An exact gamma–beta scale-mixture law for the squared norm of a complex-Gaussian-scaled sum of hyperspherical random vectors

Ki-Hun Lee; Nam-Jin Park; Ki-Hun Lee; Nam-Jin Park

doi:10.3934/math.20261129

AIMS Mathematics

2026, Volume 11, Issue 9: 28348-28382. doi: 10.3934/math.20261129

Previous Article Next Article

Research article Special Issues

An exact gamma–beta scale-mixture law for the squared norm of a complex-Gaussian-scaled sum of hyperspherical random vectors

Ki-Hun Lee ¹,
Nam-Jin Park ^{2
,
,}

1.
Department of Electronics and Information Engineering, Korea University, Sejong, 30019, South Korea
2.
Department of Control and Instrumentation Engineering, Pukyong National University, Busan, 48513, South Korea

Received: 02 June 2026 Revised: 18 August 2026 Accepted: 24 August 2026 Published: 04 September 2026
MSC : 60E05, 60E10, 62E15, 94A05

We derive the exact probability distribution of the squared norm of a sum of complex-Gaussian-scaled hyperspherical random vectors. The derived law reveals a new gamma–beta scale-mixture representation for this random norm, expressing it as the product of a gamma-distributed squared norm of an isotropic complex Gaussian vector and a compactly supported weighted sum of independent beta random variables. As a concrete engineering application, this distribution arises as the effective channel gain of fully distributed space–time block coding (FD-STBC) systems with independent complex unit-hyperspherical signatures for arbitrary numbers of transmitters $ K $ and underlying orthogonal space–time block coding (STBC) dimensions $ L $. On the basis of the representation, we obtain a fully expanded density for the mixing variable, finite-sum closed-form expressions for the probability density function (PDF) and cumulative distribution function (CDF) of the effective channel gain, an exact Laplace transform, and closed-form moments. Furthermore, we apply the derived distribution to the FD-STBC system and obtain analytical expressions for outage probability and bit error rate (BER), together with a high signal-to-noise ratio (SNR) asymptotic expansion that yields the diversity order $ \min(K, L) $. Numerical experiments show close agreement between the analytical results and Monte Carlo simulations for both the derived distribution and the associated performance metrics.
Citation: Ki-Hun Lee, Nam-Jin Park. An exact gamma–beta scale-mixture law for the squared norm of a complex-Gaussian-scaled sum of hyperspherical random vectors[J]. AIMS Mathematics, 2026, 11(9): 28348-28382. doi: 10.3934/math.20261129

Related Papers:

Abstract

We derive the exact probability distribution of the squared norm of a sum of complex-Gaussian-scaled hyperspherical random vectors. The derived law reveals a new gamma–beta scale-mixture representation for this random norm, expressing it as the product of a gamma-distributed squared norm of an isotropic complex Gaussian vector and a compactly supported weighted sum of independent beta random variables. As a concrete engineering application, this distribution arises as the effective channel gain of fully distributed space–time block coding (FD-STBC) systems with independent complex unit-hyperspherical signatures for arbitrary numbers of transmitters $ K $ and underlying orthogonal space–time block coding (STBC) dimensions $ L $. On the basis of the representation, we obtain a fully expanded density for the mixing variable, finite-sum closed-form expressions for the probability density function (PDF) and cumulative distribution function (CDF) of the effective channel gain, an exact Laplace transform, and closed-form moments. Furthermore, we apply the derived distribution to the FD-STBC system and obtain analytical expressions for outage probability and bit error rate (BER), together with a high signal-to-noise ratio (SNR) asymptotic expansion that yields the diversity order $ \min(K, L) $. Numerical experiments show close agreement between the analytical results and Monte Carlo simulations for both the derived distribution and the associated performance metrics.

