11 problems

Let $(\Omega,\sigma(\mathscr{H}))$ be the measurable space associated with the sub-linear expectation framework, and let $\varphi:\mathbb R^{\mathbb N}\to\mathbb R$ be a Borel meas…

Lower-capacity replacement conjecture. The assertion of Proposition 1.1 remains true when $\mathcal V^{\mathscr{P}}$ is replaced by $\widehat{\mathcal V}^{\ast}$. This conjecture c…

Let $Y_1,Y_2,dots$ be the identically distributed random variables and let $U_n^2=sum_{j=1}^n Y_j^2$. The self-normalized cluster set is … Under the general moment conditions of th…

Let $\{Y_n\}$ be the independent and identically distributed sequence considered in Theorem 2(b), and let $T_n=\sum_{i=1}^nY_i$. The theorem's displayed relation (2.5) involves the…

Let $\{Y_n;n\ge1\}$ be the original sequence of independent and identically distributed random variables considered in Theorem 2(b), and let $\widetilde Y_n$ denote its copy used i…

Let $\{Y_n;n\ge1\}$ be independent and identically distributed random variables in a sub-linear expectation space, and let $T_n=\sum_{i=1}^nY_i$ and $d_n=\sqrt{2n\log\log n}$. Writ…

Let $\{X_{n,i};i=1,\ldots,k_n\}$ be an array of independent random variables in the sub-linear expectation space $(\Omega,\mathscr{H},\widehat{\mathbb E})$ with … Let $x_n$ be a se…

C2 conjecture. Then

Let $(X_i)_{i\geq 1}$ be the independent random variables and let $S_n=X_1+\cdots+X_n$ occurring in (clt1), and let $\xi$ be a tight random variable, meaning that … Write…

Variance-ratio weakening conjecture. The standard variance-comparison condition can be weakened to the aggregate variance-ratio condition above, while retaining the corresponding s…

Let $\{Z_{n,k}:1\leq k\leq k_n\}$ be an array with filtrations $\mathscr{H}_{n,k}$, and let $\widehat{\mathbb E}$ and $\widehat{\mathcal E}$ denote the upper and lower conditional…