• Proceedings of the Korean Statistical Society Conference
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• 2001.11a
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• pp.137-141
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• 2001

We can obtain SLLN's for fuzzy random variables with respect to the new metric $d_s$ on the space F(R) of fuzzy numbers in R. In this paper, we obtain a SLLN for convex tight random elements taking values in F(R).

• Proceedings of the Korean Statistical Society Conference
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• 2002.11a
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• pp.215-220
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• 2002

In this paper, we establish weak laws of large numbers for weighted sums of convex-compactly uniformly integrable fuzzy random variables taking values in the space of upper-semicontinuous fuzzy sets in R$^{p}$.

• Communications for Statistical Applications and Methods
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• v.12 no.2
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• pp.275-283
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• 2005

In this paper, we give a sufficient condition for convergence in probability of weighted sums of convex-compactly uniformly integrable fuzzy random variables. As a result, we obtain weak law of large numbers for weighted sums of convexly tight fuzzy random variables.

• Communications for Statistical Applications and Methods
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• v.16 no.3
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• pp.529-540
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• 2009

In this paper, we present some results on weak laws of large numbers for weighted sums of fuzzy random variables taking values in the space of fuzzy numbers of the real line R. We first give improvements of WLLN for weighted sums of convex-compactly uniformly integrable fuzzy random variables obtained by Joo and Hyun (2005). And then, we consider the case that the averages of expectations of fuzzy random variables converges. As results, WLLN for weighted sums of convexly tight or identically distributed case is obtained.

• KSII Transactions on Internet and Information Systems (TIIS)
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• v.15 no.6
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• pp.2204-2224
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• 2021

This paper studies secure wireless transmission from a multi-antenna transmitter to a single-antenna intended receiver overheard by multiple eavesdroppers with considering the imperfect channel state information (CSI) of wiretap channel. To enhance security of communication link, the artificial noise (AN) is generated at transmitter. We first design the robust joint optimal beamforming of secret signal and AN to minimize transmit power with constraints of security quality of service (QoS), i.e., minimum allowable signal-to-interference-and-noise ratio (SINR) at receiver and maximum tolerable SINR at eavesdroppers. The formulated design problem is shown to be nonconvex and we transfer it into linear matrix inequalities (LMIs). The semidefinite relaxation (SDR) technique is used and the approximated method is proved to solve the original problem exactly. To verify the robustness and tightness of proposed beamforming, we also provide a method to calculate the worst-case SINR at eavesdroppers for a designed transmit scheme using semidefinite programming (SDP). Additionally, the secrecy rate maximization is explored for fixed total transmit power. To tackle the nonconvexity of original formulation, we develop an iterative approach employing sequential parametric convex approximation (SPCA). The simulation results illustrate that the proposed robust transmit schemes can effectively improve the transmit performance.