15 problems

Persistence exponent conjecture. One should have

Let $(X_t)_{t\geq 0}$ be a Lévy process in the domain of attraction of an $\alpha$-stable Lévy process with positivity parameter $\varrho$, and let $T_z$ be the first-passage time…

Persistence exponent conjecture. The process has persistence exponent $1-H$, meaning that

Let $(X_i)$ be centered i.i.d. increments with law $\mu$ and finite variance, let $S_n$ and $A_n$ be the associated random walk and integrated walk, and write…

Let $(\xi_i)_{i\ge0}$ be i.i.d. centered random variables, define … let $N_n$ be the number of real roots of $P_n$, and let $b$ be the constant appearing in the Gaussian persistenc…

Weighted-sum persistence conjecture. If $\mu$ has finite variance and $\sigma_n\asymp n^p$ for some $p>0$, then

Khokhlov-Molchan conjecture. One has

For each $\beta\ge0$, consider a $\beta$-fractionally integrated Lévy process and let $\theta(\beta)$ be its persistence exponent. The Lévy universality conjecture. For every…

For each $\beta\ge0$, let $\theta(\beta)$ denote the persistence exponent of a $\beta$-fractionally integrated Lévy process, when defined. The convexity conjecture. The function ……

Let $B^H$ be fractional Brownian motion with Hurst parameter $H\in(0,1)$, let $A_t^H=\int_0^tB_s^H\,\mathrm{d}s$, and define $T_1^H=\inf\{t\ge0:A_t^H>1\}$. The fractional Brownian…

Let $Z$ be a Lévy process, let $A_t=\int_0^tZ_s\,\mathrm{d}s$, and for $x>0$ define $T_x=\inf\{t\ge0:A_t>x\}$. The general Lévy persistence conjecture. If … then for every $x>0$, ……

Let $Z$ be a strictly $\alpha$-stable Lévy process, let $A_t=\int_0^tZ_s\,\mathrm{d}s$, and define $T_1=\inf\{t\ge0:A_t>1\}$. The stable Lévy persistence conjecture. If … then ther…

Let $(X_i)_{i\ge1}$ be i.i.d. with law $\mu$, let $S_n=\sum_{i=1}^nX_i$, and let $A_n=\sum_{i=1}^nS_i$. Define $T_0=\inf\{n\ge1:A_n>0\}$. The positivity-parameter conjecture. If ……

Let $(X_i)_{i\ge1}$ be i.i.d. with law $\mu$, let $S_n=\sum_{i=1}^nX_i$, and let $A_n=\sum_{i=1}^nS_i$. Define $T_0=\inf\{n\ge1:A_n>0\}$. The asymptotic positivity conjecture. If ……

Let $(X_i)_{i\ge 1}$ be i.i.d. with law $\mu$, let $S_n=\sum_{i=1}^nX_i$, and let $A_n=\sum_{i=1}^nS_i$. Define $T_0=\inf\{n\ge1:A_n>0\}$. The persistence exponent conjecture. If…