq \left\| \sum\limits_{k=1}^n a_ky_k\right\| _p \leq \beta_r B_p \left( \sum\limits_{k=1}^n a_k^2\right)^{1/2},\quad A_p,B_p>0 NEWLINE\]NEWLINE (see \textit{I. K. Matsak} and \textit{A. N. Plichko} [Math. Notes 44, No. 3--4, 690--694 (1988); translation from Mat. Zametki 44, No. 3, 378--384 (1988; Zbl 0674.60048)]).NEWLINENEWLINEThe authors prove the inequality NEWLINE\[NEWLINE \left( \frac1{2}\left[ \frac{p}2\right] \right)^{1/2} \leq B_p^{\min } \leq 2(e(p+2))^{1/2} NEWLINE\]NEWLINE for the minimal possible value \(B_p^{\min }\) of the constant \(B_p\).