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January 2019, 24(1): 297-319. doi: 10.3934/dcdsb.2017144

## Predicting and estimating probability density functions of chaotic systems

 1 Dept. of mathematics & Statistics, University of Missouri-Kansas City, 5100 Rockhill Rd, Kansas City, MO 64110, USA 2 Dept. of Mathematics & Statistics, Concordia University, 1455 de Maisonneuve Blvd West, Montreal, Quebec, Canada H3G 1M8

Received  November 2016 Revised  March 2017 Published  April 2017

Fund Project: The research of Prof. Paweł Góra was supported by NSERC grant. Also, the research of Profs. Noah Rhee and Majid Bani-Yaghoub was supported by UMKC's funding for excellence program.

In the present work, for the first time, we employ Ulam's method to estimate and to predict the existence of the probability density functions of single species populations with chaotic dynamics. In particular, given a chaotic map, we show that Ulam's method generates a sequence of density functions in L1-space that may converge weakly to a function in L1-space. In such a case we show that the limiting function generates an absolutely continuous (w.r.t. the Lebesgue measure) invariant measure (w.r.t. the given chaotic map) and therefore the limiting function is the probability density function of the chaotic map. This fact can be used to determine the existence and estimate the probability density functions of chaotic biological systems.

Citation: Noah H. Rhee, PaweŁ Góra, Majid Bani-Yaghoub. Predicting and estimating probability density functions of chaotic systems. Discrete & Continuous Dynamical Systems - B, 2019, 24 (1) : 297-319. doi: 10.3934/dcdsb.2017144
##### References:
  K. Alligood, T. Sauer and J. Yorke, Chaos: An Introduction to Dynamical Systems Springer-Verlag, New York, 1996. doi: 10. 1007/978-3-642-59281-2.  M. Anazawa, Bottom-up derivation of discrete-time population models with the Allee effect, Theoretical Population Biology, 75 (2009), 56-67. doi: 10.1016/j.tpb.2008.11.001. L. Becks, F. M. Hilker, H. Malchow, K. Jürgens and H. Arndt, Experimental demonstration of chaos in a microbial food web, Nature, 435 (2005), 1226-1229. doi: 10.1038/nature03627. A. Boyarskyand and P. Góra, Laws of Chaos Birkhauser, Boston, 1997. doi: 10. 1007/978-1-4612-2024-4. F. Brauer and C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology Texts in Applied Mathematics, 2012. doi: 10. 1007/978-1-4614-1686-9. C. Castillo-Chavez and A. Yakubu, Epidemics on Attractors, Contemporary Mathematics, 284 (2001), 23-42. doi: 10.1090/conm/284/04697. B. Cipra, Beetlemania: Chaos in ecology, in What's happening in the mathematical sciences 1998-1999 (ed. P. Zorn), Volume 4, American Mathematical Society, Providence, Rhode Island, USA (1999), 72-81.  R. F. Costantino, R. A. Desharnais, J. M. Cushing and B. Dennis, Chaotic dynamics in an insect population, Science, 275 (1997), 389-391. doi: 10.1126/science.275.5298.389. C. Dellacherie and P. A. Meyer, Probabilities and Potential, North-Holland Pub. Co. , N. Y. , 1978, (Chapter Ⅱ, Theorem T25).  B. Dennis, R. A. Desharnais, J. M. Cushing, S. M. Henson and R. F. Costantino, Estimating chaos and complex dynamics in an insect population, Ecological Monographs, 71 (2001), 277-303.  R. A. Desharnais, R. F. Costantino, J. M. Cushing and B. Dennis, Estimating chaos in an insect population, Science, 276 (1997), 1881-1882.  W. Falck, O. N. Bjornstad and N. C. Stenseth, Voles and lemmings: Chaos and uncertainty in fluctuating populations, Proceedings of the Royal Society of London B, 262 (1995), 363-370. doi: 10.1098/rspb.1995.0218. H. C. J. Godfray and S. P. Blythe, Complex dynamics in multispecies communities, Philosophical Transactions of the Royal Society of London B, 330 (1990), 221-233.  