Abstract

Analyzing foliage-penetrating (FOPEN) ultra-wideband synthetic aperture radar (SAR) images is a challenging problem owing to the noisy and impulsive nature of foliage clutter. Indeed, many target-detection algorithms for FOPEN SAR data are characterized by high false-alarm rates. In this work, a statistical–physical model for foliage clutter is proposed that explains the presence of outliers in the data and suggests the use of symmetric alpha-stable (SαS) distributions for accurate clutter modeling. Furthermore, with the use of general assumptions of the noise sources and propagation conditions, the proposed model relates the parameters of the SαS model to physical parameters such as the attenuation coefficient and foliage density.

© 2003 Optical Society of America

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References

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  1. M. Ressler, L. Happ, L. Nguyen, T. Ton, M. Bennett, “The Army Research Laboratory ultra-wideband testbed radars,” in Proceedings of the IEEE International Radar Conference (Institute of Electrical and Electronics Engineers, New York, 1995), pp. 686–691.
  2. R. O. Harger, “Harmonic radar systems for near-ground in-foliage non-linear scatterers,” IEEE Trans. Aerosp. Electron. Syst. AES-12, 230–245 (1976).
    [CrossRef]
  3. J. G. Fleischman, S. Ayasli, E. M. Adams, D. R. Gosselin, “Foliage penetration experiment: part I: foliage attenuation and backscatter analysis of SAR imagery,” IEEE Trans. Aerosp. Electron. Syst. 32, 134–144 (1996).
    [CrossRef]
  4. J. W. McCorkle, “Early results from the ARL UWB foliage penetration SAR,” in Underground and Obscured-Object Imaging and Detection, N. K. Del Grande, I. Cindrich, P. B. Johnson, eds., Proc. SPIE1942, 88–95 (1993).
    [CrossRef]
  5. E. Jakeman, P. N. Pusey, “A model for non-Rayleigh sea echo,” IEEE Trans. Antennas Propag. 24, 806–814 (1976).
    [CrossRef]
  6. K. J. Sangston, K. R. Gerlach, “Coherent detection of radar targets in non-Gaussian background,” IEEE Trans. Aerosp. Electron. Syst. AES-30, 330–340 (1978).
  7. E. Conte, M. Longo, “Characterization of radar clutter as spherically-invariant random processes,” Proc. IEEE 134, 191–197 (1987).
  8. S. Bochner, “Stable law of probability and completely monotone functions,” Duke Math. J. 3, 726–728 (1937).
    [CrossRef]
  9. J. Ilow, D. Hatzinakos, “Analytic alpha-stable noise modeling in a Poisson field of interferers or scatterers,” IEEE Trans. Signal Process. 46, 1601–1611 (1998).
    [CrossRef]
  10. R. Kapoor, A. Banerjee, G. A. Tsihrintzis, N. Nandhakumar, “Detection of targets in heavy-tailed foliage clutter using an ultra-wideband (UWB) radar and alpha-stable clutter models,” IEEE Trans. Aerosp. Electron. Syst. 35, 819–834 (1999).
    [CrossRef]
  11. A. Banerjee, P. Burlina, R. Chellappa, “Adaptive target detection in foliage-penetrating SAR images using alpha-stable models,” IEEE Trans. Image Process. 13, 1823–1831 (1999).
    [CrossRef]
  12. W. Feller, An Introduction to Probability Theory and Its Applications, (Wiley, New York, 1971), Vol. 2.
  13. U. A. Muller, M. M. Dacorogna, O. V. Pictet, “Heavy tails in high-frequency financial data,” in A Practical Guide to Heavy Tails, R. J. Adler, R. Feldman, M. Taqqu, eds. (Birkhauser, Boston, Mass., 1998), pp. 55–78.
  14. E. F. Fama, R. Roll, “Parameter estimates for symmetric stable distributions,” J. Am. Stat. Assoc. 66, 331–338 (1971).
    [CrossRef]
  15. J. H. McCulloch, “Financial applications of stable distributions,” in Statistical Methods in Finance, Handbook of Statistics (North-Holland, New York, 1996), Vol. 14, pp. 383–425.
  16. B. W. Stuck, B. Kleiner, “A statistical analysis of telephone noise,” Bell Syst. Tech. J. 53, 1263–1320 (1974).
    [CrossRef]
  17. V. M. Zolotarev, One-Dimensional Stable Distributions (American Mathematical Society, Providence, R.I., 1996).
  18. C. L. Nikias, M. Shao, Signal Processing with Alpha-Stable Distributions and Applications (Wiley, New York, 1995).
  19. G. A. Tsihrintzis, C. L. Nikias, “Incoherent receivers in alpha-stable impulsive noise,” IEEE Trans. Signal Process. 43, 2225–2229 (1995).
    [CrossRef]
  20. G. Samorodnitsky, M. Taqqu, Stable Non-Gaussian Random Processes (Chapman & Hall, New York, 1994).
  21. J. P. Nolan, “Numerical computation of stable densities and distribution function,” Commun. Stat. Stochastic Models 133, 759–774 (1997).
  22. D. Middleton, “Non-Gaussian noise models in signal processing for telecommunications: new methods and results for class A and class B noise models,” IEEE Trans. Inf. Theory 45, 1129–1149 (1999).
    [CrossRef]

