Abstract

Many physical experiments require analysis of the statistics of fluctuating radiation. In the case of an ideal single-photon detector, the contribution of photon noise to the statistics of the registered signal has been thoroughly examined. However, practical photon counters have a dead time, leading to miscounting of certain true events, and sometimes the counters generate false afterpulses. This study investigates the impact of these two effects, and it presents the theoretical relations between the statistical moments of the radiation and the registered counts while also accounting for dead time and the probability of afterpulses. Expressions for statistical moments of any order are obtained on the basis of the generalized Poisson distribution. For counters with paralyzable dead time, alternative relations for the mean and variance are derived using generally accepted formulas. As an example, the measurements of stellar scintillation and the result of a simple experiment are considered. The results of the experimental verification of the theoretical expression confirm the need to account for the nonideal nature of detectors in almost all similar measurements.

© 2013 Optical Society of America

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References

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  1. L. Mandel, “Fluctuations of photon beams and their correlation,” Proc. Phys. Soc. 72, 1037–1048 (1958).
    [CrossRef]
  2. E. Jakeman and P. N. Pusey, “Photon-counting statistics of optical scintillation,” in Inverse Scattering Problems in Optics, Vol. 20 of Topics in Current Physics (Springer-Verlag, 1980), pp. 73–116.
  3. H. Kuzmany, “Light scattering spectroscopy,” in Solid-State Spectroscopy (Springer, 2009), pp. 183–215.
  4. L. Mandel, “Fluctuations of photon beams: the distribution of the photo-electrons,” Proc. Phys. Soc. 74, 233–243 (1959).
    [CrossRef]
  5. G. Parry and J. G. Walker, “Statistics of stellar scintillation,” J. Opt. Soc. Am. 70, 1157–1159 (1980).
    [CrossRef]
  6. H. Z. Cummins and E. R. Pike, eds., Photon Correlation and Light Beating Spectroscopy, Vol. B3 of NATO Advanced Study Institutes Series (Plenum, 1974).
  7. E. Jakeman, G. Parry, E. R. Pike, and P. N. Pusey, “The twinkling of stars,” Contemp. Phys. 19, 127–145 (1978).
    [CrossRef]
  8. J. C. Dainty, B. M. Levine, B. J. Brames, and K. A. Odonnell, “Measurements of the wavelength dependence and other properties of stellar scintillation at Mauna Kea, Hawaii,” Appl. Opt. 21, 1196–1200 (1982).
    [CrossRef]
  9. B. Stecklum, “Measurements of stellar scintillation using photon counting statistics,” Astron. Nachr. 306, 145–156 (1985).
    [CrossRef]
  10. V. Kornilov, A. A. Tokovinin, O. Vozyakova, A. Zaitsev, N. Shatsky, S. F. Potanin, and M. S. Sarazin, “MASS: a monitor of the vertical turbulence distribution,” Proc. SPIE 4839, 837–845 (2003).
    [CrossRef]
  11. A. Tokovinin, V. Kornilov, N. Shatsky, and O. Voziakova, “Restoration of turbulence profile from scintillation indices,” Mon. Not. R. Astron. Soc. 343, 891–899 (2003).
    [CrossRef]
  12. G. Vannucci and M. C. Teich, “Dead-time-modified photocount mean and variance for chaotic radiation,” J. Opt. Soc. Am. 71, 164–170 (1981).
    [CrossRef]
  13. V. G. Kornilov, “A statistical description of the non-linearity of photon counts,” Astronomy Reports 52, 70–78 (2008).
    [CrossRef]
  14. P. Consul and G. Jain, “A generalization of the Poisson distribution,” Technometrics 15, 791–799 (1973).
    [CrossRef]
  15. R. S. Ambagaspitiya and N. Balakrishnan, “On the compound generalized Poisson distributions,” ASTIN Bull. 24, 255–263 (1994).
    [CrossRef]
  16. R. M. Corless, D. J. Jeffrey, and D. E. Knuth, “A sequence of series for the Lambert W function,” in Proceedings of the 1997 International Symposium on Symbolic and Algebraic Computation, W. W. Küchlin, ed. (ACM, 1997), pp. 197–204.
  17. W. Feller, An Introduction to the Theory of Probability and its Applications, Vol. 1 (Mir, 1967).
  18. V. I. Goldansky, A. V. Kutsenko, and M. I. Podgoretsky, Counter Statistics for Nuclear Particles Registration (Fizmatgiz, 1959).
  19. V. Kornilov, A. Tokovinin, N. Shatsky, O. Voziakova, S. Potanin, and B. Safonov, “Combined MASS-DIMM instruments for atmospheric turbulence studies,” Mon. Not. R. Astron. Soc. 382, 1268–1278 (2007).
    [CrossRef]
  20. G. Vannucci and M. C. Teich, “Effects of rate variation on the counting statistics of dead-time-modified Poisson processes,” Opt. Commun. 25, 267–272 (1978).
    [CrossRef]
  21. V. Kornilov, N. Shatsky, O. Voziakova, B. Safonov, S. Potanin, and M. Kornilov, “First results of a site-testing programme at Mount Shatdzhatmaz during 2007–2009,” Mon. Not. R. Astron. Soc. 408, 1233–1248 (2010).
    [CrossRef]

