Abstract

In this paper, novel analytical closed-form expressions are derived for the probability density function of the sum of identically distributed correlated gamma-gamma random variables that models an optical atmospheric channel communication with receiver spatial diversity. The mathematical expressions here proposed provide a general procedure to obtain information about the scintillation effects induced by turbulence over a diversity reception scheme implementing equal-gain combining method. Both, validity and accuracy of the obtained statistical distribution are corroborated by comparing the analytical results to numerical results obtained by Monte-Carlo simulations. These simulations are particularized for constant, exponential and circular correlation models, corresponding to three different receivers spatial configurations. In addition, the extreme situations of no correlation and fully correlated received signals are also studied. The presented expressions lead to a simple and easy-to-compute analytical procedure of analyzing atmospheric optical communications systems with correlated spatial diversity.

© 2014 Optical Society of America

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References

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  1. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 2005).
    [Crossref]
  2. M.A. Al-Habash, L.C. Andrews, and R.L. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Engineering 40, 1554–1562 (2001).
    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  10. H. Samimi, “Distribution of the sum of K-distributed random variables and applications in free-space optical communications,” IET Optoelectronics 6, 1–6 (2012).
    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  16. M. S. Alouini, A. Abdi, and M. Kaveh, “Sum of gamma variates and performance of wireless communication systems over Nakagami-fading channels,” IEEE Trans. Vehic. Tech. 50, 1471–1480 (2001).
    [Crossref]
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    [Crossref]
  18. P. Lombardo, G. Fedele, and M. M. Rao, “MRC performance for binary signals in Nakagami fading with general branch correlation,” IEEE Trans. Commun. 47, 44–52 (1999).
    [Crossref]
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    [Crossref]
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2012 (3)

Z. Chen, S. Yu, T. Wang, G. Wu, S. Wang, and W. Gu, “Channel correlation in aperture receiver diversity systems for free-space optical communication,” J. Opt. 14, 125710 (2012).
[Crossref]

H. Samimi, “Distribution of the sum of K-distributed random variables and applications in free-space optical communications,” IET Optoelectronics 6, 1–6 (2012).
[Crossref]

H. Moradi, H.H. Refai, and P.G. LoPresti, “Switch-and-stay and switch-and-examine dual diversity for high-speed free-space optics links,” IET. Optoelectronics 6, 34–42 (2012).
[Crossref]

2011 (3)

2010 (2)

N. Wang and J. Cheng, “Moment-based estimation for the shape parameters of the gamma-gamma atmospheric turbulence model,” Opt. Express 18, 12824–12831 (2010).
[Crossref] [PubMed]

H. Moradi, H.H. Refai, and P.G. LoPresti, “Thresholding-based optimal detection of wireless optical signals,” J. Opt. Commun. Net. 2, 689–700 (2010).
[Crossref]

2009 (1)

T.A. Tsiftsis, H. G. Sandalidis, G. K. Karagiannidis, and M. Uysal, “Optical wireless links with spatial diversity over strong atmospheric turbulence channels,” IEEE Trans. Wireless Commun. 8, 951–957 (2009).
[Crossref]

2005 (1)

M. Razavi and J. H. Shapiro, “Wireless optical communications via diversity reception and optical preamplification,” IEEE Trans. Wireless Commun. 4, 975–983 (2005).
[Crossref]

2004 (1)

K. Zhang, Z. Song, and Y. L. Guan, “Simulation of Nakagami fading channels with arbitrary cross-correlation and fading parameters,” IEEE Trans. Wireless Commun. 3, 1463–1468 (2004).
[Crossref]

2001 (2)

M.A. Al-Habash, L.C. Andrews, and R.L. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Engineering 40, 1554–1562 (2001).
[Crossref]

M. S. Alouini, A. Abdi, and M. Kaveh, “Sum of gamma variates and performance of wireless communication systems over Nakagami-fading channels,” IEEE Trans. Vehic. Tech. 50, 1471–1480 (2001).
[Crossref]

1999 (1)

P. Lombardo, G. Fedele, and M. M. Rao, “MRC performance for binary signals in Nakagami fading with general branch correlation,” IEEE Trans. Commun. 47, 44–52 (1999).
[Crossref]