1999

R. Kapoor, A. Banerjee, G. A. Tsihrintzis, N. Nandhakumar, “Detection of targets in heavy-tailed foliage clutter using an ultra-wideband (UWB) radar and alpha-stable clutter models,” IEEE Trans. Aerosp. Electron. Syst. 35, 819–834 (1999).
[CrossRef]

A. Banerjee, P. Burlina, R. Chellappa, “Adaptive target detection in foliage-penetrating SAR images using alpha-stable models,” IEEE Trans. Image Process. 13, 1823–1831 (1999).
[CrossRef]

D. Middleton, “Non-Gaussian noise models in signal processing for telecommunications: new methods and results for class A and class B noise models,” IEEE Trans. Inf. Theory 45, 1129–1149 (1999).
[CrossRef]

1998

J. Ilow, D. Hatzinakos, “Analytic alpha-stable noise modeling in a Poisson field of interferers or scatterers,” IEEE Trans. Signal Process. 46, 1601–1611 (1998).
[CrossRef]

1997

J. P. Nolan, “Numerical computation of stable densities and distribution function,” Commun. Stat. Stochastic Models 133, 759–774 (1997).

1996

J. G. Fleischman, S. Ayasli, E. M. Adams, D. R. Gosselin, “Foliage penetration experiment: part I: foliage attenuation and backscatter analysis of SAR imagery,” IEEE Trans. Aerosp. Electron. Syst. 32, 134–144 (1996).
[CrossRef]

1995

G. A. Tsihrintzis, C. L. Nikias, “Incoherent receivers in alpha-stable impulsive noise,” IEEE Trans. Signal Process. 43, 2225–2229 (1995).
[CrossRef]

1987

E. Conte, M. Longo, “Characterization of radar clutter as spherically-invariant random processes,” Proc. IEEE 134, 191–197 (1987).

1978

K. J. Sangston, K. R. Gerlach, “Coherent detection of radar targets in non-Gaussian background,” IEEE Trans. Aerosp. Electron. Syst. AES-30, 330–340 (1978).

1976

R. O. Harger, “Harmonic radar systems for near-ground in-foliage non-linear scatterers,” IEEE Trans. Aerosp. Electron. Syst. AES-12, 230–245 (1976).
[CrossRef]

E. Jakeman, P. N. Pusey, “A model for non-Rayleigh sea echo,” IEEE Trans. Antennas Propag. 24, 806–814 (1976).
[CrossRef]

1974

B. W. Stuck, B. Kleiner, “A statistical analysis of telephone noise,” Bell Syst. Tech. J. 53, 1263–1320 (1974).
[CrossRef]

1971

E. F. Fama, R. Roll, “Parameter estimates for symmetric stable distributions,” J. Am. Stat. Assoc. 66, 331–338 (1971).
[CrossRef]

1937

S. Bochner, “Stable law of probability and completely monotone functions,” Duke Math. J. 3, 726–728 (1937).
[CrossRef]

Adams, E. M.

J. G. Fleischman, S. Ayasli, E. M. Adams, D. R. Gosselin, “Foliage penetration experiment: part I: foliage attenuation and backscatter analysis of SAR imagery,” IEEE Trans. Aerosp. Electron. Syst. 32, 134–144 (1996).
[CrossRef]

Ayasli, S.