2010

V. Kornilov, N. Shatsky, O. Voziakova, B. Safonov, S. Potanin, and M. Kornilov, “First results of a site-testing programme at Mount Shatdzhatmaz during 2007–2009,” Mon. Not. R. Astron. Soc. 408, 1233–1248 (2010).
[CrossRef]

2008

V. G. Kornilov, “A statistical description of the non-linearity of photon counts,” Astronomy Reports 52, 70–78 (2008).
[CrossRef]

2007

V. Kornilov, A. Tokovinin, N. Shatsky, O. Voziakova, S. Potanin, and B. Safonov, “Combined MASS-DIMM instruments for atmospheric turbulence studies,” Mon. Not. R. Astron. Soc. 382, 1268–1278 (2007).
[CrossRef]

2003

V. Kornilov, A. A. Tokovinin, O. Vozyakova, A. Zaitsev, N. Shatsky, S. F. Potanin, and M. S. Sarazin, “MASS: a monitor of the vertical turbulence distribution,” Proc. SPIE 4839, 837–845 (2003).
[CrossRef]

A. Tokovinin, V. Kornilov, N. Shatsky, and O. Voziakova, “Restoration of turbulence profile from scintillation indices,” Mon. Not. R. Astron. Soc. 343, 891–899 (2003).
[CrossRef]

1994

R. S. Ambagaspitiya and N. Balakrishnan, “On the compound generalized Poisson distributions,” ASTIN Bull. 24, 255–263 (1994).
[CrossRef]

1985

B. Stecklum, “Measurements of stellar scintillation using photon counting statistics,” Astron. Nachr. 306, 145–156 (1985).
[CrossRef]

1982

1981

1980

1978

E. Jakeman, G. Parry, E. R. Pike, and P. N. Pusey, “The twinkling of stars,” Contemp. Phys. 19, 127–145 (1978).
[CrossRef]

G. Vannucci and M. C. Teich, “Effects of rate variation on the counting statistics of dead-time-modified Poisson processes,” Opt. Commun. 25, 267–272 (1978).
[CrossRef]

1973

P. Consul and G. Jain, “A generalization of the Poisson distribution,” Technometrics 15, 791–799 (1973).
[CrossRef]

1959

L. Mandel, “Fluctuations of photon beams: the distribution of the photo-electrons,” Proc. Phys. Soc. 74, 233–243 (1959).
[CrossRef]

1958

L. Mandel, “Fluctuations of photon beams and their correlation,” Proc. Phys. Soc. 72, 1037–1048 (1958).
[CrossRef]

Ambagaspitiya, R. S.

R. S. Ambagaspitiya and N. Balakrishnan, “On the compound generalized Poisson distributions,” ASTIN Bull. 24, 255–263 (1994).
[CrossRef]

Balakrishnan, N.

R. S. Ambagaspitiya and N. Balakrishnan, “On the compound generalized Poisson distributions,” ASTIN Bull. 24, 255–263 (1994).
[CrossRef]

Brames, B. J.

Consul, P.

P. Consul and G. Jain, “A generalization of the Poisson distribution,” Technometrics 15, 791–799 (1973).
[CrossRef]

Corless, R. M.

R. M. Corless, D. J. Jeffrey, and D. E. Knuth, “A sequence of series for the Lambert W function,” in Proceedings of the 1997 International Symposium on Symbolic and Algebraic Computation, W. W. Küchlin, ed. (ACM, 1997), pp. 197–204.

Dainty, J. C.

Feller, W.