1996 (1)

M. M. Ibrahim and A. M. Ibrahim, “Performance analysis of optical receivers with space diversity reception,” Proc. IEEE Commun. 143, 369–372 (1996).
[Crossref]

1995 (1)

V. A. Aalo, “Performance of maximal-ratio diversity systems in a correlated Nakagami-fading environment,” IEEE Trans. Commun. 43, 2360–2369 (1995).
[Crossref]

1985 (1)

P.G. Moschopoulos, “The distribution of the sum of independent gamma random variables,” Ann. Inst. Statist. Math. (Part A) 37, 541–544 (1985).
[Crossref]

1964 (1)

S. Kotz and J. Adams, “Distribution of sum of identically distributed exponentially correlated gamma variables,” Ann. Math. Statist. 35, 277–283 (1964).
[Crossref]

Aalo, V. A.

V. A. Aalo, “Performance of maximal-ratio diversity systems in a correlated Nakagami-fading environment,” IEEE Trans. Commun. 43, 2360–2369 (1995).
[Crossref]

Abdi, A.

M. S. Alouini, A. Abdi, and M. Kaveh, “Sum of gamma variates and performance of wireless communication systems over Nakagami-fading channels,” IEEE Trans. Vehic. Tech. 50, 1471–1480 (2001).
[Crossref]

Abramowitz, M.

M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions (Dover Publications, 1972).

Adams, J.

S. Kotz and J. Adams, “Distribution of sum of identically distributed exponentially correlated gamma variables,” Ann. Math. Statist. 35, 277–283 (1964).
[Crossref]

Al-Habash, M.A.

M.A. Al-Habash, L.C. Andrews, and R.L. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Engineering 40, 1554–1562 (2001).
[Crossref]

Alouini, M. S.

M. S. Alouini, A. Abdi, and M. Kaveh, “Sum of gamma variates and performance of wireless communication systems over Nakagami-fading channels,” IEEE Trans. Vehic. Tech. 50, 1471–1480 (2001).
[Crossref]

Andrews, L. C.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 2005).
[Crossref]

Andrews, L.C.

M.A. Al-Habash, L.C. Andrews, and R.L. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Engineering 40, 1554–1562 (2001).
[Crossref]

Balaban, P.

M.C. Jeruchim, P. Balaban, and K.S. Shanmugan, Simulation of Communication Systems (Plenum, 1992).
[Crossref]

Castillo-Vázquez, M.

Chen, Z.

Z. Chen, S. Yu, T. Wang, G. Wu, S. Wang, and W. Gu, “Channel correlation in aperture receiver diversity systems for free-space optical communication,” J. Opt. 14, 125710 (2012).
[Crossref]

Cheng, J.

Dang, A.

Erdelyi, A.

A. Erdelyi, Tables of Integrals Transforms, vol. I (McGraw Hill, 1954).

Exton, H.

H. Exton, Multiple Hypergeometric Functions and Applications (John Wiley & Sons, 1976).

Fedele, G.

P. Lombardo, G. Fedele, and M. M. Rao, “MRC performance for binary signals in Nakagami fading with general branch correlation,” IEEE Trans. Commun. 47, 44–52 (1999).
[Crossref]

Garrido-Balsells, J. M.

Gradshteyn, I. S.

I. S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series and Products (Elsevier, 2007).

Gu, W.

Z. Chen, S. Yu, T. Wang, G. Wu, S. Wang, and W. Gu, “Channel correlation in aperture receiver diversity systems for free-space optical communication,” J. Opt. 14, 125710 (2012).
[Crossref]

Guan, Y. L.

K. Zhang, Z. Song, and Y. L. Guan, “Simulation of Nakagami fading channels with arbitrary cross-correlation and fading parameters,” IEEE Trans. Wireless Commun. 3, 1463–1468 (2004).
[Crossref]

Ibrahim, A. M.

M. M. Ibrahim and A. M. Ibrahim, “Performance analysis of optical receivers with space diversity reception,” Proc. IEEE Commun. 143, 369–372 (1996).
[Crossref]

Ibrahim, M. M.

