Timeline for Determine the affine envelope of a random process's MGF

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Dec 22, 2021 at 13:18	comment	added	Iosif Pinelis		Previous comment continued: So, no, no information whatsoever about any finite number of moments of $B_1$ can guarantee even the finiteness (let alone the boundedness) of the mgf of the backward recurrence time $Z_u$ in any neighborhood of $0$. E.g., if $B_1$ has a log-normal distribution (en.wikipedia.org/wiki/Log-normal_distribution), then all the moments of $B_1$ are finite, whereas the mgf of $B_1$ is infinite in any right neighborhhod of $0$, and hence the only upper bound here on the mgf of the backward recurrence time $Z_u$ in any right neighborhhod of $0$ is $\infty$.
Dec 22, 2021 at 13:09	comment	added	Iosif Pinelis		@leeyee : As (6) shows, for the mgf of the backward recurrence time $Z_u$ to be bounded in a neighborhood of $0$, we need the mgf of $B_1$ to be finite (and hence bounded) in a neighborhood of $0$. On the other hand, by (say) expanding the mgf of any positive random variable $X$ in terms of the moments of $X$, it is clear that no information whatsoever about any finite number of moments of $X$ can guarantee that the mgf of $X$ be finite in a neighborhood of $0$.
Dec 22, 2021 at 12:55	comment	added	leeyee		Thanks for this inspiring answer! As the bound depends on $B_1$'s MGF, i.e., $Ee^{2hB_1}$, may I know what if $B_1$'s MGF is not known? Is it possible to approximate it (say, using moments)?
Dec 21, 2021 at 3:10	history	answered	Iosif Pinelis	CC BY-SA 4.0