Convergence Issues

This page addresses situations where moments are flagged as “convergence uncertain” or where results appear numerically unstable.

Understanding the “Convergence Uncertain” Flag 

When the Convergence tab shows a red “uncertain” badge for a moment, it means the backend exhausted all five convergence strategies without confirming the result. The value shown is the raw partial sum — it may or may not be close to the true moment.

This does not necessarily mean the moment does not exist. It means ToFUL could not confirm convergence within the current settings.

Fix 1 — Increase Max Series Terms 

For discrete distributions, the default maximum of 200 terms may not be enough for slowly converging distributions.

In the Advanced expander, increase Max series terms to 400 or 500 and recompute.

When this helps: Distributions where the PMF decays slowly — for example, Geometric(p=0.05), Negative Binomial with small \(p\), or custom PMFs with polynomial decay.

When this does not help: If the PMF does not decay at all (divergent series), adding more terms makes the sum larger, not more stable.

Fix 2 — Increase Calc Precision 

A higher Calc precision tightens the convergence tolerance, which can trigger acceptance by the term-magnitude test or Wynn ε algorithm when the default tolerance was too loose.

Set Calc precision to 10–12 and recompute.

When this helps: When results are very close to converged but just missing the default 1e-12 threshold.

Fix 3 — Check Whether the Moment Exists 

For some distributions, moments of order \(r \geq k\) are genuinely infinite. Adding more terms or increasing precision will not help.

Symptoms of a non-existent moment:

The partial sum grows monotonically as more terms are added.
The ratio of consecutive terms is \(\geq 1\).
The Convergence tab shows ratio-bound with “diverge” in the info.

Distributions with non-existent high-order moments:

Distribution	Finite moments of order…
Cauchy	None (no finite moments at all)
Pareto(α)	r < α only
Student-t(ν)	r < ν only
Lévy(c)	None
Log-normal	All orders finite
Geometric(p)	All orders finite for p > 0

If your distribution has a power-law tail, compute only the moments that are known to exist analytically before comparing with ToFUL.

Fix 4 — Reformulate the PMF/PDF 

Numerical cancellation can occur when the PMF/PDF involves large intermediate values that nearly cancel. Reformulating in log-space can help.

Example — Poisson with large lambda:

# Bug — exp(-50) * 50**x causes underflow for large x
(exp(-50) * 50**x) / factorial(x)  if  x >= 0  else  0

# Fix — use log-space formulation via gamma function
exp(-50 + x * log(50) - log(gamma(x+1)))  if  x >= 0  else  0

The two expressions are mathematically identical but the second avoids the catastrophic underflow of exp(-50).

Fix 5 — Use a Tighter Range 

For continuous distributions where the PDF has negligible mass outside a finite interval, use explicit finite bounds rather than infinity.

This replaces Gauss-Laguerre or Gauss-Hermite (which assume specific tail behaviour) with standard adaptive Gauss-Kronrod, which adapts freely to the actual integrand shape.

Example — Log-normal distribution on [0, 100] instead of [0, inf]:

A log-normal LN(0,1) has negligible density beyond x=50. Using [0, 50] instead of [0, inf] captures 99.9999% of the mass and may integrate more stably.

Fix 6 — Switch to mpmath Mode 

For distributions where standard float64 arithmetic accumulates too much rounding error, set Calc precision to 13 or higher. This triggers mpmath’s arbitrary-precision tanh-sinh quadrature for CRVs.

Note that for discrete series, increasing precision does not switch to arbitrary-precision arithmetic — it only tightens the tolerance.

Diagnosing Slowly Converging Series 

The Convergence tab info string contains diagnostic information. Read the method and info for the highest-order moment:

Method: wynn-epsilon
Info:   Wynn epsilon converged at column 4: 111.222...

Method: partial-sum
Info:   Convergence uncertain after 500 terms (partial sum=111.1...)

If wynn-epsilon converges but partial-sum appears for higher-order moments, the higher-order moments need more terms. The Wynn algorithm extracts the limit reliably when partial sums oscillate around it, but for series that are still actively growing, it cannot help.

Alternating Series 

If your PMF produces alternating positive and negative values, this is almost certainly a modelling error — probabilities cannot be negative.

However, when computing moments (not probabilities), the term \((x-a)^r \cdot P(x)\) can alternate in sign for odd \(r\) and negative-valued \(x\). This is expected and the Cohen-Villegas-Zagier algorithm handles it automatically.

If the Convergence tab shows cohen-villegas-zagier for odd-order central moments of a symmetric distribution centred at 0, this is normal correct behaviour (odd central moments of symmetric distributions are 0, and the CVZ algorithm efficiently confirms this).

Quick Reference — Which Fix to Try First 

Symptom	First fix to try
“Convergence uncertain” for all moments	Increase Max series terms
“Convergence uncertain” only for high-order moments	The moment may not exist; check tail behaviour
Sum slightly off from 1.0	Increase Calc precision to 10

Results look right but method shows `partial-sum` with small deviation	Normal; result is still accurate
Moments grow implausibly large	Check for non-existent moments
Results unstable across runs	Increase Calc precision