M. M. Ibrahim and A. M. Ibrahim, “Performance analysis of optical receivers with space diversity reception,” Proc. IEEE Commun. 143, 369–372 (1996).
[Crossref]

Jeruchim, M.C.

M.C. Jeruchim, P. Balaban, and K.S. Shanmugan, Simulation of Communication Systems (Plenum, 1992).
[Crossref]

Jurado-Navas, A.

Karagiannidis, G. K.

T.A. Tsiftsis, H. G. Sandalidis, G. K. Karagiannidis, and M. Uysal, “Optical wireless links with spatial diversity over strong atmospheric turbulence channels,” IEEE Trans. Wireless Commun. 8, 951–957 (2009).
[Crossref]

Kaveh, M.

M. S. Alouini, A. Abdi, and M. Kaveh, “Sum of gamma variates and performance of wireless communication systems over Nakagami-fading channels,” IEEE Trans. Vehic. Tech. 50, 1471–1480 (2001).
[Crossref]

Kotz, S.

S. Kotz and J. Adams, “Distribution of sum of identically distributed exponentially correlated gamma variables,” Ann. Math. Statist. 35, 277–283 (1964).
[Crossref]

Lombardo, P.

P. Lombardo, G. Fedele, and M. M. Rao, “MRC performance for binary signals in Nakagami fading with general branch correlation,” IEEE Trans. Commun. 47, 44–52 (1999).
[Crossref]

LoPresti, P.G.

H. Moradi, H.H. Refai, and P.G. LoPresti, “Switch-and-stay and switch-and-examine dual diversity for high-speed free-space optics links,” IET. Optoelectronics 6, 34–42 (2012).
[Crossref]

H. Moradi, H.H. Refai, and P.G. LoPresti, “Thresholding-based optimal detection of wireless optical signals,” J. Opt. Commun. Net. 2, 689–700 (2010).
[Crossref]

Moradi, H.

H. Moradi, H.H. Refai, and P.G. LoPresti, “Switch-and-stay and switch-and-examine dual diversity for high-speed free-space optics links,” IET. Optoelectronics 6, 34–42 (2012).
[Crossref]

H. Moradi, H.H. Refai, and P.G. LoPresti, “Thresholding-based optimal detection of wireless optical signals,” J. Opt. Commun. Net. 2, 689–700 (2010).
[Crossref]

Moschopoulos, P.G.

P.G. Moschopoulos, “The distribution of the sum of independent gamma random variables,” Ann. Inst. Statist. Math. (Part A) 37, 541–544 (1985).
[Crossref]

Oppenheim, A.

A. Oppenheim and R. W. Schafer, Discrete-time Signal Processing (Prentice Hall, 1999).

Paris, J. F.

Phillips, R. L.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 2005).
[Crossref]

Phillips, R.L.

M.A. Al-Habash, L.C. Andrews, and R.L. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Engineering 40, 1554–1562 (2001).
[Crossref]

Puerta-Notario, A.

Rao, M. M.

P. Lombardo, G. Fedele, and M. M. Rao, “MRC performance for binary signals in Nakagami fading with general branch correlation,” IEEE Trans. Commun. 47, 44–52 (1999).
[Crossref]

Razavi, M.

M. Razavi and J. H. Shapiro, “Wireless optical communications via diversity reception and optical preamplification,” IEEE Trans. Wireless Commun. 4, 975–983 (2005).
[Crossref]

Refai, H.H.

H. Moradi, H.H. Refai, and P.G. LoPresti, “Switch-and-stay and switch-and-examine dual diversity for high-speed free-space optics links,” IET. Optoelectronics 6, 34–42 (2012).
[Crossref]

H. Moradi, H.H. Refai, and P.G. LoPresti, “Thresholding-based optimal detection of wireless optical signals,” J. Opt. Commun. Net. 2, 689–700 (2010).
[Crossref]

Ryzhik, I.M.

I. S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series and Products (Elsevier, 2007).

Samimi, H.

H. Samimi, “Distribution of the sum of K-distributed random variables and applications in free-space optical communications,” IET Optoelectronics 6, 1–6 (2012).
[Crossref]

Sandalidis, H. G.

