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Existence of an independent factor for fractional operators

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Let WW be a random variable satisfying the support, normalization, and limiting conditions denoted by,, and. Let UU be the random variable appearing in. The relation can be written as

U=WV,U=WV,

where VV is required to be independent of WW and supported in [0,1][0,1]. Equivalently, with W~=−log⁡W\widetilde{W}=-\log W,

−log⁡W−log⁡V=−log⁡U∼Exponential⁡(1).-\log W-\log V=-\log U\sim\operatorname{Exponential}(1).

Existence conjecture. Under certain conditions on WW, there exists a random variable VV, independent of WW and with support in [0,1][0,1], such that holds. In terms of W~\widetilde{W}, this amounts to decomposing an Exponential⁡(1)\operatorname{Exponential}(1) random variable as a sum of two non-negative independent random variables, with W~\widetilde{W} fixed and satisfying

P[W~∈[0,ϵ)]>0for all ϵ>0,\mathbb{P}[\widetilde{W}\in[0,\epsilon)]>0\quad\text{for all }\epsilon>0,

and

lim⁡n→∞nE[e−nW~]=0.\lim_{n\to\infty}n\mathbb{E}[\mathrm{e}^{-n\widetilde{W}}]=0.

The problem asks for conditions on the prescribed factor WW that guarantee the existence of an independent factor VV. The preceding discussion shows that a beta-distributed choice for VV generally fails, although it works in the L-fractional-calculus case α=1\alpha=1; the general existence question remains open.

References

Primary source

Marc Jornet, “Theory on new fractional operators using normalization and probability tools”, arXiv:2403.06198 (2024).

fractional-calculusprobabilityrandom-variables
Posted

Progress summary

Refreshed
Claimed solved

A 2024 paper left the question open, but a reader-provided, unverified argument now claims a complete moment-theoretic solution and a full classification for beta examples.

Jornet’s 2024 paper formulates the conjecture: determine when a prescribed factor WW admits an independent V∈[0,1]V\in[0,1] with WVWV uniform. It explicitly leaves the general existence question open.

Known results

  • Gamma construction: −log⁡W∼Gamma⁡(α,1)-\log W\sim\operatorname{Gamma}(\alpha,1) and −log⁡V∼Gamma⁡(1−α,1)-\log V\sim\operatorname{Gamma}(1-\alpha,1) for 0<α<10<\alpha<1 (Jornet, 2024).
  • A Bernoulli–exponential mixture gives a second positive construction (Jornet, 2024).
  • A different Bernoulli–exponential mixture gives nonexistence via failure of the characteristic-function quotient (Jornet, 2024).

Posted attempt

A reader-provided argument claims a complete necessary-and-sufficient criterion using Hausdorff moments, and claims W∼Beta⁡(a,b)W\sim\operatorname{Beta}(a,b) admits a factor exactly when a≥1a\ge1 and 0<b≤10<b\le1. It also identifies a sign inconsistency in the stated limiting condition. The argument has not been independently verified.

Current status (as of August 2026): The published conjecture has positive and negative examples, while the newly posted general criterion and beta classification remain unverified.

Sources

Solutions 1

ProofThis solution needs a summarySee full solutionHide full solution

There is first a limiting-condition inconsistency: equation (4.4) assumes

lim⁡n→∞nE[Wn]=+∞,\lim_{n\to\infty}n\mathbb E[W^n]=+\infty,

whereas the later reformulation replaces this by 00. Under the displayed zero-limit condition, no factor can exist: independence and WV∼Uniform⁡(0,1)WV\sim\operatorname{Uniform}(0,1) would give

E[Vn]=1(n+1)E[Wn]⟶+∞,\mathbb E[V^n] = \frac1{(n+1)\mathbb E[W^n]} \longrightarrow+\infty,

contradicting 0≤V≤10\le V\le1.