W. Feller, An Introduction to the Theory of Probability and its Applications, Vol. 1 (Mir, 1967).

Goldansky, V. I.

V. I. Goldansky, A. V. Kutsenko, and M. I. Podgoretsky, Counter Statistics for Nuclear Particles Registration (Fizmatgiz, 1959).

Jain, G.

P. Consul and G. Jain, “A generalization of the Poisson distribution,” Technometrics 15, 791–799 (1973).
[CrossRef]

Jakeman, E.

E. Jakeman, G. Parry, E. R. Pike, and P. N. Pusey, “The twinkling of stars,” Contemp. Phys. 19, 127–145 (1978).
[CrossRef]

E. Jakeman and P. N. Pusey, “Photon-counting statistics of optical scintillation,” in Inverse Scattering Problems in Optics, Vol. 20 of Topics in Current Physics (Springer-Verlag, 1980), pp. 73–116.

Jeffrey, D. J.

R. M. Corless, D. J. Jeffrey, and D. E. Knuth, “A sequence of series for the Lambert W function,” in Proceedings of the 1997 International Symposium on Symbolic and Algebraic Computation, W. W. Küchlin, ed. (ACM, 1997), pp. 197–204.

Knuth, D. E.

R. M. Corless, D. J. Jeffrey, and D. E. Knuth, “A sequence of series for the Lambert W function,” in Proceedings of the 1997 International Symposium on Symbolic and Algebraic Computation, W. W. Küchlin, ed. (ACM, 1997), pp. 197–204.

Kornilov, M.

V. Kornilov, N. Shatsky, O. Voziakova, B. Safonov, S. Potanin, and M. Kornilov, “First results of a site-testing programme at Mount Shatdzhatmaz during 2007–2009,” Mon. Not. R. Astron. Soc. 408, 1233–1248 (2010).
[CrossRef]

Kornilov, V.

V. Kornilov, N. Shatsky, O. Voziakova, B. Safonov, S. Potanin, and M. Kornilov, “First results of a site-testing programme at Mount Shatdzhatmaz during 2007–2009,” Mon. Not. R. Astron. Soc. 408, 1233–1248 (2010).
[CrossRef]

V. Kornilov, A. Tokovinin, N. Shatsky, O. Voziakova, S. Potanin, and B. Safonov, “Combined MASS-DIMM instruments for atmospheric turbulence studies,” Mon. Not. R. Astron. Soc. 382, 1268–1278 (2007).
[CrossRef]

A. Tokovinin, V. Kornilov, N. Shatsky, and O. Voziakova, “Restoration of turbulence profile from scintillation indices,” Mon. Not. R. Astron. Soc. 343, 891–899 (2003).
[CrossRef]

V. Kornilov, A. A. Tokovinin, O. Vozyakova, A. Zaitsev, N. Shatsky, S. F. Potanin, and M. S. Sarazin, “MASS: a monitor of the vertical turbulence distribution,” Proc. SPIE 4839, 837–845 (2003).
[CrossRef]

Kornilov, V. G.

V. G. Kornilov, “A statistical description of the non-linearity of photon counts,” Astronomy Reports 52, 70–78 (2008).
[CrossRef]

Kutsenko, A. V.

V. I. Goldansky, A. V. Kutsenko, and M. I. Podgoretsky, Counter Statistics for Nuclear Particles Registration (Fizmatgiz, 1959).

Kuzmany, H.

H. Kuzmany, “Light scattering spectroscopy,” in Solid-State Spectroscopy (Springer, 2009), pp. 183–215.

Levine, B. M.

Mandel, L.

L. Mandel, “Fluctuations of photon beams: the distribution of the photo-electrons,” Proc. Phys. Soc. 74, 233–243 (1959).
[CrossRef]

L. Mandel, “Fluctuations of photon beams and their correlation,” Proc. Phys. Soc. 72, 1037–1048 (1958).
[CrossRef]

Odonnell, K. A.

Parry, G.

G. Parry and J. G. Walker, “Statistics of stellar scintillation,” J. Opt. Soc. Am. 70, 1157–1159 (1980).
[CrossRef]

E. Jakeman, G. Parry, E. R. Pike, and P. N. Pusey, “The twinkling of stars,” Contemp. Phys. 19, 127–145 (1978).
[CrossRef]

Pike, E. R.