J. G. Fleischman, S. Ayasli, E. M. Adams, D. R. Gosselin, “Foliage penetration experiment: part I: foliage attenuation and backscatter analysis of SAR imagery,” IEEE Trans. Aerosp. Electron. Syst. 32, 134–144 (1996).
[CrossRef]

Banerjee, A.

R. Kapoor, A. Banerjee, G. A. Tsihrintzis, N. Nandhakumar, “Detection of targets in heavy-tailed foliage clutter using an ultra-wideband (UWB) radar and alpha-stable clutter models,” IEEE Trans. Aerosp. Electron. Syst. 35, 819–834 (1999).
[CrossRef]

A. Banerjee, P. Burlina, R. Chellappa, “Adaptive target detection in foliage-penetrating SAR images using alpha-stable models,” IEEE Trans. Image Process. 13, 1823–1831 (1999).
[CrossRef]

Bennett, M.

M. Ressler, L. Happ, L. Nguyen, T. Ton, M. Bennett, “The Army Research Laboratory ultra-wideband testbed radars,” in Proceedings of the IEEE International Radar Conference (Institute of Electrical and Electronics Engineers, New York, 1995), pp. 686–691.

Bochner, S.

S. Bochner, “Stable law of probability and completely monotone functions,” Duke Math. J. 3, 726–728 (1937).
[CrossRef]

Burlina, P.

A. Banerjee, P. Burlina, R. Chellappa, “Adaptive target detection in foliage-penetrating SAR images using alpha-stable models,” IEEE Trans. Image Process. 13, 1823–1831 (1999).
[CrossRef]

Chellappa, R.

A. Banerjee, P. Burlina, R. Chellappa, “Adaptive target detection in foliage-penetrating SAR images using alpha-stable models,” IEEE Trans. Image Process. 13, 1823–1831 (1999).
[CrossRef]

Conte, E.

E. Conte, M. Longo, “Characterization of radar clutter as spherically-invariant random processes,” Proc. IEEE 134, 191–197 (1987).

Dacorogna, M. M.

U. A. Muller, M. M. Dacorogna, O. V. Pictet, “Heavy tails in high-frequency financial data,” in A Practical Guide to Heavy Tails, R. J. Adler, R. Feldman, M. Taqqu, eds. (Birkhauser, Boston, Mass., 1998), pp. 55–78.

Fama, E. F.

E. F. Fama, R. Roll, “Parameter estimates for symmetric stable distributions,” J. Am. Stat. Assoc. 66, 331–338 (1971).
[CrossRef]

Feller, W.

W. Feller, An Introduction to Probability Theory and Its Applications, (Wiley, New York, 1971), Vol. 2.

Fleischman, J. G.

J. G. Fleischman, S. Ayasli, E. M. Adams, D. R. Gosselin, “Foliage penetration experiment: part I: foliage attenuation and backscatter analysis of SAR imagery,” IEEE Trans. Aerosp. Electron. Syst. 32, 134–144 (1996).
[CrossRef]

Gerlach, K. R.

K. J. Sangston, K. R. Gerlach, “Coherent detection of radar targets in non-Gaussian background,” IEEE Trans. Aerosp. Electron. Syst. AES-30, 330–340 (1978).

Gosselin, D. R.

J. G. Fleischman, S. Ayasli, E. M. Adams, D. R. Gosselin, “Foliage penetration experiment: part I: foliage attenuation and backscatter analysis of SAR imagery,” IEEE Trans. Aerosp. Electron. Syst. 32, 134–144 (1996).
[CrossRef]

Happ, L.

M. Ressler, L. Happ, L. Nguyen, T. Ton, M. Bennett, “The Army Research Laboratory ultra-wideband testbed radars,” in Proceedings of the IEEE International Radar Conference (Institute of Electrical and Electronics Engineers, New York, 1995), pp. 686–691.

Harger, R. O.

R. O. Harger, “Harmonic radar systems for near-ground in-foliage non-linear scatterers,” IEEE Trans. Aerosp. Electron. Syst. AES-12, 230–245 (1976).
[CrossRef]

Hatzinakos, D.

J. Ilow, D. Hatzinakos, “Analytic alpha-stable noise modeling in a Poisson field of interferers or scatterers,” IEEE Trans. Signal Process. 46, 1601–1611 (1998).
[CrossRef]

Ilow, J.