T.A. Tsiftsis, H. G. Sandalidis, G. K. Karagiannidis, and M. Uysal, “Optical wireless links with spatial diversity over strong atmospheric turbulence channels,” IEEE Trans. Wireless Commun. 8, 951–957 (2009).
[Crossref]

Schafer, R. W.

A. Oppenheim and R. W. Schafer, Discrete-time Signal Processing (Prentice Hall, 1999).

Shanmugan, K.S.

M.C. Jeruchim, P. Balaban, and K.S. Shanmugan, Simulation of Communication Systems (Plenum, 1992).
[Crossref]

Shapiro, J. H.

M. Razavi and J. H. Shapiro, “Wireless optical communications via diversity reception and optical preamplification,” IEEE Trans. Wireless Commun. 4, 975–983 (2005).
[Crossref]

Song, Z.

K. Zhang, Z. Song, and Y. L. Guan, “Simulation of Nakagami fading channels with arbitrary cross-correlation and fading parameters,” IEEE Trans. Wireless Commun. 3, 1463–1468 (2004).
[Crossref]

Stegun, I.A.

M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions (Dover Publications, 1972).

Takayama, Y.

Takenaka, H.

Toyoshima, M.

Tsiftsis, T.A.

T.A. Tsiftsis, H. G. Sandalidis, G. K. Karagiannidis, and M. Uysal, “Optical wireless links with spatial diversity over strong atmospheric turbulence channels,” IEEE Trans. Wireless Commun. 8, 951–957 (2009).
[Crossref]

Uysal, M.

T.A. Tsiftsis, H. G. Sandalidis, G. K. Karagiannidis, and M. Uysal, “Optical wireless links with spatial diversity over strong atmospheric turbulence channels,” IEEE Trans. Wireless Commun. 8, 951–957 (2009).
[Crossref]

Wang, N.

Wang, S.

Z. Chen, S. Yu, T. Wang, G. Wu, S. Wang, and W. Gu, “Channel correlation in aperture receiver diversity systems for free-space optical communication,” J. Opt. 14, 125710 (2012).
[Crossref]

Wang, T.

Z. Chen, S. Yu, T. Wang, G. Wu, S. Wang, and W. Gu, “Channel correlation in aperture receiver diversity systems for free-space optical communication,” J. Opt. 14, 125710 (2012).
[Crossref]

Wu, G.

Z. Chen, S. Yu, T. Wang, G. Wu, S. Wang, and W. Gu, “Channel correlation in aperture receiver diversity systems for free-space optical communication,” J. Opt. 14, 125710 (2012).
[Crossref]

Yu, S.

Z. Chen, S. Yu, T. Wang, G. Wu, S. Wang, and W. Gu, “Channel correlation in aperture receiver diversity systems for free-space optical communication,” J. Opt. 14, 125710 (2012).
[Crossref]

Zhang, K.

K. Zhang, Z. Song, and Y. L. Guan, “Simulation of Nakagami fading channels with arbitrary cross-correlation and fading parameters,” IEEE Trans. Wireless Commun. 3, 1463–1468 (2004).
[Crossref]

Ann. Inst. Statist. Math. (Part A) (1)

P.G. Moschopoulos, “The distribution of the sum of independent gamma random variables,” Ann. Inst. Statist. Math. (Part A) 37, 541–544 (1985).
[Crossref]

Ann. Math. Statist. (1)

S. Kotz and J. Adams, “Distribution of sum of identically distributed exponentially correlated gamma variables,” Ann. Math. Statist. 35, 277–283 (1964).
[Crossref]

IEEE Trans. Commun. (2)

V. A. Aalo, “Performance of maximal-ratio diversity systems in a correlated Nakagami-fading environment,” IEEE Trans. Commun. 43, 2360–2369 (1995).
[Crossref]

P. Lombardo, G. Fedele, and M. M. Rao, “MRC performance for binary signals in Nakagami fading with general branch correlation,” IEEE Trans. Commun. 47, 44–52 (1999).
[Crossref]

IEEE Trans. Vehic. Tech. (1)

