Complete left-tail asymptotic for branching processes with immigration

Anton A Kutsenko

REVIEW 3 major objections 5 minor 13 references

Complete left-tail asymptotic for branching processes with immigration

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper proves an exact, globally convergent left-tail series for the density of the martingale limit of a Galton-Watson process with immigration.

desk verdict Genuinely new theorem for branching processes with immigration, but the written proof of the key exponential decay bound (8) has a contour-direction error and a misapplied Stirling estimate. read the letter →

arxiv 2506.03823 v1 pith:6GZWKV72 submitted 2025-06-04 math.PR math.FA

Anton A Kutsenko This is my paper

classification math.PRmath.FA MSC 60J8030D05

keywords Galton-Watsonprocesswithimmigrationmartingalelimitleft-tailasymptoticSchröderfunctionalequationPoincaréone-periodicfunctionGammaFouriercoefficients

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Galton-Watson processes with immigration have a martingale limit whose density describes the small-size tail of the population. This paper proves that this density, in the Schröder case, has a complete left-tail asymptotic: an infinite series of explicit terms that converges for every positive $x$, not just in a small-$x$ limit. Each term combines a power of $x$, a one-periodic function of $\log_E x$, and a Gamma-function factor. Because the series converges everywhere, it simultaneously provides a full asymptotic expansion and a quick numerical approximation. Earlier work for immigration covered only the first asymptotic terms; the result here is the first complete statement for the left tails.

What carries the argument

Four functional equations carry the argument: $P(\Pi(z))=\Pi(E z)$, $\Phi(P(z))=p_1\Phi(z)$, $R(E z)=R(z)Q(\Pi(z))$, and $Q(z)\Psi(P(z))=q_0\Psi(z)$ for the analytic functions $\Pi$, $\Phi$, $R$, $\Psi$. From these one defines the one-periodic functions $K(z)=p_1^{-z}\Phi(\Pi(E^z))$ and $L(z)=q_0^{-z}R(E^z)\Psi(\Pi(E^z))$, and the analytic series $A(z)=\Phi^{-1}(z)/(z\Psi(\Phi^{-1}(z)))=\sum A_n z^n$. The load-bearing identity is $\Pi_{\mathrm{imm}}(z)=z^{\log_E(p_1q_0)}(K\cdot L)(\log_E z)A(z^{\log_E p_1}K(\log_E z))$, which turns the integral for the density into a Gamma-function evaluation. The strip of analyticity of $K$ and $L$ is controlled by the critical angle $\theta^*$ of the filled Julia set of $P$.

What would settle it

For the paper's illustrative example $P(z)=0.3z+0.7z^2$, $Q(z)=0.5+0.5z$, compute $p_{\mathrm{imm}}(x)$ by high-precision numerical inversion of (4) on a grid such as $x=10^{-6},10^{-5},\ldots,3$, and compare with the $M=10$ truncated series (9). The theorem predicts that the difference shrinks according to the bound (8); a systematic discrepancy that does not shrink as $M$ grows would contradict the central claim.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.1: if the offspring generating function $P$ satisfies $p_0=0$, $1<p_1<\infty$ and has critical angle $\theta^*>\pi$, and the immigration generating function $Q$ has $q_0\neq 0$, then for $x>0$ the density is $p_{\mathrm{imm}}(x)=\sum_{n=0}^\infty A_n x^{-\log_E(p_1^{n+1}q_0)-1}B_n(-\log_E x)$, where the $B_n$ are one-periodic functions whose Fourier coefficients are divided by Gamma functions. The theorem states the condition $\log_E(p_1q_0)<-1$; the remark after the theorem notes that it can be removed by discarding finitely many initial terms. The double series converges super-exponentially in $n$ and exponentially in $m$, with the explicit bound (8). The proof writes the Fourier transform of the density as $\Pi(z)R(z)$, expresses each factor through one-periodic functions and the analytic function $A$, shifts the contour, and uses the Gamma-function contour integral to evaluate the resulting Mellin-type integrals. The paper also derives an $M$-term approximation that is accurate in real time for moderate $x$.

