Capacity of MIMO Channels with Antenna Selection

Transcription

1 1 Capaity of MIMO Channels with Antenna Seletion Shahab Sanayei and Aria Nosratinia Department of Eletrial Engineering The University of Texas at Dallas, Rihardson, TX Abstrat We explore the apaity of MIMO hannels in the presene of antenna seletion. Antenna seletion redues the omplexity of the radio devies and requires only a small amount of hannel state feedbak to the transmit side. For high SNR, we define the apaity gain as the onstant term in the expansion of the ergodi apaity in terms of SNR. We show that this value is representative of the hannel state information (CSI) at the transmitter. We investigate the asymptoti behavior of the apaity gain for three ases: omplete CSI, no CSI, and partial CSI at transmitter (antenna seletion). We show that while water-filling provides a apaity gain that inreases logarithmially in M (the number of transmit antennas), the apaity gain of transmit antenna seletion behaves only like log(log M). For the low SNR ase, we use the onept of hannel gain, a measure introdued by Verdu [1. We show that hannel gain for antenna seletion inreases only logarithmially in M as opposed to water-filling hannel gain whih inreases linearly in M. The methodology developed in this paper, although motivated by antenna seletion, is fairly general and is useful whenever partial CSI is available at the transmitter. The same tehniques are also applied to the reeive seletion, and orresponding results are noted in hight- and low-snr regimes. I. INTRODUCTION In MIMO systems, the ost and omplexity of multiple RF-hains, power amplifiers, and low noise amplifiers (LNA) is a serious pratial issue. One solution to the ost/omplexity problem is antenna subset seletion [2, [3, [4, [5. At the reeive side, antenna subset seletion redues the omplexity. At the transmit side, antenna subset seletion not only redues the omplexity, but also improves the apaity of the MIMO system [6, [7,

2 2 [8, [9 at the ost of a minimal amount of feedbak. Fast and effiient algorithms have been devised to determine the seleted antenna subsets [10, [11, [12. However, despite the pratial importane of antenna seletion, the information theoreti properties of the resulting hannels remains a mostly unexplored territory. A few notable exeptions exist [13, [14, however, to date losed form expressions for apaity have not been available. In this paper, we analyze the apaity of the antenna seletion hannel in the high-snr and low-snr regimes. In the high-snr regime, we define the notion of apaity gain as the onstant term in the high- SNR expansion of the apaity expression, and demonstrate that it is diretly related to transmit-side hannel state information. This onept was first introdued in [8, [9 and is losely related to a similar onept that was independently proposed by Lozano et al. [15. We are able to draw onlusions about the behavior of the system in the asymptote of large number of transmit antennas, and draw omparisons between the antenna seletion apaity and water-filling apaity. Our results have interesting impliations in the design and analysis of all MIMO systems (not just antenna seletion). The waterfilling apaity (C wf ) has the same growth rate as the apaity of the uninformed transmitter (C) [16, but nevertheless C wf > C with non-vanishing differene at high SNR 1. The differene is an exess rate that is due to hannel state feedbak. The exess rate is not limited to waterfilling; when partial CSI is available at the transmitter the exess rate is still there, but is smaller. Examples of partial CSI inlude transmit antenna seletion and hannel ovariane feedbak. The growth of this exess rate with the number of transmit antennas is a measure of how effetively CSI is being used by a given method. The above developments, the reader may reall, were in the high-snr regime. In the low-snr regime we employ a similar methodology for analysis. In partiular, we look at the power series expansion of the apaity around SNR=0, where the oeffiient of the first-order term is alled hannel gain 2, a quantity that is related to the hannel state information. Using this notion, we show that at low SNR, the optimal seletion strategy at the transmitter is to selet exatly one transmit antenna, regardless of other parameters. This is ew result that is reminisent of, but distint from, the well-known water-filling result at low SNR. Finally, motivated by the symmetry inherent in the problem, we analyze the reeive side antenna seletion in a manner similar to transmit seletion. Even if no multiplexing is lost due to seletion, 1 We assume there are more transmit than reeive antennas. 2 Terminology due to Verdu [1