M. S. Alouini, A. Abdi, and M. Kaveh, “Sum of gamma variates and performance of wireless communication systems over Nakagami-fading channels,” IEEE Trans. Vehic. Tech. 50, 1471–1480 (2001).
[Crossref]

IEEE Trans. Wireless Commun. (3)

T.A. Tsiftsis, H. G. Sandalidis, G. K. Karagiannidis, and M. Uysal, “Optical wireless links with spatial diversity over strong atmospheric turbulence channels,” IEEE Trans. Wireless Commun. 8, 951–957 (2009).
[Crossref]

M. Razavi and J. H. Shapiro, “Wireless optical communications via diversity reception and optical preamplification,” IEEE Trans. Wireless Commun. 4, 975–983 (2005).
[Crossref]

K. Zhang, Z. Song, and Y. L. Guan, “Simulation of Nakagami fading channels with arbitrary cross-correlation and fading parameters,” IEEE Trans. Wireless Commun. 3, 1463–1468 (2004).
[Crossref]

IET Optoelectronics (1)

H. Samimi, “Distribution of the sum of K-distributed random variables and applications in free-space optical communications,” IET Optoelectronics 6, 1–6 (2012).
[Crossref]

IET. Optoelectronics (1)

H. Moradi, H.H. Refai, and P.G. LoPresti, “Switch-and-stay and switch-and-examine dual diversity for high-speed free-space optics links,” IET. Optoelectronics 6, 34–42 (2012).
[Crossref]

J. Opt. (1)

Z. Chen, S. Yu, T. Wang, G. Wu, S. Wang, and W. Gu, “Channel correlation in aperture receiver diversity systems for free-space optical communication,” J. Opt. 14, 125710 (2012).
[Crossref]

J. Opt. Commun. Net. (1)

H. Moradi, H.H. Refai, and P.G. LoPresti, “Thresholding-based optimal detection of wireless optical signals,” J. Opt. Commun. Net. 2, 689–700 (2010).
[Crossref]

Opt. Engineering (1)

M.A. Al-Habash, L.C. Andrews, and R.L. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Engineering 40, 1554–1562 (2001).
[Crossref]

Opt. Express (3)

Opt. Lett. (1)

Proc. IEEE Commun. (1)

M. M. Ibrahim and A. M. Ibrahim, “Performance analysis of optical receivers with space diversity reception,” Proc. IEEE Commun. 143, 369–372 (1996).
[Crossref]

Other (7)

A. Erdelyi, Tables of Integrals Transforms, vol. I (McGraw Hill, 1954).

H. Exton, Multiple Hypergeometric Functions and Applications (John Wiley & Sons, 1976).

M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions (Dover Publications, 1972).

A. Oppenheim and R. W. Schafer, Discrete-time Signal Processing (Prentice Hall, 1999).

I. S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series and Products (Elsevier, 2007).

M.C. Jeruchim, P. Balaban, and K.S. Shanmugan, Simulation of Communication Systems (Plenum, 1992).
[Crossref]

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 2005).
[Crossref]

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Figures (5)

Fig. 1
Fig. 1 Single-input multiple-output (SIMO) system model with a laser transmitter and N optical receivers.
Fig. 2
Fig. 2 Analytical and simulation results for the pdf of the normalized irradiance I for the extreme correlation cases of ρ = 0 and ρ = 1. N = 4 optical receivers and typical moderate turbulence intensity with αx = 10 and αy = 5 are assumed.
Fig. 3
Fig. 3 Analytical and simulation results for the pdf of V and I RVs obtained with the constant correlation model and N = 3 equidistant optical receivers. In (a) and (b), the results are shown for different correlation levels with a typical turbulence intensity σ I 2 = 0.32. In (c) and (d) the results are shown as a function of turbulence conditions (weak, moderate and strong) with a correlation level given by ρ = 0.6.
Fig. 4
Fig. 4 Analytical and simulation results for the pdf of V and I RVs obtained with the exponential correlation model and N = 5 optical receivers. In (a) and (b), the results are shown for different correlation levels with a typical turbulence intensity σ I 2 = 0.32. In (c) and (d) the results are shown as a function of turbulence conditions (weak, moderate and strong) with a correlation level given by ρ = 0.6.
Fig. 5
Fig. 5 Analytical and simulation results for the pdf of V and I RVs obtained with the circular correlation model and N = 5 optical receivers. In (a) and (b), the results are shown for different correlation levels with a typical turbulence intensity σ I 2 = 0.32. In (c) and (d) the results are shown as a function of turbulence conditions (weak, moderate and strong) with a correlation level given by ρ2 = ρ12 = ρ15 = 0.8 and ρ3 = ρ13 = ρ14 = 0.6.