Load-bearing premise

The load-bearing premise is geometric: the offspring generating function must have a filled Julia set (points that never escape under repeated iteration) that opens around the fixed point $1$ at an angle strictly wider than a half-circle, so the periodic functions used in the series remain analytic in a sufficiently wide strip.

Editorial extensions

If this is right

The density of the martingale limit can be evaluated to any desired accuracy by truncating the double series, with explicit super-exponential and exponential error bounds from (8).
The series gives the full asymptotic hierarchy as $x\to 0$: each $n$ contributes a power $x^{-\log_E(p_1^{n+1}q_0)-1}$ multiplied by a one-periodic function of $\log_E x$, so the small-$x$ behavior is a superposition of log-periodic oscillations.
The $M$-term approximation (9) computes the density in real time for moderate $x$, while direct quadrature of the Fourier integral takes minutes on the paper's test cases.
In the no-immigration case $Q\equiv 1$, the construction collapses to the previously known complete left-tail series for ordinary Galton-Watson processes.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The double-series convergence is strong enough that term-by-term differentiation of (6) should yield analogous left-tail series for derivatives of the density with respect to $x$; the paper does not state this, but the same bounds (8) control the differentiated terms.
The condition $\theta^*>\pi$ enters only through the Fourier-coefficient decay estimate (54). If a concrete process with $\theta^*\le\pi$ still shows numerical convergence of the same series, that would show the geometric hypothesis is stronger than necessary and the strip estimate is the only bottleneck.
The formulas make it direct to compare the Fourier coefficients in the immigration case with those of the no-immigration case, so one could isolate how immigration reshapes the small-$x$ log-periodic oscillations.
Because the truncated series is real-time, it could serve as a likelihood term in statistical inference for offspring and immigration parameters, replacing simulation or first-term asymptotics in the small-tail regime.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

You should know two things about this paper. First, the result is genuinely new: a complete left-tail asymptotic series for the density of the martingale limit in supercritical Galton–Watson processes with immigration, converging everywhere rather than just asymptotically. The construction with the one-periodic function L, the combined coefficients ϑ_nm, and the product A(z) is a real extension of [4], and the numerical comparison is honest. Second, the proof as written does not establish the advertised quantitative bound (8). Section 2.2 has two load-bearing sign errors.

The first is in (51)–(54). For m > 0, the contour in (52) is shifted to [0,1] + i(s−ε), so |e^{-2πimz}| = e^{+2πm(s−ε)}, which grows. You need the opposite shift, z = x − i(s−ε), to get decay. Hence (54) is a growth bound, not a decay bound. The exponential decay of ϑ_nm that the rest of the proof relies on is not derived.

The second is (58). Stirling gives |Γ(−a+iB)| ~ sqrt(2π) B^{-a−1/2} e^{-πB/2}, which is exponentially small in B. The paper states a lower bound that is polynomial, even growing, in m. That is incompatible with the actual behavior of the Gamma factor, so (59) does not follow.

The good news: the theorem statement may well be true. With the corrected Fourier decay e^{-π|m|θ*/ln E} and the correct asymptotic 1/|Γ| ~ e^{+π^2|m|/ln E}, the condition θ* > π is exactly what you need to make the double series converge exponentially. So the architecture of the proof is coherent; the written estimates are just wrong.

There are also smaller issues: the interchange of sum and integral in (50) is done before the decay bound is available, and the Remark after Theorem 1.1 defers the removal of log_E(p1q0) < −1 without a proof. These are minor compared to the two main gaps.

Who gets value from this? Specialists in branching processes and the Karlin–McGregor function. The flaws look repairable by a careful referee who knows complex analysis, but as posted the proof is not rigorous. Send it to peer review anyway; the result is important enough for the subfield and the fixes are likely within reach.

Referee Report

3 major / 5 minor

Summary. The paper derives a complete left-tail asymptotic series for the density of the martingale limit of a supercritical Galton-Watson process with immigration, in the Schröder case p0=0. The main result (Theorem 1.1) represents the density p_imm(x) as a sum over n of terms involving one-periodic functions B_n defined through Fourier coefficients and Gamma functions, and asserts the double-series bound (8) that is super-exponential in n and exponential in m. The proof extends the no-immigration machinery of [4] by introducing a second one-periodic function L and an analytic function A, then applies contour integration and Fourier analysis. A numerical comparison is provided between the truncated series and a direct Fourier integral.