J. Ilow, D. Hatzinakos, “Analytic alpha-stable noise modeling in a Poisson field of interferers or scatterers,” IEEE Trans. Signal Process. 46, 1601–1611 (1998).
[CrossRef]

Jakeman, E.

E. Jakeman, P. N. Pusey, “A model for non-Rayleigh sea echo,” IEEE Trans. Antennas Propag. 24, 806–814 (1976).
[CrossRef]

Kapoor, R.

R. Kapoor, A. Banerjee, G. A. Tsihrintzis, N. Nandhakumar, “Detection of targets in heavy-tailed foliage clutter using an ultra-wideband (UWB) radar and alpha-stable clutter models,” IEEE Trans. Aerosp. Electron. Syst. 35, 819–834 (1999).
[CrossRef]

Kleiner, B.

B. W. Stuck, B. Kleiner, “A statistical analysis of telephone noise,” Bell Syst. Tech. J. 53, 1263–1320 (1974).
[CrossRef]

Longo, M.

E. Conte, M. Longo, “Characterization of radar clutter as spherically-invariant random processes,” Proc. IEEE 134, 191–197 (1987).

McCorkle, J. W.

J. W. McCorkle, “Early results from the ARL UWB foliage penetration SAR,” in Underground and Obscured-Object Imaging and Detection, N. K. Del Grande, I. Cindrich, P. B. Johnson, eds., Proc. SPIE1942, 88–95 (1993).
[CrossRef]

McCulloch, J. H.

J. H. McCulloch, “Financial applications of stable distributions,” in Statistical Methods in Finance, Handbook of Statistics (North-Holland, New York, 1996), Vol. 14, pp. 383–425.

Middleton, D.

D. Middleton, “Non-Gaussian noise models in signal processing for telecommunications: new methods and results for class A and class B noise models,” IEEE Trans. Inf. Theory 45, 1129–1149 (1999).
[CrossRef]

Muller, U. A.

U. A. Muller, M. M. Dacorogna, O. V. Pictet, “Heavy tails in high-frequency financial data,” in A Practical Guide to Heavy Tails, R. J. Adler, R. Feldman, M. Taqqu, eds. (Birkhauser, Boston, Mass., 1998), pp. 55–78.

Nandhakumar, N.

R. Kapoor, A. Banerjee, G. A. Tsihrintzis, N. Nandhakumar, “Detection of targets in heavy-tailed foliage clutter using an ultra-wideband (UWB) radar and alpha-stable clutter models,” IEEE Trans. Aerosp. Electron. Syst. 35, 819–834 (1999).
[CrossRef]

Nguyen, L.

M. Ressler, L. Happ, L. Nguyen, T. Ton, M. Bennett, “The Army Research Laboratory ultra-wideband testbed radars,” in Proceedings of the IEEE International Radar Conference (Institute of Electrical and Electronics Engineers, New York, 1995), pp. 686–691.

Nikias, C. L.

G. A. Tsihrintzis, C. L. Nikias, “Incoherent receivers in alpha-stable impulsive noise,” IEEE Trans. Signal Process. 43, 2225–2229 (1995).
[CrossRef]

C. L. Nikias, M. Shao, Signal Processing with Alpha-Stable Distributions and Applications (Wiley, New York, 1995).

Nolan, J. P.

J. P. Nolan, “Numerical computation of stable densities and distribution function,” Commun. Stat. Stochastic Models 133, 759–774 (1997).

Pictet, O. V.

U. A. Muller, M. M. Dacorogna, O. V. Pictet, “Heavy tails in high-frequency financial data,” in A Practical Guide to Heavy Tails, R. J. Adler, R. Feldman, M. Taqqu, eds. (Birkhauser, Boston, Mass., 1998), pp. 55–78.

Pusey, P. N.

E. Jakeman, P. N. Pusey, “A model for non-Rayleigh sea echo,” IEEE Trans. Antennas Propag. 24, 806–814 (1976).
[CrossRef]

Ressler, M.

M. Ressler, L. Happ, L. Nguyen, T. Ton, M. Bennett, “The Army Research Laboratory ultra-wideband testbed radars,” in Proceedings of the IEEE International Radar Conference (Institute of Electrical and Electronics Engineers, New York, 1995), pp. 686–691.

Roll, R.

E. F. Fama, R. Roll, “Parameter estimates for symmetric stable distributions,” J. Am. Stat. Assoc. 66, 331–338 (1971).
[CrossRef]

Samorodnitsky, G.