3 3 reeive antenna seletion inurs a loss of reieve power due to de-seleted antennas, thus, unlike transmit seletion whih may inrease apaity (under onditions mentioned earlier), reeive seletion always inurs a apaity loss. In eah of the low- and high-snr regimes, we fous on three important ases: full CSI at transmitter, no CSI at transmitter, and antenna-seletion CSI at transmitter. In the latter ase, apaity analysis naturally depends on the antenna seletion algorithm. Optimal antenna seletion is not only pratially diffiult, it also indues hannel distributions that do not lead to a tratable formulation. Thankfully there exist antenna seletion algorithms that deliver apaities almost indistinguishable from optimal seletion [11, [8. In this paper we undertake the analysis of antenna seletion using these algorithms. We use the following notation. E[ refers to expeted value of a random variable, I N denotes the N N identity matrix, (x) + = max{x, 0}, and γ is the Euler-Masheroni onstant. We use = bn to denote the asymptoti equivalene of and defined as: lim n bn = 1. We use the natural logarithm throughout this paper so the apaity unit is in Nats/Se/Hz. The hi-square distribution with 2p degrees of freedom is shown by χ 2 2p, and the maximum of n independent χ2 2p distributions is denoted by χ 2 2p,n. With an abuse of notation, we show random variables following the latter distribution also with χ 2 2p,n. II. SYSTEM MODEL We assume a frequeny non-seletive (flat) linear time invariant fading hannel between M transmit and N reeive antennas. The signal model is: y(t) = Hx(t) + n(t) (1) where y(t) represents the N 1 reeived vetor sampled at time t, and x(t) represents the M 1 vetor transmitted by the antennas with power onstrain E[x H x ρ, where ρ is the average SNR (per hannel use), n(t) is the N 1 additive irularly symmetri omplex Gaussian noise vetor with zero mean and ovariane matrix equal to I N (the N N identity matrix) and H is the N M hannel matrix, whose ij-th element is the salar hannel between the i-th reeive and j-th transmit antenna. We assume that the elements of H are independent and have omplex Gaussian distribution with zero mean and unit variane. We also assume that H is perfetly known at the reeiver but it is not neessarily known at the transmitter. For antenna seletion, we assume there is a rate-limited feedbak hannel from reeiver to transmitter so that a subset of transmit antennas an be seleted by the reeiver and furthermore we assume that the feedbak hannel is without error or delay.

4 4 1) Let S 1 = {all olumns of H} and P 1 = I N. 2) hoose h 1 = arg max h S1 h 2, H1 = h 1. 3) for i = 2 : L a) S i = S i 1 { h i 1 } b) Pi = I ρ H L i 1 (I + ρ H L i 1 H H i 1 ) 1 HH i 1 ) h i = arg max h Si h H Pi h d) Hi = [ H i 1 hi 4) H = HL Fig. 1. Antenna seletion algorithm III. TRANSMIT ANTENNA SELECTION We onsider a transmit antenna seletion sheme where a subset of transmit antennas are used for transmission with equal power. Optimal transmit antenna seletion via exhaustive searh among all M L ombinations has omplexity O(M L ), whih is impratial for large number of transmit antennas. One may redue this omplexity by employing a suessive seletion sheme, i.e., a greedy algorithm that at eah step maximizes the apaity of the seleted sub-hannel. A very similar methodology was mentioned in [10 for reeive antenna seletion. In this work, starting from the original hannel matrix, the algorithm removes antennas one after another in a way that the apaity loss is minimized. Gharavi-Alkhansari and Greshman [12 showed that an inremental suessive seletion leads to less omputational omplexity. Simulation results show that this suessive seletion aptures almost all the apaity of optimal antenna seletion in a wide range of SNRs. Therefore, we adopt the latter algorithm for our analysis. The input of the algorithm onsists of ρ (the given SNR), L (the desired number of transmit antennas to be seleted) and H (the original hannel matrix). The output of the algorithm is H (the hannel matrix assoiated with seleted transmit antennas). This algorithm, shown in Figure 1, is the basis for the following analysis.

