Seven states of randomness

If the probability density of ${\displaystyle U=U'+U''}$ is denoted ${\displaystyle p_{2}(u)}$, then it can be obtained by the double convolution ${\displaystyle p_{2}(x)=\int p(u)p(x-u)\,du}$.

When u is known, the conditional probability density of u′ is given by the portioning ratio:

${\displaystyle {\frac {p(u')p(u-u')}{p_{2}(u)}}}$

In many important cases, the maximum of ${\displaystyle p(u')p(u-u')}$ occurs near ${\displaystyle u'=u/2}$, or near ${\displaystyle u'=0}$ and ${\displaystyle u'=u}$. Take the logarithm of ${\displaystyle p(u')p(u-u')}$ and write:

${\displaystyle \Delta (u)=2\log p(u/2)-[\log p(0)+\log p(u)]}$

• If ${\displaystyle \log p(u)}$ is cap-convex, the portioning ratio is maximal for ${\displaystyle u'=u/2}$
• If ${\displaystyle \log p(u)}$ is straight, the portioning ratio is constant
• If ${\displaystyle \log p(u)}$ is cup-convex, the portioning ratio is minimal for ${\displaystyle u'=u/2}$

Splitting the doubling convolution into three parts gives:

${\displaystyle p_{2}(x)=\int _{0}^{x}p(u)p(x-u)\,du=\left\{\int _{0}^{\tilde {x}}+\int _{\tilde {x}}^{x-{\tilde {x}}}+\int _{x-{\tilde {x}}}^{x}\right\}p(u)p(x-u)\,du=I_{L}+I_{0}+I_{R}}$

p(u) is short-run concentrated in probability if it is possible to select ${\displaystyle {\tilde {u}}(u)}$ so that the middle interval of (${\displaystyle {\tilde {u}},u-{\tilde {u}}}$) has the following two properties as u→∞:

• I₀/p₂(u) → 0
• ${\displaystyle (u-2{\tilde {u}})u}$ does not → 0

Consider the formula ${\displaystyle \operatorname {E} [U^{q}]=\int _{0}^{\infty }u^{q}p(u)\,du}$, if p(u) is the scaling distribution the integrand is maximum at 0 and ∞, on other cases the integrand may have a sharp global maximum for some value ${\displaystyle {\tilde {u}}_{q}}$ defined by the following equation:

${\displaystyle 0={\frac {d}{du}}(q\log u+\log p(u))={\frac {q}{u}}-\left|{\frac {d\log p(u)}{du}}\right|}$

One must also know ${\displaystyle u^{q}p(u)}$ in the neighborhood of ${\displaystyle {\tilde {u}}_{q}}$. The function ${\displaystyle u^{q}p(u)}$ often admits a "Gaussian" approximation given by:

${\displaystyle \log[u^{q}p(u)]=\log p(u)+qu={\text{constant}}-(u-{\tilde {u}}_{q})^{2}{\tilde {\sigma }}_{q}^{-2/2}}$

When ${\displaystyle u^{q}p(u)}$ is well-approximated by a Gaussian density, the bulk of ${\displaystyle \operatorname {E} [U^{q}]}$ originates in the "q-interval" defined as ${\displaystyle [{\tilde {u}}_{q}-{\tilde {\sigma }}_{q},{\tilde {u}}_{q}+{\tilde {\sigma }}_{q}]}$. The Gaussian q-intervals greatly overlap for all values of ${\displaystyle \sigma }$. The Gaussian moments are called delocalized. The lognormal's q-intervals are uniformly spaced and their width is independent of q; therefore if the log-normal is sufficiently skew, the q-interval and (q + 1)-interval do not overlap. The lognormal moments are called uniformly localized. In other cases, neighboring q-intervals cease to overlap for sufficiently high q, such moments are called asymptotically localized.