G. Samorodnitsky, M. Taqqu, Stable Non-Gaussian Random Processes (Chapman & Hall, New York, 1994).

Sangston, K. J.

K. J. Sangston, K. R. Gerlach, “Coherent detection of radar targets in non-Gaussian background,” IEEE Trans. Aerosp. Electron. Syst. AES-30, 330–340 (1978).

Shao, M.

C. L. Nikias, M. Shao, Signal Processing with Alpha-Stable Distributions and Applications (Wiley, New York, 1995).

Stuck, B. W.

B. W. Stuck, B. Kleiner, “A statistical analysis of telephone noise,” Bell Syst. Tech. J. 53, 1263–1320 (1974).
[CrossRef]

Taqqu, M.

G. Samorodnitsky, M. Taqqu, Stable Non-Gaussian Random Processes (Chapman & Hall, New York, 1994).

Ton, T.

M. Ressler, L. Happ, L. Nguyen, T. Ton, M. Bennett, “The Army Research Laboratory ultra-wideband testbed radars,” in Proceedings of the IEEE International Radar Conference (Institute of Electrical and Electronics Engineers, New York, 1995), pp. 686–691.

Tsihrintzis, G. A.

R. Kapoor, A. Banerjee, G. A. Tsihrintzis, N. Nandhakumar, “Detection of targets in heavy-tailed foliage clutter using an ultra-wideband (UWB) radar and alpha-stable clutter models,” IEEE Trans. Aerosp. Electron. Syst. 35, 819–834 (1999).
[CrossRef]

G. A. Tsihrintzis, C. L. Nikias, “Incoherent receivers in alpha-stable impulsive noise,” IEEE Trans. Signal Process. 43, 2225–2229 (1995).
[CrossRef]

Zolotarev, V. M.

V. M. Zolotarev, One-Dimensional Stable Distributions (American Mathematical Society, Providence, R.I., 1996).

Bell Syst. Tech. J.

B. W. Stuck, B. Kleiner, “A statistical analysis of telephone noise,” Bell Syst. Tech. J. 53, 1263–1320 (1974).
[CrossRef]

Commun. Stat. Stochastic Models

J. P. Nolan, “Numerical computation of stable densities and distribution function,” Commun. Stat. Stochastic Models 133, 759–774 (1997).

Duke Math. J.

S. Bochner, “Stable law of probability and completely monotone functions,” Duke Math. J. 3, 726–728 (1937).
[CrossRef]

IEEE Trans. Aerosp. Electron. Syst.

K. J. Sangston, K. R. Gerlach, “Coherent detection of radar targets in non-Gaussian background,” IEEE Trans. Aerosp. Electron. Syst. AES-30, 330–340 (1978).

R. O. Harger, “Harmonic radar systems for near-ground in-foliage non-linear scatterers,” IEEE Trans. Aerosp. Electron. Syst. AES-12, 230–245 (1976).
[CrossRef]

J. G. Fleischman, S. Ayasli, E. M. Adams, D. R. Gosselin, “Foliage penetration experiment: part I: foliage attenuation and backscatter analysis of SAR imagery,” IEEE Trans. Aerosp. Electron. Syst. 32, 134–144 (1996).
[CrossRef]

R. Kapoor, A. Banerjee, G. A. Tsihrintzis, N. Nandhakumar, “Detection of targets in heavy-tailed foliage clutter using an ultra-wideband (UWB) radar and alpha-stable clutter models,” IEEE Trans. Aerosp. Electron. Syst. 35, 819–834 (1999).
[CrossRef]

IEEE Trans. Antennas Propag.

E. Jakeman, P. N. Pusey, “A model for non-Rayleigh sea echo,” IEEE Trans. Antennas Propag. 24, 806–814 (1976).
[CrossRef]

IEEE Trans. Image Process.

A. Banerjee, P. Burlina, R. Chellappa, “Adaptive target detection in foliage-penetrating SAR images using alpha-stable models,” IEEE Trans. Image Process. 13, 1823–1831 (1999).
[CrossRef]

IEEE Trans. Inf. Theory

D. Middleton, “Non-Gaussian noise models in signal processing for telecommunications: new methods and results for class A and class B noise models,” IEEE Trans. Inf. Theory 45, 1129–1149 (1999).
[CrossRef]

IEEE Trans. Signal Process.