C. Godfray and M. Hassell, Chaotic beetles, Science, 275 (1997), 323-324. doi: 10.1126/science.275.5298.323. I. Hanski, P. Turchin, E. Korpimaki and H. Henttonen, Population oscillations of boreal rodents: Regulation by mustelid predators leads to chaos, Nature, 364 (1993), 232-235. doi: 10.1038/364232a0. M. P. Hassell, H. N. Comins and R. M. May, Spatial structure and chaos in insect population dynamics, Nature, 353 (1991), 255-258. doi: 10.1038/353255a0. H. Henttonen and I. Hanski, Population dynamics of small rodents in northern Fennoscandia, in Chaos in real data: The analysis of non-linear dynamics from short ecological time series (ed. J. N. Perry, R. H. Smith, I. P. Woiwod and D. R. Morse), Kluwer Academic, Dordrecht, The Netherlands, (2000), 73-96. doi: 10. 1007/978-94-011-4010-2_4.  M. V. Jakobson, Absolutely continuous invariant measures for one-parameter families of one-dimensional maps, Communications in Mathematical Physics, 81 (1981), 39-88. doi: 10.1007/BF01941800.  M. Jakobson, Piecewise smooth maps with absolutely continuous invariant measures and uniformly scaled Markov partitions, Proceedings of Symposia in Pure Mathematics, 69 (2001), 825-881. doi: 10.1090/pspum/069/1858558.  P. Kareiva, Predicting and producing chaos, Nature, 375 (1995), 189-190. doi: 10.1038/375189a0.  G. Keller and T. Nowicki, Spectral theory, zeta functions and the distribution of periodic points for Collet-Eckmann maps, Communications in Mathematical Physics, 149 (1992), 31-69. doi: 10.1007/BF02096623. B. E. Kendall, C. J. Briggs, W. W. Murdoch, P. Turchin, S. P. Ellner, E. McCauley, R. M. Nisbet and S. N. Wood, Why do populations cycle? A synthesis of statistical and mechanistic modeling approaches, Ecology, 80 (1999), 1789-1805.  A. Lasota and M. C. Mackey, Chaos, Fractals, and Noise Springer-Verlag, New York, 1994, Chapter V, page 87. doi: 10. 1007/978-1-4612-4286-4. T. Y. Li, Finite approximation for the Frobenius-Perron operator, a solution to Ulam's conjecture, Journal of Approximation Theory, 17 (1976), 177-186. doi: 10.1016/0021-9045(76)90037-X. J. A. Logan and F. P. Hain (eds. ), Chaos and Insect Ecology Virginia Experiment Station Information Series, 91-3, Virginia Polytechnic Institute and State University, Blacksburg, Virginia, USA, 1991.  S. Luzzatto and H. Takahasi, Computable conditions for the occurrence of non-uniform hyperbolicity in families of one-dimensional maps, Nonlinearity, 19 (2006), 1657-1695. doi: 10.1088/0951-7715/19/7/013. W. de Melo and S. van Strien, One-dimensional Dynamics in Ergebnisse der Mathematik und ihrer Grenzgebiete Volume 3, Springer-Verlag, Berlin, 1993, Chapter V, section 6. doi: 10. 1007/978-3-642-78043-1. M. Misiurewicz, Absolutely continuous measures for certain maps of an interval, Institute Hautes Etudes Sci. Publ. Math., 53 (1981), 17-51.  R. M. Nisbet and W. S. C. Gurney, Modelling fluctuating populations, Wiley, 1982,285-308.  T. Nowicki, Some dynamical properties of S-unimodal maps, Fundamenta. Mathematicae, 142 (1993), 45-57.  T. Nowicki and S. van Strien, Nonhyperbolicity and invariant measures for unimodal maps, Ann. Inst. H. Poincare Phys. Theor., 53 (1990), 427-429.  L. Oksanen and T. Oksanen, Long-term microtine dynamics in north Fennoscandian tundra: The vole cycle and lemming chaos, Ecography, 15 (1992), 226-236. doi: 10.1111/j.1600-0587.1992.tb00029.x. L. F. Olsen, G. L. Truty and W. M. Schaffer, Oscillations and chaos in epidemics: A nonlinear dynamic study of six childhood diseases in Copenhagen, Denmark, Theoretical Population Biology, 33 (1988), 344-370. doi: 10.1016/0040-5809(88)90019-6. L. F. Olsen and W. M. Schaffer, Chaos vs. noisy periodicity: Alternative hypotheses for childhood epidemics, Science, 249 (1990), 499-504.  