For the corrected problem there is a necessary-and-sufficient criterion. Put

μn=E[Wn]>0,bn=1(n+1)μn.\mu_n=\mathbb E[W^n]>0, \qquad b_n=\frac1{(n+1)\mu_n}.

An independent V∈[0,1]V\in[0,1] with WV∼Uniform⁡(0,1)WV\sim\operatorname{Uniform}(0,1) exists if and only if

∑j=0k(−1)j(kj)(n+j+1)E[Wn+j]≥0(n,k≥0).\boxed{ \sum_{j=0}^k \frac{(-1)^j\binom{k}{j}} {(n+j+1)\mathbb E[W^{n+j}]} \ge0 \qquad(n,k\ge0). }

Necessity follows because this sum equals

E[Vn(1−V)k].\mathbb E[V^n(1-V)^k].

Conversely, the Hausdorff moment theorem gives a probability law on [0,1][0,1] with moments bnb_n. Take VV with this law independently of WW. Then

E[(WV)n]=1n+1\mathbb E[(WV)^n]=\frac1{n+1}

for every nn, and compact moment determinacy implies that WVWV is uniform.

For beta distributions there is a complete explicit classification:

W∼Beta⁡(a,b) admits such a factor  ⟺  a≥1,0<b≤1.\boxed{ W\sim\operatorname{Beta}(a,b) \text{ admits such a factor} \iff a\ge1,\quad 0<b\le1. }

Indeed, WV≤WWV\le W forces

P(W≤t)≤t.\mathbb P(W\le t)\le t.

For a<1a<1,

P(W≤t)∼taa B(a,b)>t(t↓0),\mathbb P(W\le t) \sim\frac{t^a}{a\,\mathrm B(a,b)}>t \qquad(t\downarrow0),

which is impossible. Moreover, the forced moments are

E[Vn]=(a+b)n(n+1)(a)n∼Γ(a)Γ(a+b)nb−1,\mathbb E[V^n] = \frac{(a+b)_n}{(n+1)(a)_n} \sim \frac{\Gamma(a)}{\Gamma(a+b)}n^{b-1},

ruling out b>1b>1.

Conversely, let a≥1a\ge1 and 0<b≤10<b\le1. If a+b≤2a+b\le2, take independent

X∼Beta⁡(1,a−1),Y∼Beta⁡(a+b,2−a−b),V=XY,X\sim\operatorname{Beta}(1,a-1), \qquad Y\sim\operatorname{Beta}(a+b,2-a-b), \qquad V=XY,

interpreting a zero second parameter as a point mass at 11. Then

E[Vn]=(1)n(a)n(a+b)n(2)n=(a+b)n(n+1)(a)n.\mathbb E[V^n] = \frac{(1)_n}{(a)_n} \frac{(a+b)_n}{(2)_n} = \frac{(a+b)_n}{(n+1)(a)_n}.

If a+b≥2a+b\ge2, set c=a+b−1c=a+b-1, let

Z∼Beta⁡(c,1−b),Z\sim\operatorname{Beta}(c,1-b),

and independently take

T∼1cδ1+(1−1c)Uniform⁡(0,1).T\sim \frac1c\delta_1+ \left(1-\frac1c\right) \operatorname{Uniform}(0,1).

For V=ZTV=ZT,

E[Vn]=(c)n(a)nn+cc(n+1)=(a+b)n(n+1)(a)n.\mathbb E[V^n] = \frac{(c)_n}{(a)_n} \frac{n+c}{c(n+1)} = \frac{(a+b)_n}{(n+1)(a)_n}.

Both constructions give WV∼Uniform⁡(0,1)WV\sim\operatorname{Uniform}(0,1).

In particular, the entire proposed family

W∼Beta⁡(a,b),0<a,b<1,W\sim\operatorname{Beta}(a,b), \qquad0<a,b<1,

admits no such factor, even without independence. Thus the limiting typo, the exact general existence condition, and the complete beta-family classification are all resolved.

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Shivam Patel Posted