G. A. Tsihrintzis, C. L. Nikias, “Incoherent receivers in alpha-stable impulsive noise,” IEEE Trans. Signal Process. 43, 2225–2229 (1995).
[CrossRef]

J. Ilow, D. Hatzinakos, “Analytic alpha-stable noise modeling in a Poisson field of interferers or scatterers,” IEEE Trans. Signal Process. 46, 1601–1611 (1998).
[CrossRef]

J. Am. Stat. Assoc.

E. F. Fama, R. Roll, “Parameter estimates for symmetric stable distributions,” J. Am. Stat. Assoc. 66, 331–338 (1971).
[CrossRef]

Proc. IEEE

E. Conte, M. Longo, “Characterization of radar clutter as spherically-invariant random processes,” Proc. IEEE 134, 191–197 (1987).

Other

M. Ressler, L. Happ, L. Nguyen, T. Ton, M. Bennett, “The Army Research Laboratory ultra-wideband testbed radars,” in Proceedings of the IEEE International Radar Conference (Institute of Electrical and Electronics Engineers, New York, 1995), pp. 686–691.

J. W. McCorkle, “Early results from the ARL UWB foliage penetration SAR,” in Underground and Obscured-Object Imaging and Detection, N. K. Del Grande, I. Cindrich, P. B. Johnson, eds., Proc. SPIE1942, 88–95 (1993).
[CrossRef]

J. H. McCulloch, “Financial applications of stable distributions,” in Statistical Methods in Finance, Handbook of Statistics (North-Holland, New York, 1996), Vol. 14, pp. 383–425.

V. M. Zolotarev, One-Dimensional Stable Distributions (American Mathematical Society, Providence, R.I., 1996).

C. L. Nikias, M. Shao, Signal Processing with Alpha-Stable Distributions and Applications (Wiley, New York, 1995).

W. Feller, An Introduction to Probability Theory and Its Applications, (Wiley, New York, 1971), Vol. 2.

U. A. Muller, M. M. Dacorogna, O. V. Pictet, “Heavy tails in high-frequency financial data,” in A Practical Guide to Heavy Tails, R. J. Adler, R. Feldman, M. Taqqu, eds. (Birkhauser, Boston, Mass., 1998), pp. 55–78.

G. Samorodnitsky, M. Taqqu, Stable Non-Gaussian Random Processes (Chapman & Hall, New York, 1994).

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Figures (4)

Fig. 1
Fig. 1

Example of impulsive foliage clutter: original FOPEN SAR image.

Fig. 2
Fig. 2

Histogram of foliage clutter pixels showing that their distribution is symmetric and bell shaped.

Fig. 3
Fig. 3

Exploratory data-analysis tests for heavy-tail and stability properties. (a) Complementary distribution function showing that data are heavy tailed. (b) Limit-distribution test suggesting that data are stable.

Fig. 4
Fig. 4

Empirical tests to compare the goodness of fit of foliage clutter with various distributions. (a) Amplitude probability distribution, (b) comparison of theoretical and empirical complementary distribution functions.

Equations (20)

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

AX1+BX2=dCX+D,
Cα=Aα+Bα,0<α<2.
X1+X2++Xn=dCnX+Dn,
Y1+Y2++Yndn+andX.
Φ(t)=exp(jat-γ|t|α[1+jβ sign(t)ω(t, α)]),
ω(t, α)=tan(απ/2)forα12π log|t|forα=1.
Φ(t)=exp(jat-γ|t|α).
y(t)=i=1Na(ri)xi(t),
Y=i=1NPrimXi.
Pr[kinR]=exp(-λA)(λA)kk!,
Y=Ki=1N1(ri2)m/2Xi.
S=i=0τi-1/αRi,
AS+BS=i=1(A-ατi)-1/αRi+i=1(B-ατi)-1/αRi=di=1τi1/αRi.
P[A-α(τi+1-τi)>x]=P[τi+1-τi>Aαx]=exp(-Aαx)
γ=λPαE[Xα]Cα,
Y=i=1NPrim exp(σGi)Xi.
γ=λPαE[(exp(σG)X)α]=λPαE[Xα]exp(1/2α2σ2).
P[X>x]x-αasx.
d log F¯(x)d log(x)-α,
Xt(m)=i=(t-1)m+1tmXi,

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