References

[1]	M. Kang, M.-S. Alouini, Largest eigenvalue of complex Wishart matrices and performance analysis of MIMO MRC systems, IEEE J. Sel. Areas Commun., 21 (2003), 418–426. https://doi.org/10.1109/JSAC.2003.809720 doi: 10.1109/JSAC.2003.809720
[2]	S. Kotz, N. Balakrishnan, N. L. Johnson, Continuous multivariate distributions: Models and applications, John Wiley & Sons, Inc., 2000. https://doi.org/10.1002/0471722065
[3]	R. J. Muirhead, Aspects of multivariate statistical theory, John Wiley & Sons, Inc., 1982. https://doi.org/10.1002/9780470316559
[4]	V. Tarokh, H. Jafarkhani, A. R. Calderbank, Space-time block codes from orthogonal designs, IEEE Trans. Inf. Theory, 45 (1999), 1456–1467. https://doi.org/10.1109/18.771146 doi: 10.1109/18.771146
[5]	S. M. Alamouti, A simple transmit diversity technique for wireless communications, IEEE J. Sel. Areas Commun., 16 (1998), 1451–1458. https://doi.org/10.1109/49.730453 doi: 10.1109/49.730453
[6]	J. N. Laneman, G. W. Wornell, Distributed space-time-coded protocols for exploiting cooperative diversity in wireless networks, IEEE Trans. Inf. Theory, 49 (2003), 2415–2425. https://doi.org/10.1109/TIT.2003.817829 doi: 10.1109/TIT.2003.817829
[7]	J. N. Laneman, D. N. C. Tse, G. W. Wornell, Cooperative diversity in wireless networks: Efficient protocols and outage behavior, IEEE Trans. Inf. Theory, 50 (2004), 3062–3080. https://doi.org/10.1109/TIT.2004.838089 doi: 10.1109/TIT.2004.838089
[8]	S. Yiu, R. Schober, L. Lampe, Distributed space-time block coding, IEEE Trans. Commun., 54 (2006), 1195–1206. https://doi.org/10.1109/TCOMM.2006.877947 doi: 10.1109/TCOMM.2006.877947
[9]	B. Sirkeci-Mergen, A. Scaglione, Randomized space-time coding for distributed cooperative communication, IEEE Trans. Signal Process., 55 (2007), 5003–5017. https://doi.org/10.1109/TSP.2007.896061 doi: 10.1109/TSP.2007.896061
[10]	K.-H. Lee, H. Lee, J. Choi, S. Park, B. C. Jung, Distributed space-time block coding for barrage relay networks, In: MILCOM 2023 - 2023 IEEE military communications conference (MILCOM), Boston: IEEE, 2023,292–297. https://doi.org/10.1109/MILCOM58377.2023.10356386
[11]	K. V. Mardia, P. E. Jupp, Directional statistics, John Wiley & Sons, Inc., 2000. https://doi.org/10.1002/9780470316979
[12]	M. D. Springer, W. E. Thompson, The distribution of products of beta, gamma and Gaussian random variables, SIAM J. Appl. Math., 18 (1970), 721–737. https://doi.org/10.1137/0118065 doi: 10.1137/0118065
[13]	R. E. Gaunt, Products of normal, beta and gamma random variables: Stein operators and distributional theory, Braz. J. Probab. Stat., 32 (2018), 437–466. https://doi.org/10.1214/16-BJPS349 doi: 10.1214/16-BJPS349
[14]	J. Pitman, M. Z. Rácz, Beta-gamma tail asymptotics, Electron. Commun. Probab., 20 (c), 1–7. https://doi.org/10.1214/ECP.v20-4545 doi: 10.1214/ECP.v20-4545
[15]	S. B. Provost, Y.-H. Cheong, On the distribution of linear combinations of the components of a Dirichlet random vector, Can. J. Stat., 28 (2000), 417–425. https://doi.org/10.2307/3315988 doi: 10.2307/3315988
[16]	T. Pham-Gia, N. Turkkan, Distribution of the linear combination of two general beta variables and applications, Commun. Stat. Theory Methods, 27 (1998), 1851–1869. https://doi.org/10.1080/03610929808832194 doi: 10.1080/03610929808832194
[17]	S. Nadarajah, X. Jiang, J. Chu, A saddlepoint approximation to the distribution of the sum of independent non-identically beta random variables, Stat. Neerl., 69 (2015), 102–114. https://doi.org/10.1111/stan.12051 doi: 10.1111/stan.12051
[18]	H. Homei, S. Nadarajah, On products and mixed sums of gamma and beta random variables motivated by availability, Methodol. Comput. Appl. Probab., 20 (2018), 799–810. https://doi.org/10.1007/s11009-017-9591-2 doi: 10.1007/s11009-017-9591-2
[19]	F. Verde, D. Darsena, A. Scaglione, Cooperative randomized MIMO-OFDM downlink for multicell networks: Design and analysis, IEEE Trans. Signal Process., 58 (2010), 384–402. https://doi.org/10.1109/TSP.2009.2030837 doi: 10.1109/TSP.2009.2030837