5 5 A. A Framework for High SNR Analysis Using the Sherman-Morisson formula for determinants [17, for the seleted hannel H we have: det(i N + ρ L H H L H ) = (1 + L h ρ H i P i hi ) (2) As ρ, P i P i, where, P i = I N H i 1 ( H H i 1 H i 1 ) 1 Hi 1 is a projetion matrix of rank N i+1. When M is also large, at eah seletion step, the distribution of the remaining hannel vetors an still be well approximated by a irularly symmetri Gaussian distribution. Our simulations verify that for large M the Gaussianity assumption provides a good approximation for the atual distribution of the remaining olumns. 3 Using this assumption we an approximate the statistis of the right side of (2). We know that for an unorrelated omplex Gaussian vetor x and a projetion matrix P, x H P x has χ 2 distribution with rank(p ) degrees of freedom. Hene for large ρ and large M, we have: det( H H H ) L χ 2 2(N i+1),m i+1 (3) where χ 2 2p,n stands for a random variable whih is the maximum of n independent χ2 2p random variables. The pdf of this random variable an be omputed in losed form [18: where e p (x) = p 1 k=0 xk k!. f χ 2 2p,n (x) = nxp 1 (p 1)! e x (1 e x e p (x)) n 1 (4) B. Capaity gain of MIMO systems We introdue the onept of apaity gain as a measure of effetiveness of hannel state information at the transmitter. We start with the apaity expression for a general MIMO system. Under the flat fading assumption, given a general hannel matrix, the ergodi apaity of the MIMO hannel is alulated as follows [19: C = E[ max tr(q) ρ log(det(i N + HQH H )) (5) where Q = E[xx H is the ovariane of the transmitted vetor x. We onsider three different ases: First, uninformed transmission in whih CSI is perfetly known at reeiver, but not at transmitter. Seond, informed transmission in whih CSI is perfetly known both at transmitter and reeiver. Third, transmission using an optimal subset of transmit antennas seleted by the reeiver. The only information 3 We have no formal proof for this, however, and even for very simple ases it is an open problem and there is no known analytial result.

6 G wf 12 G as Spetral effiieny Bits/se/Hz m log(ρ) 2 No Tx Ant. Sel. (optimal) Ant. Sel. (suessive) Water filling SNR (db) Fig. 2. Capaity gain of antenna seletion: M=8 N=L=2 available at the transmitter is the indies of the seleted transmit antennas. Figure 2 shows the ergodi apaity and the apaity gain for the above three ases. Uninformed transmitter: As shown in [19, when CSI is available only at the reeiver, the ovariane matrix that maximizes the apaity is of the form Q = 1 M I N, hene, the ergodi apaity is: [ C = E[log(det(I N + ρ m M HHH )) = E log(1 + ρ M λ i) where where λ 1,..., λ m are ordered nonzero eigenvalues of the Wishart matrix HH H [19, and m = rank(h) = min{m, N} is the degrees of freedom of the MIMO hannel, assuming hannel is full-rank. As shown in [20, C = m log ρ + O(1). So the ergodi apaity grows linearly with m. Now we notie that in the asymptoti expansion of C there is a onstant term that does not vanish as ρ. Thus we define the apaity gain as follows: For uninformed transmission we have: G = lim (C m log ρ) (6) ρ

7 7 ( [ m ) G = lim E log(1 + ρ ρ M λ i) m log ρ [ m = lim E log( 1 ρ ρ + λ i M ) [ m = E log λ i m log M (7) in the last step, the exhange of expetation and limit is allowed by the monotone onvergene theorem. Informed transmitter (water-filling apaity): In this ase the hannel state information is available at transmitter. As addressed in [19 the water-filling apaity of the MIMO hannel is: [ m C wf = E (log(µλ i )) + where µ should satisfy ρ = m (µ λ 1 i ) +. In large SNR senario, all the eigenmodes of the hannel are used by the beamformer, hene µ = ρ+ m λ 1 i m and the water-filling apaity is equal to: [ m C wf = E (log(µλ i )) = me [ log(ρ + [ m m λ 1 i ) + E log λ i m log m (9) For large ρ we have: C wf m log ρ. In other words, availability of CSI at transmitter side has no impat on the logarithmi growth rate of the ergodi apaity, beause the growth rate only depends on the rank of the hannel matrix. Now we similarly alulate the apaity gain for informed transmission: [ m G wf = E log λ i m log m (10) (8) G = G wf G = m log(m/m) (11) In the asymptote of large SNR, this is the maximum amount of exess rate obtainable by providing hannel state information at the transmitter. We note that if M N then G = 0, thus hannel state information at transmitter annot provide any exess rate asymptotially. This result agrees with one s intuition that beamforming is effetive only when the number of transmit antennas is large. In the sequel, we only onsider the interesting ase of M > N. In partiular, we are interested to understand the behavior of the apaity gain when M N. In these ases, Equation (11) suggests that at high SNR, the

