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Generalized Schröder-type functional equations for Galton-Watson processes in random environments

Anton A. Kutsenko

Abstract

The classical Galton--Watson process works with a fixed probability of fission at each time step. One of the generalizations is that the probabilities depend on time. We consider one of the most complex and interesting cases when we do not know the exact probabilities of fission at each time step - these probabilities are random variables themselves. The limit distributions of the number of descendants are described in terms of generalized integral and differential functional equations of the Schröder type. There are no more analogs of periodic Karlin-McGregor functions, which were very helpful in the analysis of the asymptotic behavior of limit distributions for the classical case. We propose some approximate asymptotic methods. Even simple cases of random families with one or two members lead to nice asymptotics involving, interesting problems related to special functions and special constants. One of them, Example 2 is already announced on \href{https://math.stackexchange.com/questions/4748129/asymptotics-of-sequence-of-rational-numbers}{this} and \href{https://mathoverflow.net/questions/458885/simple-integral-equation}{this} sites. Finally, the phenomenon of why the oscillations in the main asymptotic term usual for the classical case become rare in the case of random environments is discussed.

Generalized Schröder-type functional equations for Galton-Watson processes in random environments

Abstract

The classical Galton--Watson process works with a fixed probability of fission at each time step. One of the generalizations is that the probabilities depend on time. We consider one of the most complex and interesting cases when we do not know the exact probabilities of fission at each time step - these probabilities are random variables themselves. The limit distributions of the number of descendants are described in terms of generalized integral and differential functional equations of the Schröder type. There are no more analogs of periodic Karlin-McGregor functions, which were very helpful in the analysis of the asymptotic behavior of limit distributions for the classical case. We propose some approximate asymptotic methods. Even simple cases of random families with one or two members lead to nice asymptotics involving, interesting problems related to special functions and special constants. One of them, Example 2 is already announced on \href{https://math.stackexchange.com/questions/4748129/asymptotics-of-sequence-of-rational-numbers}{this} and \href{https://mathoverflow.net/questions/458885/simple-integral-equation}{this} sites. Finally, the phenomenon of why the oscillations in the main asymptotic term usual for the classical case become rare in the case of random environments is discussed.
Paper Structure (7 sections, 86 equations, 13 figures)

This paper contains 7 sections, 86 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Examples
  3. Example 0.
  4. Example 1.
  5. Example 2.
  6. Example 3.
  7. Conclusion

Figures (13)

  • Figure 1: Example of distribution of $\alpha$ in (\ref{['019']}). Large blue points are zeros of (\ref{['015']}).
  • Figure 2: Typical distributions of zeros of (\ref{['015']}) in three common cases: classical - one polynomial; finite - finite number of polynomials; general - all other cases, e.g. $\mu$ is continuous.
  • Figure 3: Comparison between exact values and their approximations, see (\ref{['111']}).
  • Figure 4: Correction of Fig. \ref{['fig1c']} with double-double precision and more precise $C$, see (\ref{['110dd']}).
  • Figure 5: Comparison between exact values and their approximations for the fixed offspring distribution, see (\ref{['1110']}) and (\ref{['1111']}).
  • ...and 8 more figures