E. Jakeman, G. Parry, E. R. Pike, and P. N. Pusey, “The twinkling of stars,” Contemp. Phys. 19, 127–145 (1978).
[CrossRef]

Podgoretsky, M. I.

V. I. Goldansky, A. V. Kutsenko, and M. I. Podgoretsky, Counter Statistics for Nuclear Particles Registration (Fizmatgiz, 1959).

Potanin, S.

V. Kornilov, N. Shatsky, O. Voziakova, B. Safonov, S. Potanin, and M. Kornilov, “First results of a site-testing programme at Mount Shatdzhatmaz during 2007–2009,” Mon. Not. R. Astron. Soc. 408, 1233–1248 (2010).
[CrossRef]

V. Kornilov, A. Tokovinin, N. Shatsky, O. Voziakova, S. Potanin, and B. Safonov, “Combined MASS-DIMM instruments for atmospheric turbulence studies,” Mon. Not. R. Astron. Soc. 382, 1268–1278 (2007).
[CrossRef]

Potanin, S. F.

V. Kornilov, A. A. Tokovinin, O. Vozyakova, A. Zaitsev, N. Shatsky, S. F. Potanin, and M. S. Sarazin, “MASS: a monitor of the vertical turbulence distribution,” Proc. SPIE 4839, 837–845 (2003).
[CrossRef]

Pusey, P. N.

E. Jakeman, G. Parry, E. R. Pike, and P. N. Pusey, “The twinkling of stars,” Contemp. Phys. 19, 127–145 (1978).
[CrossRef]

E. Jakeman and P. N. Pusey, “Photon-counting statistics of optical scintillation,” in Inverse Scattering Problems in Optics, Vol. 20 of Topics in Current Physics (Springer-Verlag, 1980), pp. 73–116.

Safonov, B.

V. Kornilov, N. Shatsky, O. Voziakova, B. Safonov, S. Potanin, and M. Kornilov, “First results of a site-testing programme at Mount Shatdzhatmaz during 2007–2009,” Mon. Not. R. Astron. Soc. 408, 1233–1248 (2010).
[CrossRef]

V. Kornilov, A. Tokovinin, N. Shatsky, O. Voziakova, S. Potanin, and B. Safonov, “Combined MASS-DIMM instruments for atmospheric turbulence studies,” Mon. Not. R. Astron. Soc. 382, 1268–1278 (2007).
[CrossRef]

Sarazin, M. S.

V. Kornilov, A. A. Tokovinin, O. Vozyakova, A. Zaitsev, N. Shatsky, S. F. Potanin, and M. S. Sarazin, “MASS: a monitor of the vertical turbulence distribution,” Proc. SPIE 4839, 837–845 (2003).
[CrossRef]

Shatsky, N.

V. Kornilov, N. Shatsky, O. Voziakova, B. Safonov, S. Potanin, and M. Kornilov, “First results of a site-testing programme at Mount Shatdzhatmaz during 2007–2009,” Mon. Not. R. Astron. Soc. 408, 1233–1248 (2010).
[CrossRef]

V. Kornilov, A. Tokovinin, N. Shatsky, O. Voziakova, S. Potanin, and B. Safonov, “Combined MASS-DIMM instruments for atmospheric turbulence studies,” Mon. Not. R. Astron. Soc. 382, 1268–1278 (2007).
[CrossRef]

A. Tokovinin, V. Kornilov, N. Shatsky, and O. Voziakova, “Restoration of turbulence profile from scintillation indices,” Mon. Not. R. Astron. Soc. 343, 891–899 (2003).
[CrossRef]

V. Kornilov, A. A. Tokovinin, O. Vozyakova, A. Zaitsev, N. Shatsky, S. F. Potanin, and M. S. Sarazin, “MASS: a monitor of the vertical turbulence distribution,” Proc. SPIE 4839, 837–845 (2003).
[CrossRef]

Stecklum, B.

B. Stecklum, “Measurements of stellar scintillation using photon counting statistics,” Astron. Nachr. 306, 145–156 (1985).
[CrossRef]

Teich, M. C.

G. Vannucci and M. C. Teich, “Dead-time-modified photocount mean and variance for chaotic radiation,” J. Opt. Soc. Am. 71, 164–170 (1981).
[CrossRef]

G. Vannucci and M. C. Teich, “Effects of rate variation on the counting statistics of dead-time-modified Poisson processes,” Opt. Commun. 25, 267–272 (1978).
[CrossRef]

Tokovinin, A.