8 8 apaity gain an be used as an information-theoreti metri to evaluate any method that uses hannel state information at the transmitter. Antenna seletion: In the high SNR regime, one is interested in the ase L N, to maintain the degrees of freedom of the hannel and prevent exessive rate loss. Suppose we have seleted L (L N) out of M transmit antennas (M N) then if the seleted hannel is H, the apaity gain is: [ ( G = lim E log det(i N + ρ ρ L H H ) H ) N log ρ [ ( = lim log det( 1 ρ I N + 1 ) L H H H ) = E ρ E [ ( log det( H H ) H ) N log L (12) IV. ASYMPTOTIC BEHAVIOR OF CAPACITY GAIN In this setion we explore the behavior of apaity gain, for large M, in the ase of informed, uninformed, and antenna seletion transmitter. Uninformed transmitter: in the ase M > N, Equation (7) an be rewritten as: G = E [ log det(hh H ) N log M (13) It is known [20 that det(hh H ) N χ2 2(M i+1) therefore [21: where ψ(n) = γ + n 1 k=1 1 k G = N (ψ(m i + 1) log M) (14) is the di-gamma funtion. We have [20: lim G = 0 (15) M Informed transmitter: Using Equations (11) and (15) for large M we have: G wf = N log( M N ) = N log M (16) Antenna seletion: Using the results of Setion III-A, we an evaluate the apaity gain for antenna seletion. Using (12): [ G = E log det( H H H ) N log L = L E[log( χ 2 2(N i+1),m i+1 ) N log L (17)

9 9 Equation (17) suggests that for large M, seleting more than N antennas does not provide any further gain. In the previous setion we argued that L annot be less than N, hene for large M the optimal value for L is N. Heneforth we assume L = N. To evaluate the asymptoti behavior of G, we only need to evaluate E[log X, where X χ 2 p,n. We use the following result from order statistis [18: Definition 1: A df F is said to belong to the domain of maximal attration of ondegenerate df U if there exist sequenes { } and { > 0} suh that at all ontinuity points of U(x). lim F n ( + x) = U(x) (18) n Theorem 1: Let X (n) be the maximum of n i.i.d. random variables {X i } n, eah with df F. If F belongs to the domain of maximal attration of U, then: X (n) d W (19) where W is a random variable whose df is U. Moreover U an only be one of the following three distributions: U 1 (x) = exp( e x ) < x <, exp( x α ) x > 0, α > 0 U 2 (x) =, 0 x 0 1 x > 0 U 3 (x) =, exp( ( x) α ) x 0, α > 0 these distributions are also known as Gumbel, Fréhet and Weilbull distributions, repetively. Theorem 1 is analogue to the entral limit theorem whih was for the normalized sum of iid random variables. Although unlink the entral limit theorem, the normaliziation ontrants and are not neessarily unique. One way to find onstant is to solve the following equation [18: = F 1 (1 1 n ) = F 1 (1 1 ne ) F 1 (1 1 n ) where F 1 is the inverse funtion. It is known that the df of a χ 2 2p random variable is equal to F (x) = 1 e x e p (x), where e p (x) = p 1 k=0 xk k!. Choosing = log n + log(log( np 1 (p 1)! )) and = 1 we have: lim F n ( + x) = exp( e x ) (20) n

10 10 We use (20) to evaluate the logarithmi moment of χ 2 p,n whih is key to our analysis. In the above formulation we hoose F to be the df of a hi-square random variable with p degrees of freedom. Jensen s inequality provides an upper bound on the logarithmi moment and furthermore the following theorem states that this bound is asymptotially tight. Theorem 2: Let {X n } n be positive random variables with finite mean and variane, X (n), W, { } and { > 0} are defined as in Theorem 1 and furthermore, an, then as n : Proof: See Appendix. log(e[x (n) ) E[log X (n) 0 So to alulate the logarithmi moment it is suffiient to only evaluate the mean of the above random variable. It is known that when the limiting distribution is of the first kind (Gumbel distribution) then Theorem 1 an also be used to evaluate the moments of X (n) (in fat for this ase, onvergene in distribution implies onvergene in moments [22). In partiular we have [ X(n) E E[W = γ (21) E [ (X(n) ) a 2 n E[W 2 = π2 6 Hene, the asymptoti growth of the logarithmi moment of the extreme order statistis of hi-square random variables is given by (22) E[log(X (n) ) = log(e[x (n) ) = log (log n + log(log( np 1 (p 1)! )) + γ = log(log n) (23) ) Now we an evaluate the behavior of G in (17): Hene G = = N E[log( χ 2 2(N i+1),m i+1 ) N log N (M i+1)n i N log(m i + 1) + log(log( log (N i)! )) + γ (24) N G = N log(log M)