E. Renshaw, Chaos in biometry, IMA Journal of Mathematics Applied in Medicine and Biology, 11 (1994), 17-44. doi: 10.1093/imammb/11.1.17. P. Rohani and D. J. D. Earn, Chaos in a cup of flour Trends in Ecology and Evolution 12 (1997), p171. doi: 10. 1016/S0169-5347(97)01055-0. M. R. Rychlik, Another proof of Jakobson's theorem and related results, Ergodic Theory and Dynamical Systems, 8 (1988), 93-109. doi: 10.1017/S014338570000434X. S. H. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology and Chemistry, Perseus Publishing, 2001.  P. Turchin, Chaos and stability in rodent population dynamics: evidence from nonlinear time-series analysis, Oikos, 68 (1993), 167-172. doi: 10.2307/3545323. P. Turchin, Chaos in microtine populations, Proceedings of the Royal Society of London B, 262 (1995), 357-361. doi: 10.1098/rspb.1995.0217. P. Turchin and S. P. Ellner, Living on the edge of chaos: Population dynamics of Fennoscandian voles, Ecology, 81 (2000), 3099-3116.  S. M. Ulam, Problems in Modern Mathematics, Interscience, New York, 1964. S. N. Wood, Statistical inference for noisy nonlinear ecological dynamic systems, Nature, 466 (2010), 1102-1104. doi: 10.1038/nature09319.  J. -C. Yoccoz, A Proof of Jakobson's Theorem https://matheuscmss.wordpress.com/category/mathematics/expository/.  C. Zimmer, Life after chaos, Science, 284 (1999), 83-86. show all references

##### References:
  K. Alligood, T. Sauer and J. Yorke, Chaos: An Introduction to Dynamical Systems Springer-Verlag, New York, 1996. doi: 10. 1007/978-3-642-59281-2.  M. Anazawa, Bottom-up derivation of discrete-time population models with the Allee effect, Theoretical Population Biology, 75 (2009), 56-67. doi: 10.1016/j.tpb.2008.11.001. L. Becks, F. M. Hilker, H. Malchow, K. Jürgens and H. Arndt, Experimental demonstration of chaos in a microbial food web, Nature, 435 (2005), 1226-1229. doi: 10.1038/nature03627. A. Boyarskyand and P. Góra, Laws of Chaos Birkhauser, Boston, 1997. doi: 10. 1007/978-1-4612-2024-4. F. Brauer and C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology Texts in Applied Mathematics, 2012. doi: 10. 1007/978-1-4614-1686-9. C. Castillo-Chavez and A. Yakubu, Epidemics on Attractors, Contemporary Mathematics, 284 (2001), 23-42. doi: 10.1090/conm/284/04697. B. Cipra, Beetlemania: Chaos in ecology, in What's happening in the mathematical sciences 1998-1999 (ed. P. Zorn), Volume 4, American Mathematical Society, Providence, Rhode Island, USA (1999), 72-81.  R. F. Costantino, R. A. Desharnais, J. M. Cushing and B. Dennis, Chaotic dynamics in an insect population, Science, 275 (1997), 389-391. doi: 10.1126/science.275.5298.389. C. Dellacherie and P. A. Meyer, Probabilities and Potential, North-Holland Pub. Co. , N. Y. , 1978, (Chapter Ⅱ, Theorem T25).  B. Dennis, R. A. Desharnais, J. M. Cushing, S. M. Henson and R. F. Costantino, Estimating chaos and complex dynamics in an insect population, Ecological Monographs, 71 (2001), 277-303.  R. A. Desharnais, R. F. Costantino, J. M. Cushing and B. Dennis, Estimating chaos in an insect population, Science, 276 (1997), 1881-1882.  W. Falck, O. N. Bjornstad and N. C. Stenseth, Voles and lemmings: Chaos and uncertainty in fluctuating populations, Proceedings of the Royal Society of London B, 262 (1995), 363-370. doi: 10.1098/rspb.1995.0218. H. C. J. Godfray and S. P. Blythe, Complex dynamics in multispecies communities, Philosophical Transactions of the Royal Society of London B, 330 (1990), 221-233.  