Average Nikolskii factors for random trigonometric polynomials

For $1\le p,q\le \infty $, the Nikolskii factor for a trigonometric polynomial $T_{{\mathbf{a}}}$ is defined by $$ N_{p,q}(T_{{\mathbf{a}}})= \frac{\|T_{{\mathbf{a}}}\|_{q}}{\|T_{{\mathbf{a}}}\|_{p}},\ \ T_{{ \mathbf{a}}}(x)=a_{1}+\sum \limits ^{n}_{k=1}(a_{2k}\sqrt{2}\cos kx+a_{2k+1} \sqrt{2}\sin kx). $$ We study this average Nikolskii factor for random trigonometric polynomials with independent $N(0,\sigma ^{2})$ coefficients and obtain the exact orders. For $1\leq p< q<\infty $, the average Nikolskii factor is of order $n^{0}$ (i.e., constant), as compared to the worst case bound of order $n^{1/p-1/q}$, and for $1\leq p< q=\infty $, the average Nikolskii factor is of order $(\ln n)^{1/2}$ as compared to the worst case bound of order $n^{1/p}$. We also give the generalization to random multivariate trigonometric polynomials.