11 Capaity gain (N=2, SNR=30 db) 2 Spetral effiieny (Nats/Se/Hz) Asymptoti Approximation Monte Carlo simulation Number of transmit antennas (M) Fig. 3. Capaity gain of antenna seletion (N=2 and SNR=30 db) Thus the apaity gain for transmit antenna seletion behaves like O (log(log M)). Also, for large number of transmit antennas Eq. 24 an be used as an approximate formula for ergodi apaity of transmit antenna seletion at high SNR, in fat C N log ρ + G (25) Figure 3 ompares our result with omputer simulations. We run the simulation for SNR=30 db and N=2. For eah point on the plot, the apaity gain is alulated by averaging over 5000 different hannel realizations and ompared to the results obtained from equation (24). Simulations math our analysis very well, thus the asymptoti formula proves to be a useful tool for the evaluating of the apaity of transmit antenna seletion. V. TRANSMIT ANTENNA SELECTION: LOW SNR CASE For low SNR ase, we use the onept of hannel gain, introdued by Verdu [1, whih is essentially the slope of the linear term in the Taylor expansion of the ergodi apaity, namely, Γ = C(ρ) ρ (26) ρ=0 sine C = Γ ρ+o(ρ 2 ), Γ an be onsidered as an information theoreti metri for evaluating the spetral effiieny at low SNR [1.

12 12 Uninformed transmitter: In this ase the hannel gain [1 is: [ H 2 Γ = E F = N (27) M we notie that in this ase the hannel gain does not depend on M, so inreasing the number of transmit antenna will not affet the apaity. Informed transmitter: When CSI is fully provided at the transmitter, at low SNR, the beamformer only uses the eigen mode of the hannel assoiated with the largest eigen value of H H H hene the hannel gain is [1: Γ wf = E[λ max (H H H) (28) It was first shown in [23 that λ max = ( M + N) 2 when M or N are large. Thus for the ase M N, we have Γ wf = M. Antenna seletion: Suppose we are seleting L transmit antennas with equal power splitting among them. In low SNR senario, the hannel gain is: [ Γ = H 2 F E L = E[ L h i 2 2 L where H is the seleted hannel, and h 1,..., h L are the L olumns of H. Thus antenna seletion in low SNR ase leads to seleting L antennas with highest norm. This is also onsistent with the suessive antenna seletion algorithm presented in Figure 1 for ρ 0. Moreover, the hannel gain is maximized when L = 1, beause E[ L h i 2 2 L max i { h i 2 2 }. This suggests that the optimal transmit antenna seletion strategy in low SNR ase is to selet only one transmit antenna whose hannel vetor is of the highest norm. In other words H = h j where h j = max{ h 1,..., h M }. Now in order to evaluate the hannel gain for low SNR transmit antenna seletion we need to evaluate E[ H 2 F. The random variable H 2 F is distributed aording to X 2N,M defined in Setion III-A. For M N, using Theorem 1 we have: Γ opt = log M + log(log( M N 1 (N 1)! )) + γ = log M (29) VI. RECEIVE ANTENNA SELECTION Up to this point, the emphasis of the paper has been on the transmitter side. However, it is not diffiult to see that due to the reiproity of eletromagneti propagation, the problem is highly symmetri, therefore we an use the framework developed thus far to also address reeive antenna seletion.

13 13 In partiular, onsider the following setup: A MIMO system with M transmit and N reeive antennas, suh that N M. We wish to hoose L out of N reeive antennas in a way to maximize the retained apaity. The algorithm depited in Figure 1 will perform the antenna seletion, with the notable differene that we now must selet rows and not olumns of H. As before, we all the seleted hannel H. Assuming reeive antenna seletion with L = M and no CSI at transmitter, the apaity of the system is: Similarly to the previous ase, we an write: C = log det(i L + ρ M H H H ) log det(i + ρ M H H M H ) log (1 + ρ M χ2 2M,N i+1) M log ρ M log M + = M log ρ + Ĝ M log χ 2 2M,N i+1 where χ 2 2M,N i+1 is as earlier defined, the maximum of N i + 1 hi-square random variables. Reall that in transmit seletion the SNR sales by the fator ρ/l in Equation (2), i.e., the fewer the seleted antennas, the more power an be sent through eah antenna. The result was that the apaity atually inreases through transmit antenna seletion, 4 an inrease that was haraterized by G. In reeive seletion, seleting fewer antennas will result in smaller reeive power, but antennas that are seleted enjoy better hannel distributions than the original MIMO hannel. However, the loss of power annot be made up by the improvement in the hannel distributions. Therefore the apaity of the reeive antenna seletion is less than the apaity of the full-sale system. Unlike transmit seletion, no additional apaity is obtained by seleting down to the best antennas, a result that is not surprising beause information is being lost by reeive seletion. The above results are demonstrated on a 2 8 system in Figure 4. In the low-snr regime, we one again use the onept of hannel gain, whih for the reeive antenna seletion leads to: [ Γ = H [ L 2 F E = E h i 2 2 M M 4 Assuming the multiplexing gain of the system is unaffeted by seletion, whih we ensured via our transmit-seletion assumption of N L M