Significance. If the result is correct, it is the first global convergent series representation for the left tail of the density in the Schröder case with immigration, generalizing the complete asymptotic of [4] and giving a quickly computable approximation. The paper explicitly identifies the needed geometric condition θ* > π on the filled Julia set of the offspring generating function. The main theoretical weakness is that the proof of the key estimate (8) rests on two sign/magnitude errors; these are likely correctable, but as written the central convergence guarantee is not established.

major comments (3)

[§2.2, Eqs. (51)–(54)] The Fourier-coefficient bound has the wrong sign. For m>0, integrating over [0,1]+i(s−ε) gives e^{2πm(s−ε)}, a growth factor, not a decay factor; the correct contour is the lower boundary, yielding |τ_m| ≤ e^{−2π|m|(s−ε)} max|T|. Consequently, (54) as printed has a positive exponent and does not provide exponential decay of ϑ_nm. Since this decay is the source of the exponential |m|-decay in (8), this is a load-bearing gap. The error appears typographical, but the proof must be corrected before the estimate (8) can be accepted.
[§2.2, Eq. (58)] The lower bound on the Gamma factor is incompatible with Stirling's formula. For z = −a + iB, |Γ(z)| ~ sqrt(2π)|B|^{−a−1/2} e^{−π|B|/2}; the printed (58) contains an additional factor e^{−(π²|m|/lnE)√(ln²(...)+4π²m²)} ≈ e^{−2π³m²/lnE}, which is super-exponentially small. Even if the inequality in (58) holds as a very weak lower bound, it is far too weak to control 1/Γ: combining it with any reasonable bound on ϑ_nm would give growth, not decay, of the quotient. A correct lower bound with exponent −π|B|/2 is required to obtain (59).
[§2.2, Eq. (50) and Remark after Theorem 1.1] The interchange of the infinite sum over n with the contour integral in (50) is asserted rather than proved. The desired bound (60) is established only later, so a dominated convergence argument using the Taylor coefficients A_n and the decay of |z|^{log_E(p1^{n+1}q0)} must be written out. In addition, the Remark claims that the condition log_E(p1 q0) < −1 can be removed, but delegates the proof to an "interesting exercise"; this is an omitted proof of a strengthening of the theorem and should either be supplied or the remark removed.

minor comments (5)

[Eq. (9)] The approximation formula is not well-defined as written: it contains 2πim with a free m and uses (K^{n+1}·L)(0) where the zeroth Fourier coefficient ϑ_n0 is meant. Please clarify that only the m=0 term of B_n is kept and that ϑ_n0 is the mean value of (K^{n+1}·L), not its value at 0.
[Section 1, p.2] The condition "1< p1 <1" is impossible; it should read "0<p1<1" (with E>1) or similar.
[§2.2, Eqs. (51)–(54)] After correcting the sign, the exponent in (54) should be −2π|m|(θ*/(2lnE)−ε), i.e., −π|m|θ*/lnE + 2π|m|ε; the printed form with π(θ*−ε)/lnE is not the precise consequence of the strip width.
[Eq. (7)] The functions B_n may have poles if ln(p1^{n+1}q0)/lnE is a positive integer for some n (with m=0). The paper does not discuss this special case; it should be mentioned that the formula is interpreted by continuity or that such parameter values are excluded.
[§2.2, Eq. (58)] The displayed lower bound is unnecessarily complicated and, as discussed, not useful. A clean statement of Stirling's bound with explicit constants valid uniformly for the ranges of n and m would be clearer.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the series representation is derived from functional equations and complex analysis, with prior self-citations used only as supporting external results.