V. Kornilov, A. Tokovinin, N. Shatsky, O. Voziakova, S. Potanin, and B. Safonov, “Combined MASS-DIMM instruments for atmospheric turbulence studies,” Mon. Not. R. Astron. Soc. 382, 1268–1278 (2007).
[CrossRef]

A. Tokovinin, V. Kornilov, N. Shatsky, and O. Voziakova, “Restoration of turbulence profile from scintillation indices,” Mon. Not. R. Astron. Soc. 343, 891–899 (2003).
[CrossRef]

Tokovinin, A. A.

V. Kornilov, A. A. Tokovinin, O. Vozyakova, A. Zaitsev, N. Shatsky, S. F. Potanin, and M. S. Sarazin, “MASS: a monitor of the vertical turbulence distribution,” Proc. SPIE 4839, 837–845 (2003).
[CrossRef]

Vannucci, G.

G. Vannucci and M. C. Teich, “Dead-time-modified photocount mean and variance for chaotic radiation,” J. Opt. Soc. Am. 71, 164–170 (1981).
[CrossRef]

G. Vannucci and M. C. Teich, “Effects of rate variation on the counting statistics of dead-time-modified Poisson processes,” Opt. Commun. 25, 267–272 (1978).
[CrossRef]

Voziakova, O.

V. Kornilov, N. Shatsky, O. Voziakova, B. Safonov, S. Potanin, and M. Kornilov, “First results of a site-testing programme at Mount Shatdzhatmaz during 2007–2009,” Mon. Not. R. Astron. Soc. 408, 1233–1248 (2010).
[CrossRef]

V. Kornilov, A. Tokovinin, N. Shatsky, O. Voziakova, S. Potanin, and B. Safonov, “Combined MASS-DIMM instruments for atmospheric turbulence studies,” Mon. Not. R. Astron. Soc. 382, 1268–1278 (2007).
[CrossRef]

A. Tokovinin, V. Kornilov, N. Shatsky, and O. Voziakova, “Restoration of turbulence profile from scintillation indices,” Mon. Not. R. Astron. Soc. 343, 891–899 (2003).
[CrossRef]

Vozyakova, O.

V. Kornilov, A. A. Tokovinin, O. Vozyakova, A. Zaitsev, N. Shatsky, S. F. Potanin, and M. S. Sarazin, “MASS: a monitor of the vertical turbulence distribution,” Proc. SPIE 4839, 837–845 (2003).
[CrossRef]

Walker, J. G.

Zaitsev, A.

V. Kornilov, A. A. Tokovinin, O. Vozyakova, A. Zaitsev, N. Shatsky, S. F. Potanin, and M. S. Sarazin, “MASS: a monitor of the vertical turbulence distribution,” Proc. SPIE 4839, 837–845 (2003).
[CrossRef]

Appl. Opt.

ASTIN Bull.

R. S. Ambagaspitiya and N. Balakrishnan, “On the compound generalized Poisson distributions,” ASTIN Bull. 24, 255–263 (1994).
[CrossRef]

Astron. Nachr.

B. Stecklum, “Measurements of stellar scintillation using photon counting statistics,” Astron. Nachr. 306, 145–156 (1985).
[CrossRef]

Astronomy Reports

V. G. Kornilov, “A statistical description of the non-linearity of photon counts,” Astronomy Reports 52, 70–78 (2008).
[CrossRef]

Contemp. Phys.

E. Jakeman, G. Parry, E. R. Pike, and P. N. Pusey, “The twinkling of stars,” Contemp. Phys. 19, 127–145 (1978).
[CrossRef]

J. Opt. Soc. Am.

Mon. Not. R. Astron. Soc.

A. Tokovinin, V. Kornilov, N. Shatsky, and O. Voziakova, “Restoration of turbulence profile from scintillation indices,” Mon. Not. R. Astron. Soc. 343, 891–899 (2003).
[CrossRef]

V. Kornilov, A. Tokovinin, N. Shatsky, O. Voziakova, S. Potanin, and B. Safonov, “Combined MASS-DIMM instruments for atmospheric turbulence studies,” Mon. Not. R. Astron. Soc. 382, 1268–1278 (2007).
[CrossRef]

V. Kornilov, N. Shatsky, O. Voziakova, B. Safonov, S. Potanin, and M. Kornilov, “First results of a site-testing programme at Mount Shatdzhatmaz during 2007–2009,” Mon. Not. R. Astron. Soc. 408, 1233–1248 (2010).
[CrossRef]

Opt. Commun.