C. Godfray and M. Hassell, Chaotic beetles, Science, 275 (1997), 323-324. doi: 10.1126/science.275.5298.323. I. Hanski, P. Turchin, E. Korpimaki and H. Henttonen, Population oscillations of boreal rodents: Regulation by mustelid predators leads to chaos, Nature, 364 (1993), 232-235. doi: 10.1038/364232a0. M. P. Hassell, H. N. Comins and R. M. May, Spatial structure and chaos in insect population dynamics, Nature, 353 (1991), 255-258. doi: 10.1038/353255a0. H. Henttonen and I. Hanski, Population dynamics of small rodents in northern Fennoscandia, in Chaos in real data: The analysis of non-linear dynamics from short ecological time series (ed. J. N. Perry, R. H. Smith, I. P. Woiwod and D. R. Morse), Kluwer Academic, Dordrecht, The Netherlands, (2000), 73-96. doi: 10. 1007/978-94-011-4010-2_4.  M. V. Jakobson, Absolutely continuous invariant measures for one-parameter families of one-dimensional maps, Communications in Mathematical Physics, 81 (1981), 39-88. doi: 10.1007/BF01941800.  M. Jakobson, Piecewise smooth maps with absolutely continuous invariant measures and uniformly scaled Markov partitions, Proceedings of Symposia in Pure Mathematics, 69 (2001), 825-881. doi: 10.1090/pspum/069/1858558.  P. Kareiva, Predicting and producing chaos, Nature, 375 (1995), 189-190. doi: 10.1038/375189a0.  G. Keller and T. Nowicki, Spectral theory, zeta functions and the distribution of periodic points for Collet-Eckmann maps, Communications in Mathematical Physics, 149 (1992), 31-69. doi: 10.1007/BF02096623. B. E. Kendall, C. J. Briggs, W. W. Murdoch, P. Turchin, S. P. Ellner, E. McCauley, R. M. Nisbet and S. N. Wood, Why do populations cycle? A synthesis of statistical and mechanistic modeling approaches, Ecology, 80 (1999), 1789-1805.  A. Lasota and M. C. Mackey, Chaos, Fractals, and Noise Springer-Verlag, New York, 1994, Chapter V, page 87. doi: 10. 1007/978-1-4612-4286-4. T. Y. Li, Finite approximation for the Frobenius-Perron operator, a solution to Ulam's conjecture, Journal of Approximation Theory, 17 (1976), 177-186. doi: 10.1016/0021-9045(76)90037-X. J. A. Logan and F. P. Hain (eds. ), Chaos and Insect Ecology Virginia Experiment Station Information Series, 91-3, Virginia Polytechnic Institute and State University, Blacksburg, Virginia, USA, 1991.  S. Luzzatto and H. Takahasi, Computable conditions for the occurrence of non-uniform hyperbolicity in families of one-dimensional maps, Nonlinearity, 19 (2006), 1657-1695. doi: 10.1088/0951-7715/19/7/013. W. de Melo and S. van Strien, One-dimensional Dynamics in Ergebnisse der Mathematik und ihrer Grenzgebiete Volume 3, Springer-Verlag, Berlin, 1993, Chapter V, section 6. doi: 10. 1007/978-3-642-78043-1. M. Misiurewicz, Absolutely continuous measures for certain maps of an interval, Institute Hautes Etudes Sci. Publ. Math., 53 (1981), 17-51.  R. M. Nisbet and W. S. C. Gurney, Modelling fluctuating populations, Wiley, 1982,285-308.  T. Nowicki, Some dynamical properties of S-unimodal maps, Fundamenta. Mathematicae, 142 (1993), 45-57.  T. Nowicki and S. van Strien, Nonhyperbolicity and invariant measures for unimodal maps, Ann. Inst. H. Poincare Phys. Theor., 53 (1990), 427-429.  L. Oksanen and T. Oksanen, Long-term microtine dynamics in north Fennoscandian tundra: The vole cycle and lemming chaos, Ecography, 15 (1992), 226-236. doi: 10.1111/j.1600-0587.1992.tb00029.x. L. F. Olsen, G. L. Truty and W. M. Schaffer, Oscillations and chaos in epidemics: A nonlinear dynamic study of six childhood diseases in Copenhagen, Denmark, Theoretical Population Biology, 33 (1988), 344-370. doi: 10.1016/0040-5809(88)90019-6. L. F. Olsen and W. M. Schaffer, Chaos vs. noisy periodicity: Alternative hypotheses for childhood epidemics, Science, 249 (1990), 499-504.  