[20]	T. Q. Duong, Ö. Alay, E. Erkip, H.-J. Zepernick, End-to-end performance of randomized distributed space-time codes, In: 21st Annual IEEE international symposium on personal, indoor and mobile radio communications, IEEE, Istanbul, 2010,988–993. https://doi.org/10.1109/PIMRC.2010.5671575
[21]	J. Soffritti, H.-J. Zepernick, M. L. Merani, Performance analysis of randomized distributed space-time codes in Nakagami-m fading, In: 2013 International conference on advanced technologies for communications (ATC 2013), Vietnam: IEEE, 2013,408–413. https://doi.org/10.1109/ATC.2013.6698146
[22]	J. Soffritti, T. Q. Duong, M. L. Merani, H.-J. Zepernick, Performance analysis of randomized distributed space-time codes over composite Gamma/lognormal fading channels, In: 2014 IEEE 80th vehicular technology conference (VTC2014-Fall), Canada: IEEE, 2014, 1–5. https://doi.org/10.1109/VTCFall.2014.6966117
[23]	Ö. Alay, P. Liu, Y. Wang, E. Erkip, S. S. Panwar, Cooperative layered video multicast using randomized distributed space time codes, IEEE Trans. Multimedia, 13 (2011), 1127–1140. https://doi.org/10.1109/TMM.2011.2158088 doi: 10.1109/TMM.2011.2158088
[24]	F. Verde, Performance analysis of randomised space-time block codes for amplify-and-forward cooperative relaying, IET Commun., 7 (2013), 1883–1898. https://doi.org/10.1049/iet-com.2013.0170 doi: 10.1049/iet-com.2013.0170
[25]	D. Gregoratti, W. Hachem, X. Mestre, Randomized isometric linear-dispersion space-time block coding for the DF relay channel, IEEE Trans. Signal Process., 60 (2012), 426–442. https://doi.org/10.1109/TSP.2011.2171685 doi: 10.1109/TSP.2011.2171685
[26]	K. Lu, S. Fu, X.-G. Xia, Closed-form designs of complex orthogonal space-time block codes of rates $(k+1)/(2k)$ for $2k-1$ or $2k$ transmit antennas, IEEE Trans. Inf. Theory, 51 (2005), 4340–4347. https://doi.org/10.1109/TIT.2005.858943 doi: 10.1109/TIT.2005.858943
[27]	K.-H. Lee, K. Lim, B. C. Jung, STEALTH: Space-time encryption via constellation hiding for MIMO two-way untrusted relay systems, IEEE Internet Things J., 12 (2025), 30319–30334. https://doi.org/10.1109/JIOT.2025.3571736 doi: 10.1109/JIOT.2025.3571736
[28]	A. M. Mathai, S. B. Provost, Quadratic forms in random variables: Theory and applications, New York: Marcel Dekker, 1992.
[29]	A. W. van der Vaart, Asymptotic statistics, Cambridge University Press, 1998. https://doi.org/10.1017/CBO9780511802256
[30]	I. S. Gradshteyn, I. M. Ryzhik, Table of integrals, series, and products, Academic Press, 2014. https://doi.org/10.1016/C2010-0-64839-5
[31]	F. W. J. Olver, D. W. Lozier, R. F. Boisvert, C. W. Clark, NIST handbook of mathematical functions, New York: Cambridge University Press, 2010.
[32]	H. M. Srivastava, P. W. Karlsson, Multiple Gaussian hypergeometric series, 1985.
[33]	Q. Shi, Y. Karasawa, Some applications of Lauricella hypergeometric function $F_A$ in performance analysis of wireless communications, IEEE Commun. Lett., 16 (2012), 581–584. https://doi.org/10.1109/LCOMM.2012.030912.112454 doi: 10.1109/LCOMM.2012.030912.112454
[34]	K. Cho, D. Yoon, On the general BER expression of one- and two-dimensional amplitude modulations, IEEE Trans. Commun., 50 (2002), 1074–1080. https://doi.org/10.1109/TCOMM.2002.800818 doi: 10.1109/TCOMM.2002.800818
[35]	D. Tse, P. Viswanath, Fundamentals of wireless communication, Cambridge University Press, 2005. https://doi.org/10.1017/CBO9780511807213
[36]	J. W. Craig, A new, simple and exact result for calculating the probability of error for two-dimensional signal constellations, In: MILCOM 91 - Conference record, McLean: IEEE, 2 (1991), 571–575. https://doi.org/10.1109/MILCOM.1991.258319
[37]	M. K. Simon, M.-S. Alouini, Digital communication over fading channels, John Wiley & Sons, Inc., 2004. https://doi.org/10.1002/0471715220

Reader Comments

Your name:*

Email:*
© 2026 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)