G. Vannucci and M. C. Teich, “Effects of rate variation on the counting statistics of dead-time-modified Poisson processes,” Opt. Commun. 25, 267–272 (1978).
[CrossRef]

Proc. Phys. Soc.

L. Mandel, “Fluctuations of photon beams and their correlation,” Proc. Phys. Soc. 72, 1037–1048 (1958).
[CrossRef]

L. Mandel, “Fluctuations of photon beams: the distribution of the photo-electrons,” Proc. Phys. Soc. 74, 233–243 (1959).
[CrossRef]

Proc. SPIE

V. Kornilov, A. A. Tokovinin, O. Vozyakova, A. Zaitsev, N. Shatsky, S. F. Potanin, and M. S. Sarazin, “MASS: a monitor of the vertical turbulence distribution,” Proc. SPIE 4839, 837–845 (2003).
[CrossRef]

Technometrics

P. Consul and G. Jain, “A generalization of the Poisson distribution,” Technometrics 15, 791–799 (1973).
[CrossRef]

Other

R. M. Corless, D. J. Jeffrey, and D. E. Knuth, “A sequence of series for the Lambert W function,” in Proceedings of the 1997 International Symposium on Symbolic and Algebraic Computation, W. W. Küchlin, ed. (ACM, 1997), pp. 197–204.

W. Feller, An Introduction to the Theory of Probability and its Applications, Vol. 1 (Mir, 1967).

V. I. Goldansky, A. V. Kutsenko, and M. I. Podgoretsky, Counter Statistics for Nuclear Particles Registration (Fizmatgiz, 1959).

E. Jakeman and P. N. Pusey, “Photon-counting statistics of optical scintillation,” in Inverse Scattering Problems in Optics, Vol. 20 of Topics in Current Physics (Springer-Verlag, 1980), pp. 73–116.

H. Kuzmany, “Light scattering spectroscopy,” in Solid-State Spectroscopy (Springer, 2009), pp. 183–215.

H. Z. Cummins and E. R. Pike, eds., Photon Correlation and Light Beating Spectroscopy, Vol. B3 of NATO Advanced Study Institutes Series (Plenum, 1974).

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Figures (3)

Fig. 1.
Fig. 1.

Measured normalized moments α˜[r] of different orders r=2,,7 as a function of the mean α[1]. Top, low-intensity illumination. Bottom, large fluxes. The vertical segments indicate the estimate of the error of the measurement. The thin lines indicate the curves for each order r, as calculated using Eq. (22), with afterpulse probability q=0.043 and τ=7.4·105.

Fig. 2.
Fig. 2.

Dependencies of measured moments νr (r=3,,9) on the power of the scintillation in the form of ν2 for the 2 cm aperture. For comparison, the dashed lines show the dependencies νr(ν2) for the case of the log–normal distribution.

Fig. 3.
Fig. 3.

Relative contribution ϵ of consecutive terms of the series given by Eq. (20) for the moments of order r, calculated using the scintillation data. The allowed nonlinearity parameters are specified in each panel. The thick curves depict the contributions of the linear, quadratic, and cubic terms from top downward. The relative contributions for Eqs. (28) and (31) are shown by dashed curves.

Equations (44)

Equations on this page are rendered with MathJax. Learn more.