E. Renshaw, Chaos in biometry, IMA Journal of Mathematics Applied in Medicine and Biology, 11 (1994), 17-44. doi: 10.1093/imammb/11.1.17. P. Rohani and D. J. D. Earn, Chaos in a cup of flour Trends in Ecology and Evolution 12 (1997), p171. doi: 10. 1016/S0169-5347(97)01055-0. M. R. Rychlik, Another proof of Jakobson's theorem and related results, Ergodic Theory and Dynamical Systems, 8 (1988), 93-109. doi: 10.1017/S014338570000434X. S. H. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology and Chemistry, Perseus Publishing, 2001.  P. Turchin, Chaos and stability in rodent population dynamics: evidence from nonlinear time-series analysis, Oikos, 68 (1993), 167-172. doi: 10.2307/3545323. P. Turchin, Chaos in microtine populations, Proceedings of the Royal Society of London B, 262 (1995), 357-361. doi: 10.1098/rspb.1995.0217. P. Turchin and S. P. Ellner, Living on the edge of chaos: Population dynamics of Fennoscandian voles, Ecology, 81 (2000), 3099-3116.  S. M. Ulam, Problems in Modern Mathematics, Interscience, New York, 1964. S. N. Wood, Statistical inference for noisy nonlinear ecological dynamic systems, Nature, 466 (2010), 1102-1104. doi: 10.1038/nature09319.  J. -C. Yoccoz, A Proof of Jakobson's Theorem https://matheuscmss.wordpress.com/category/mathematics/expository/.  C. Zimmer, Life after chaos, Science, 284 (1999), 83-86.  Graphs of the probability density function $f^*$ and its estimation $f_{2^5}$ for Example 2. Since $\{ f_{2^n} \}_{n \ge 1}$ is bounded by 2, by Theorem 4 $\{ f_{2^n} \}_{n \ge 1}$ is weakly pre-compact, and hence there exists a subsequence $\{ f_{2^{n_k}} \}_{n \geq 1}$ of $\{ f_{2^n} \}_{n \geq 1}$ which converges weakly to $f^* \in L^1[a,b]$. In fact, we may expect that $\{ f_{2^n} \}_{n \ge 1}$ converges strongly to $f^*$. Graphs of the probability density function $f^*$ and its estimation $f_{2^9}$ for Example 3. Again we may expect that $\{ f_{2^n} \}_{n \ge 1}$ converges strongly to $f^*$. Bifurcation diagram of Smith-Slatkin map for $\alpha = 1.5, \beta = 1, \gamma = 0.1$, as $p$ changes from 5 to 19. Graph of Smith-Slatkin map for $\alpha = 1.5, \beta = 1, \gamma = 0.1$ and $p=18$ with $x^*=0.977725973073732$, $x_c=0.857817481013884$, $b=\tau(x_c)=1.295956066342802$, and $a=\tau^2(x_c)=0.147708628629491$. Note that $\tau: [a,b] \rightarrow [a,b]$. Graph of $\frac{p^3-3p^2+5p-3}{6(p^2+1).}$ For $\alpha=1.5$ and $\gamma=0.1$, the condition $\frac\gamma\alpha < \frac{p^3-3p^2+5p-3}{6(p^2+1)}$ in Proposition 3 is fulfilled for all $p > 2.$ Graph of Smith-Slatkin map for $\alpha = 1.5, \beta = 1, \gamma = 0.1$ and $p=8$. Three first iterates of the critical point are shown. Graph of Smith-Slatkin map for $\alpha = 1.5, \beta = 1, \gamma = 0.1$ and $p=8.5$. Three first iterates of the critical point are shown. Graph of $f_{2^8}$ for Example 5. There are six peak values, where the first peak value occurs at $x=a$ and the last peak value at $x=b$. Although the closed form of the probability density function $f^*$ is unknown, the numerical simulations strongly show that $\{ f_{2^n} \}_{n \geq 1}$ is bounded by an integrable function. So, by Theorem 4, $\{ f_{2^n} \}_{n \ge 1}$ is weakly pre-compact, and hence there exists a sub-sequence $\{ f_{2^{n_k}} \}_{n \geq 1}$ of $\{ f_{2^n} \}_{n \geq 1}$ which converges weakly to $f^* \in L^1[a,b]$. By Proposition 2 the measure $\mu$ with density function $f^*$ is an absolutely continuous $\tau$-invariant measure. So $f^*$ is the PDF of the chaotic map $\tau$. In fact, we may expect that $\{ f_{2^n} \}_{n \geq 1}$ converges strongly to $f^*$.