full rationale

The derivation chain in Theorem 1.1 is self-contained in the sense relevant to circularity. The proof starts from the functional equations (19), (34), and (39), factors the Fourier transform as Π_imm(z) = Π(z)R(z), rewrites it through the one-periodic functions K and L in (46), expands A(z) = Σ A_n z^n, shifts the contour, and applies the Hankel representation of the Gamma function in (55). The final series (61) is an identity, not a fitted ansatz: the coefficients A_n, ϑ_nm, and the exponents are determined solely by the model functions P and Q. No parameter is fitted to data, no prediction coincides by construction with an input, and no uniqueness theorem is invoked to force the representation. The author's prior articles [4] and [12] are cited for the no-immigration series, the strip of analyticity of K, and numerical tools; those are established external results and do not assume the immigration theorem being proved. The conditions θ* > π and log_E(p1 q0) < -1 are structural assumptions entering the convergence proof at (54) and (50); they do not make the conclusion definitional. The possible sign or growth issue in (54)/(58) raised by the skeptic is a correctness or gap concern, not a circularity concern, since the theorem's content is not equivalent to its assumptions.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters or invented entities. The derivation relies on standard complex analysis, the known no-immigration framework from [4], and the theorem's own conditions on P and Q (regularity, θ*>π). The most load-bearing unproved input is the analyticity of Φ inside the open filled Julia set, taken from [4].

assumptions (5)

domain assumption The martingale limit W of a supercritical Galton-Watson process with immigration exists and has a density given by the Fourier inversion formula (36).
Formula (36) is taken from the classical theory (see [2] for the no-immigration case); the existence requires E>1 and Q regular at 1, conditions stated in Section 1.
domain assumption The function Φ(z) is analytic in the open filled Julia set J_P and solves Φ(P(z)) = p1 Φ(z), with Φ(z) ~ z as z→0.
This is the main input from the author's prior work [4], invoked at equation (16).
domain assumption The infinite product Ψ(z) = ∏_{t≥0} q0^{-1} Q(P_t(z)) converges to an analytic function inside J_P.
Convergence is argued in Section 2.2 using exponential decay of P_t(z) in J_P; the argument is sketched, not fully detailed.
standard math The Hankel representation of the Gamma function applies to the exponent values ln(p1^{n+1} q0)/ln E + i 2πm/ln E.
Used to evaluate the contour integral in (55), cited to [13], p. 254.
domain assumption The critical angle θ* of P satisfies θ* > π.
This is a condition in Theorem 1.1. It guarantees a strip of analyticity of width sufficient for the Fourier decay estimate (54) to beat the Gamma growth (58). The paper notes θ* ≥ π always and says the strict inequality can be weakened.

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Cite this review

Pith. "Pith review of Complete left-tail asymptotic for branching processes with immigration." pith.science (2026). https://pith.science/paper/6GZWKV72

@misc{pith2026250603823,
  author       = {Pith},
  title        = {Pith review of: Complete left-tail asymptotic for branching processes with immigration},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6GZWKV72}},
  note         = {Machine review of arXiv:2506.03823}
}

read the original abstract

We derive a complete left-tail asymptotic series for the density of the {\it martingale limit} of a Galton-Watson process with immigration. We show that the series converges everywhere, not only for small arguments. This is the first complete result regarding the left tails of branching processes with immigration. A good, quickly computed approximation for the density will also be derived from the series.

Figures

Figures reproduced from arXiv: 2506.03823 by the authors.

**Figure 2.** Julia set for the polynomial P(z) = 0.1z + 0.5z 2 + 0.4z 3 . It is the boundary of the open filled Julia set JP , which is the domain of definition for the analytic function Φ, as seen in Equations (15) and (16). The unit disc belongs to the filled Julia set. The figure is taken from my article [12]. This function is analytic and satisfies the Poincaré-type functional equation P(Π(z)) = Π(Ez), Π(0) = 1, Π ′ (0) = −1… view at source ↗

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Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

[4]

A.: Complete Left Tail Asymptotic for the Density of Branching Processes in the Schröder Case.J

Kutsenko, A. A.: Complete Left Tail Asymptotic for the Density of Branching Processes in the Schröder Case.J. Fourier Anal. Appl.30, 39 (2024)

work page 2024
[1]

Harris T.E.: Branching processes. Ann. Math. Stat.41, 474-494 (1948)

work page 1948
[2]

Dubuc, S.: La densité de la loi-limite d’un processus en cascade expansif.Z. Wahrsch. Verw. Gebiete19, 281-290 (1971)

work page 1971
[3]