ϕ(x;ξ,s2)=0P(x;ξS)f(S;s2)dS.
E[x]=x=0xϕ(x;ξ,s2)=0x=0xP(x,ξS)f(S;s2)dS=0ξSf(S;s2)dS=ξ.
α[r]=x=0(x)r0P(x;ξS)f(S;s2)dS,
α[r]=0(ξS)rf(S;s2)dS=ξrνr,
Var[x]=ξ+ξ2s2.
γx(t)=x=0xt0P(x;ξS)f(S;s2)dS=0eSξ(t1)f(S;s2)dS=MS(ξtξ),
P(k;θ,λ)=θ(θ+kλ)k1eθkλk!,k=0n,P(k;θ,λ)=0,k>n,
E[k]=θ1λ,Var[k]=θ(1λ)3.
γ(t)=eθ(W(λeλt)/λ1),
μ[1]=θ1λ,μ[2]μ[1]2=1+λ(2λ)θ(1λ),μ[3]μ[1]3=1+3λ(2λ)θ(1λ)+λ2(2λ28λ+9)θ2(1λ)2,μ[4]μ[1]4=1+6λ(2λ)θ(1λ)+λ2(11λ244λ+48)θ2(1λ)2+.
μ[r]=μ[1]r(1ωr(r1)τ)=Grμ[1]r.
γx(t)=γx2(t)γx1(t2)=e(1q)θ(t1)+qθ(t21).
μ[1]=θ(1+q),μ[r]=θr(1+q)r+θr1r(r1)q(1+q)r2+O(q2),
μ[r]μ[1]r=1+r(r1)qμ[1](1+q)+O(q2/μ[1]2).
γx(t)=e(1q)θ(W(λeλt)/λ1)+qθ(W(λeλt2)/λ1).
μ[1]=θ(1+q)1λ,μ[2]μ[1]2=1+λ(2λ)μ[1](1λ)2+2qμ[1](1λ)2(1+q),μ[3]μ[1]3=1+3λ(2λ)μ[1](1λ)2+6qμ[1](1λ)2(1+q)+O(λ2θ2).
μ[r]μ[1]r=1+r(r1)λ(2λ)2θ(1λ)(1+q)+r(r1)qθ(1λ)(1+q)2+O(λ2θ2)+O(q2θ2).
μ[r]=Grμ[1]r+Hrμ[1]r1.
μ[r]=Gr(1+q)rk=0r(k)k!(ξS)r+k(τ)k+Hr(1+q)r1k=0(r+1)(k)k!(ξS)r+k1(τ)k,
α[r]=Gr(1+q)rξrk=0r(k)k!(ξτ)kνr+k+Hr(1+q)r1ξr1k=0(r+1)(k)k!(ξτ)kνr+k1.
α[1]=(1+q)ξk=0(ξτ)kk!νk+1=(1+q)ξ(1τξs2τξ+12ν3τ2ξ216ν4τ3ξ3+).
α˜[r]=Gr+Hrα[1].
α˜[r]=νrr(νr+1νrν2)τξ+,
α1=E[x]=0ξSeτξSf(S;s2)dS.
MS(u)=E[eu(S1)]=0eu(S1)f(S;s2)dS,
MS(u)=k=0νkukk!,MS(u)u=k=0νk+1ukk!,.
α1=eτξ(ξMS(τξ)MS(τξ)τ)=ξeτξ(MS(u)+MS(u)u).
α1=ξeτξk=0(νk+νk+1)(τξ)kk!=ξeτξ(1τξs2+12τ2ξ2(s2+ν3)).
μ[2]=μ[1]2(12τ).
α[2]=G20ξ2S2e2τξSf(S;s2)dS.
α[2]=G2ξ2e2τξk=0(νk+2νk+1+νk+2)(2τξ)kk!=G2ξ2e2τξ(1+s22(2s2+ν3)τξ+).
α12=E[x1x2]=0ξ1ξ2S1S2eτξ1S1τξ2S2f(S1,S2))dS1dS2,
α12=eτ1ξ1τ2ξ2(ξ1ξ2MS(u,v)ξ2MS(u,v)τ1ξ1MS(u,v)τ2+2MS(u,v)τ1τ2),
α12=ξ1ξ2eτ1ξ1τ2ξ2k=0l=0(νk,l+νk,l+1+νk+1,l+νk+1,l+1)(τ1ξ1)kk!(τ2ξ2)ll!.
α12=ξ1ξ2eτ1ξ1τ2ξ2[1+ν1,1(ν0,2+ν1,1+ν1,2)ξ2τ2(ν2,0+ν1,1+ν2,1)ξ1τ1+].
νn+1<nr+1nτξνnandνn+1<nr+1rτξνn,
νn+1<nr+1rτξ(νn+2νn1+νn2).
MS1(t)=MS*1((1b)t),
νr*=ξr(ξβ)rνr.
MS*((1b)t)=MS(t)ebt,
(1b)rνr*=νr+rbνr1+.
ξeτξ=α1(1+q)Σ1,ξτ=W(τα1(1+q)Σ1),
α˜[2]=(12τ)ν2(14τξ+2τ2ξ2)+Σ2Σ12+2qα1.
α˜12Σ1(1)Σ1(2)1=ν1,1(1τ1ξ1τ2ξ2+τ1ξ1τ2ξ2)+Σ12.

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