Maximum values of $f_{2^n}$
 $n$ $2^n$ maximum values of$f_{2^n}$ $1$ $2$ $1.25$ $2$ $4$ $1.5075$ $3$ $8$ $1.7232$ $4$ $16$ $1.8157$ $5$ $32$ $1.9126$ $6$ $64$ $1.9465$ $7$ $128$ $1.9748$ $8$ $256$ $1.9871$ $9$ $512$ $1.9930$ $10$ $1024$ $1.9964$
 $n$ $2^n$ maximum values of$f_{2^n}$ $1$ $2$ $1.25$ $2$ $4$ $1.5075$ $3$ $8$ $1.7232$ $4$ $16$ $1.8157$ $5$ $32$ $1.9126$ $6$ $64$ $1.9465$ $7$ $128$ $1.9748$ $8$ $256$ $1.9871$ $9$ $512$ $1.9930$ $10$ $1024$ $1.9964$
Peak values of $f_{2^n}$ for $\tau$ in Example 3
 $n$ $2^n$ 1st peak value of $f_{2^n}$ 2nd peak value of $f_{2^n}$ $5$ $32$ $1.2899$ $3.8291$ $6$ $64$ $1.8337$ $5.4724$ $7$ $128$ $2.4475$ $7.3235$ $8$ $256$ $3.5149$ $10.5309$ $9$ $512$ $4.8498$ $14.5401$ $10$ $1024$ $6.9200$ $20.7534$
 $n$ $2^n$ 1st peak value of $f_{2^n}$ 2nd peak value of $f_{2^n}$ $5$ $32$ $1.2899$ $3.8291$ $6$ $64$ $1.8337$ $5.4724$ $7$ $128$ $2.4475$ $7.3235$ $8$ $256$ $3.5149$ $10.5309$ $9$ $512$ $4.8498$ $14.5401$ $10$ $1024$ $6.9200$ $20.7534$
The consecutive ratio of peak values of $f_{2^n}$ in Example 3
 $n$ $2^n$ $f_{2^n}/{f_{2^{n-1}}}$ at 1st peak value $f_{2^n}/{f_{2^{n-1}}}$ at 2nd peak value $6$ $64$ $1.4216$ $1.4292$ $7$ $128$ $1.3347$ $1.3382$ $8$ $256$ $1.4361$ $1.4380$ $9$ $512$ $1.3798$ $1.3807$ $10$ $1024$ $1.4269$ $1.4273$
 $n$ $2^n$ $f_{2^n}/{f_{2^{n-1}}}$ at 1st peak value $f_{2^n}/{f_{2^{n-1}}}$ at 2nd peak value $6$ $64$ $1.4216$ $1.4292$ $7$ $128$ $1.3347$ $1.3382$ $8$ $256$ $1.4361$ $1.4380$ $9$ $512$ $1.3798$ $1.3807$ $10$ $1024$ $1.4269$ $1.4273$
Peak values of $f_{2^n}$ for $\tau$ in Example 5
 $n$ $2^n$ The 1st peak values The 2nd peak values $8$ $256$ $9.9001$ $6.5190$ $9$ $512$ $13.987$ $9.0862$ $10$ $1024$ $20.609$ $12.770$ $11$ $2048$ $28.397$ $16.860$
 $n$ $2^n$ The 1st peak values The 2nd peak values $8$ $256$ $9.9001$ $6.5190$ $9$ $512$ $13.987$ $9.0862$ $10$ $1024$ $20.609$ $12.770$ $11$ $2048$ $28.397$ $16.860$
Peak values of $f_{2^n}$ for $\tau$ in Example 5
 $n$ $2^n$ The 3rd peak values The 4th peak values $8$ $256$ $4.2530$ $2.7992$ $9$ $512$ $5.8493$ $3.8154$ $10$ $1024$ $8.1477$ $5.2147$ $11$ $2048$ $10.711$ $6.8142$