D., Bingham, N.H.: Large deviations in the supercritical branching process

Biggings, J. D., Bingham, N.H.: Large deviations in the supercritical branching process. Adv. Appl. Prob.25, 757-772 (1993) 12

work page 1993
[5]

Karlin, S., McGregor, J.: Embeddability of discrete-time branching processes into continuous-time branching processes. Trans. Am. Math. Soc.132, 115-136 (1968)

work page 1968
[6]

Karlin, S., McGregor, J.: Embedding iterates of analytic functions with two fixed points into continuous groups. Trans. Am. Math. Soc.132, 137-145 (1968)

work page 1968
[7]

V., Ren, Y.-X.: Small value probabilities for supercritical branching processes with immigration

Chu, W., Li W. V., Ren, Y.-X.: Small value probabilities for supercritical branching processes with immigration. Bernoulli20, 377-393 (2014)

work page 2014
[8]

Galton–Watson trees with vanish- ing Martingale limit

Berestycki, N., Gantert, N., Mörters, P., Sidorova, N. Galton–Watson trees with vanish- ing Martingale limit. J. Stat. Phys.155, 737–762 (2014)

work page 2014

Show all 13 references

[9]

Oscillatory critical amplitudes in hierarchical models

B. Derrida, C. Itzykson, and J. M. Luck, “Oscillatory critical amplitudes in hierarchical models”.Comm. Math. Phys.94, 115–132 (1984)

1984
[10]

Distribution of repetitions of ancestors in genealogical trees

B. Derrida, S. C. Manrubia, D. H. Zanette. “Distribution of repetitions of ancestors in genealogical trees”.Physica A281, 1-16 (2000)

2000
[11]

Oscillatory critical amplitudes in hierarchical models and the Harris function of branching processes

O. Costin and G. Giacomin, “Oscillatory critical amplitudes in hierarchical models and the Harris function of branching processes”.J. Stat. Phys.150, 471–486 (2013)

2013
[12]

Kutsenko, A.A.: Approximation of the number of descendants in branching processes. J. Stat. Phys.190, 68 (2023)

2023
[13]

Fifth Edition by Moll, V.H

Whittaker, E.T., Watson, J.N.: A Course of Modern Analysis. Fifth Edition by Moll, V.H. (ed). Cambridge Univ. Press, Cambridge (2021) 13

2021

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [4]

A.: Complete Left Tail Asymptotic for the Density of Branching Processes in the Schröder Case.J

Kutsenko, A. A.: Complete Left Tail Asymptotic for the Density of Branching Processes in the Schröder Case.J. Fourier Anal. Appl.30, 39 (2024)

work page 2024

[2] [1]

Harris T.E.: Branching processes. Ann. Math. Stat.41, 474-494 (1948)

work page 1948

[3] [2]

Dubuc, S.: La densité de la loi-limite d’un processus en cascade expansif.Z. Wahrsch. Verw. Gebiete19, 281-290 (1971)

work page 1971

[4] [3]

D., Bingham, N.H.: Large deviations in the supercritical branching process

Biggings, J. D., Bingham, N.H.: Large deviations in the supercritical branching process. Adv. Appl. Prob.25, 757-772 (1993) 12

work page 1993

[5] [5]

Karlin, S., McGregor, J.: Embeddability of discrete-time branching processes into continuous-time branching processes. Trans. Am. Math. Soc.132, 115-136 (1968)

work page 1968

[6] [6]

Karlin, S., McGregor, J.: Embedding iterates of analytic functions with two fixed points into continuous groups. Trans. Am. Math. Soc.132, 137-145 (1968)

work page 1968

[7] [7]

V., Ren, Y.-X.: Small value probabilities for supercritical branching processes with immigration

Chu, W., Li W. V., Ren, Y.-X.: Small value probabilities for supercritical branching processes with immigration. Bernoulli20, 377-393 (2014)

work page 2014

[8] [8]

Galton–Watson trees with vanish- ing Martingale limit

Berestycki, N., Gantert, N., Mörters, P., Sidorova, N. Galton–Watson trees with vanish- ing Martingale limit. J. Stat. Phys.155, 737–762 (2014)

work page 2014