Equations (50)

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

I r x = i = 1 N I r x , i .
I = i = 1 N I i .
f γ ( γ ) = γ a 1 b a Γ ( a ) exp ( γ b ) ,
I = X i = 1 N Y i = X V .
ρ i j = cov ( Y i , Y j ) var ( Y i ) var ( Y j ) ,
C y = [ 1 ρ 12 ρ 1 N ρ 21 1 ρ 2 N ρ N 1 ρ N 2 1 ] N × N ,
M V ( s ) = E [ e s V ] = 𝔏 { f V ( v ) ; s } = i = 1 N ( 1 λ i s ) α ,
C = [ 1 ρ 12 ρ 1 N ρ 21 1 ρ 2 N ρ N 1 ρ N 2 1 ] N × N ,
𝔏 { F V ( v ) ; s } = 1 s 𝔏 { f V ( v ) ; s } = 1 s M V ( s ) = 1 s i = 1 N ( 1 + λ i s ) α ,
i = 1 N ( 1 + λ i s ) α = i = 1 N ( 1 λ i ) α Γ ( N α ) . Γ ( N α ) s N α i = 1 N ( 1 ( 1 λ i ) s ) α
i = 1 N ( 1 + λ i s ) α = 1 det ( A ) α Γ ( N α ) 𝔏 { v N α 1 Φ 2 ( N ) ( α , , α ; N α ; v λ 1 1 , , v λ i 1 ) } ,
f V ( v ) = 𝔏 1 { M V ( s ) ; v } = { v N α 1 [ det ( A ) ] α Γ ( N α ) } × × Φ 2 ( N ) ( α , , α ; N α ; v λ 1 , , v λ N ) ,
F V ( v ) = 𝔏 1 { 1 s M V ( s ) ; v } = { v N α [ det ( A ) ] α Γ ( N α + 1 ) } × × Φ 2 ( N ) ( α , , α ; N α + 1 ; v λ 1 , , v λ N ) .
f X ( x ) = α x ( α x x ) α x 1 Γ ( α x ) exp ( α x x ) ,
f I ( I | x ) = f V ( I | x ) | I V | = 1 x f V ( I | x ) ,
f I ( I | x ) = { I N α 1 x N α [ det ( A ) ] α Γ ( N α ) } × × Φ 2 ( N ) ( α , , α ; N α ; I x λ 1 , , I x λ N ) .
f I ( I ) = 0 f I ( I | x ) f X ( x ) d x = I N α 1 [ det ( A ) ] α Γ ( N α ) α x α x Γ ( α x ) 𝕀 ,
𝕀 = 0 x α x N α 1 Φ 2 ( N ) exp ( α x x ) dx
f I ( I ) I N α 1 [ det ( A ) ] α Γ ( N α ) α x α x Γ ( α x ) i = 1 n x i α x N α ( n + 1 ) 2 [ L n + 1 ( x i ) ] 2 × × Φ 2 ( N ) ( α , , α ; N α ; I α x x i λ 1 , , I α x x i λ N )
𝕀 = 0 f ( t ) t α x exp ( α x t ) dt ,
F ( s ) = 𝔏 { f ( t ) ; s } = Γ ( N α ) s N α i = 1 N ( 1 + I λ i s ) α ,
F ( s ) = Γ ( N α ) i = 1 N ( s ( I λ i ) ) α .
i = 1 N 1 ( s d i ) α i = i = 1 N m = 1 α i c m i ( s d i ) m ,
F ( s ) = Γ ( N α ) i = 1 N m = 1 α i c m i ( s d i ) m .
𝔏 1 { 1 ( s + a ) n + 1 ; t } = t n n ! exp ( a t ) u ( t ) ,
𝔏 1 { 1 ( s d i ) m ; t } = t m 1 ( m 1 ) ! exp ( d i t ) u ( t ) ,