 $n$ $2^n$ The 3rd peak values The 4th peak values $8$ $256$ $4.2530$ $2.7992$ $9$ $512$ $5.8493$ $3.8154$ $10$ $1024$ $8.1477$ $5.2147$ $11$ $2048$ $10.711$ $6.8142$
Peak values of $f_{2^n}$ for $\tau$ in Example 5
 $n$ $2^n$ The 5th peak values The 6th peak values $8$ $256$ $2.0871$ $4.1312$ $9$ $512$ $2.8524$ $5.5220$ $10$ $1024$ $3.5875$ $7.8931$ $11$ $2048$ $4.5694$ $10.607$
 $n$ $2^n$ The 5th peak values The 6th peak values $8$ $256$ $2.0871$ $4.1312$ $9$ $512$ $2.8524$ $5.5220$ $10$ $1024$ $3.5875$ $7.8931$ $11$ $2048$ $4.5694$ $10.607$
The consecutive ratio of the first two peak values of $f_{2^n}$ for $\tau$ in Example 5
 $n$ $2^n$ $f_{2^n}/{f_{2^{n-1}}}$ at 1st peak value $f_{2^n}/{f_{2^{n-1}}}$ at 2nd peak value $9$ $512$ $1.4128$ $1.3938$ $10$ $1024$ $1.4735$ $1.4055$ $11$ $2048$ $1.3779$ $1.3203$
 $n$ $2^n$ $f_{2^n}/{f_{2^{n-1}}}$ at 1st peak value $f_{2^n}/{f_{2^{n-1}}}$ at 2nd peak value $9$ $512$ $1.4128$ $1.3938$ $10$ $1024$ $1.4735$ $1.4055$ $11$ $2048$ $1.3779$ $1.3203$
The consecutive ratio of the middle two peak values of $f_{2^n}$ for $\tau$ in Example 5
 $n$ $2^n$ $f_{2^n}/{f_{2^{n-1}}}$ at 3rd peak value $f_{2^n}/{f_{2^{n-1}}}$ at 4th peak value $9$ $512$ $1.3753$ $1.3630$ $10$ $1024$ $1.3929$ $1.3668$ $11$ $2048$ $1.3146$ $1.3067$
 $n$ $2^n$ $f_{2^n}/{f_{2^{n-1}}}$ at 3rd peak value $f_{2^n}/{f_{2^{n-1}}}$ at 4th peak value $9$ $512$ $1.3753$ $1.3630$ $10$ $1024$ $1.3929$ $1.3668$ $11$ $2048$ $1.3146$ $1.3067$
The consecutive ratio of the last two peak values of $f_{2^n}$ for $\tau$ in Example 5
 $n$ $2^n$ $f_{2^n}/{f_{2^{n-1}}}$ at 5th peak value $f_{2^n}/{f_{2^{n-1}}}$ at 6th peak value $9$ $512$ $1.3667$ $1.3367$ $10$ $1024$ $1.2577$ $1.4294$ $11$ $2048$ $1.2737$ $1.3439$
 $n$ $2^n$ $f_{2^n}/{f_{2^{n-1}}}$ at 5th peak value $f_{2^n}/{f_{2^{n-1}}}$ at 6th peak value $9$ $512$ $1.3667$ $1.3367$ $10$ $1024$ $1.2577$ $1.4294$ $11$ $2048$ $1.2737$ $1.3439$
The estimates of ${\bar X}$ and $\sigma$ for Example 5
 $N$ ${\bar x}$ s $10^5$ $0.6658$ $0.3570$ $10^6$ $0.6657$ $0.3574$ $10^7$ $0.6657$ $0.3575$
 $N$ ${\bar x}$ s $10^5$ $0.6658$ $0.3570$ $10^6$ $0.6657$ $0.3574$ $10^7$ $0.6657$ $0.3575$
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