f ( t ) = Γ ( N α ) i = 1 N m = 1 α i c m i t m 1 ( m 1 ) ! exp ( I λ i t ) .
𝕀 = 2 Γ ( N α ) i = 1 N m = 1 α i c m i ( m 1 ) ! λ i m α x 2 α x m α x 2 I m α x 2 K m α x ( 2 α x I λ i ) .
f I ( I ) = 2 [ det ( A ) ] α Γ ( α x ) i = 1 N m = 1 α i c m i Γ ( m ) λ i m α x 2 α x m + α x 2 × × I N α 1 m α x 2 K m α x ( 2 α x I λ i ) ,
f V ( v ) = 𝔏 1 { 1 [ det ( A ) ] α i = 1 N m = 1 α i c m i ( s d i ) m ; t } ,
f V ( v ) = 1 [ det ( A ) ] α i = 1 N m = 1 α i c m i Γ ( m ) v m 1 exp ( v λ i ) .
f V ( v ) = i = 1 N m = 1 α i c m i [ det ( A ) ] α λ i m g V m , i ( v ) ,
i = 1 N m = 1 α i c m i [ det ( A ) ] α λ i m = 1 ,
f I ( I ) = 2 Γ ( N α ) Γ ( α x ) α x N α + α x 2 λ N α + α x 2 × × I N α + α x 2 1 K N α α x ( 2 α x I λ ) .
f I ( I ) = 2 ( α x N α ) N α + α x 2 Γ ( N α ) Γ ( α x ) I N α + α x 2 1 K N α α x ( 2 α x N α I ) .
f I ( I ) = 2 ( α x x ) α + α x 2 Γ ( α ) Γ ( α x ) I α + α x 2 1 K α α x ( 2 α x α I ) .
y i = k = 1 i l i k g k ,
E [ y i ] = k = 1 i l i k E [ g k ] ,
var [ y i ] = k = 1 i l i k 2 var [ g k ] .
E [ g i ] = α β k = 1 i 1 l i k E [ g k ] l i i = α g , i β g , i ,
var [ g i ] = α β 2 k = 1 i 1 l i k 2 var [ g k ] l i i 2 = α g , i β g , i 2 ,
C y = [ 1 ρ ρ ρ 1 ρ ρ ρ 1 ] .
C y = [ 1 ρ ρ 2 ρ 3 ρ 4 ρ 1 ρ ρ 2 ρ 3 ρ 2 ρ 1 ρ ρ 2 ρ 3 ρ 2 ρ 1 ρ ρ 4 ρ 3 ρ 2 ρ 1 ] .
C y = [ 1 ρ 2 ρ 3 ρ 4 ρ 5 ρ 5 1 ρ 2 ρ 3 ρ 4 ρ 4 ρ 5 1 ρ 2 ρ 3 ρ 3 ρ 4 ρ 5 1 ρ 2 ρ 2 ρ 3 ρ 4 ρ 5 1 ] ,
c m i = 1 ( α i m ) ! d α i m d w α i m [ j = 1 j i N 1 ( w d j ) α j ] w = d i ,
d n ( u v ) d x n = ( u + v ) ( n )
( u 1 + + u m ) n = k 1 + + k m = n ( n k 1 , , k m ) j = 1 N u t k t ,
d n d x n ( j = 1 N u j ) = ( u 1 + + u N ) ( n ) = k 1 + + k N = n ( n k 1 , , k N ) j = 1 N u j ( k j ) .
j = 1 j i N u j ( k j ) = j = 1 j i N [ ( 1 ) k j ( α j ) k j ( w d j ) α j k j ] ,
c m i = 1 ( α i m ) ! k 1 + + k N = α i m i ^ ( α i m k 1 , , k N i ^ ) × × j = 1 j i N [ ( 1 ) k j ( α j ) k j ( d i d j ) α j k j ] ,

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