14 Spetral Effiieny (bits/se/hz) SNR (db) No Seletion Rx Seletion Fig. 4. Reeive antenna seletion in a 2 8 system It is evident that, unlike the transmit-seletion ase, there is no penalty for seleting more antennas, in fat the more antennas seleted, the higher the apaity. Calulating this value using Theorem 1 we have Γ = L M log N (30) VII. CONCLUSION In this paper, we investigate the behavior of the apaity of antenna seletion in the asymptote of large number of transmit antennas. For high SNR ase, we introdue the onept of apaity gain, an information theoreti metri for shemes that use hannel state information at the transmitter. It is shown that this quantity desribes the advantage gained by having hannel state information at the transmitter. We present new results in order statistis that are useful for asymptoti analysis of antenna seletion problems. By exploring the behavior of the apaity gain, we show that the optimal number of seleted antennas for large M is exatly N. Simulations show that the analysis is aurate and an be used for approximation of the apaity of antenna seletion. For low SNR ase, we first show that the optimal seletion strategy is to selet only one transmit antenna with the highest hannel norm. Also, we evaluate the hannel gain and show its logarithmi behavior in the asymptote of large number of antennas. VIII. APPENDIX Lemma 1: If an ( ) X(n) d, then log 0.

15 15 Proof: For every ɛ > 0 and δ > 0 [ [ X (n) P r 1 > ɛ X (n) = P r > ɛ [ ( ) X(n) a 2 E n ɛ 2 ( an ) 2 < E[W 2 + δ ɛ 2 ( an ) 2 0 as n. Thus X(n) onlude that log i.p. 1, hene X(n) ( X(n) ) d 0. Lemma 2: Let µ n = E[X (n), if Proof: From Eq. (21) we have µn an b ( ) n µ n 1 0, hene log an µ n 0. d 1. Now using the ontinous mapping theorem [24 we ( ), then log µn 0. 0, also we have bn 0. By multiplying two sides we get Definition 2: The sequene of random varibales {X n } is alled uniformly integrable if [24: lim lim sup x df Xn (x) < (31) n X n > Lemma 3: If the random variables {X n } have finite mean, then the uniform integrability is equivalent to the following ondition: lim lim sup n X n > Proof: Using integration by part we have x df Xn (x) = x(1 F Xn (x)) + X n > Pr[ X n > xdx < (32) Sine E[X n <, we have lim (1 F Xn ()) = 0 and this proves the lemma. Pr[ X n > xdx (33) Theorem 3: If the sequane of random variables {X n } is uniformly integrable, then X n E[X n E[X. Proof: See [24. d X implies ( ) Proof of Theorem 2: Let Y n = log Xn d, then by Lemma 1 we have Y n 0, we show that Y n is uniformly integrable, and to do so we use the alternative form of uniform integrability provided by Lemma 2: Pr[ Y n > ydy = Pr[Y n > ydy + Pr[Y n < ydy = I 1 + I 2 (34)

16 16 We evaluate eah integral, Let η n = bn I 1 = = = = α Pr[Y n > ydy Pr[X n > e y dy [ Xn Pr > a n (e y 1) dy [ Xn Pr > a n dt t, t := e y 1, α = e 1 (35) t + 1 [ ( ) 2 Xn π2 6. Thus using therefore η n as n. From Eq. (22) we have E Chebyshev s inequality, for every δ > 0 we have: Also in a similar way for all δ > 0 we have, I 2 = = = = ( bn ( bn ( bn [ ( ) 2 I 1 1 E Xn ηn 2 α t 2 dt (t + 1) 1 π 2 /6 + δ ηn 2 α t 2 (t + 1) dt 1 η 2 n(α + 1) α π 2 /6 + δ t 2 dt = π2 /6 + δ ηnα(α 2 0 (36) + 1) Pr[Y n < ydy Pr[X n < e y dy [ an X n Pr ) 2 ) 2 ) 2 ( π2 6 + δ) > a n (1 e y ) dy E[( an X n ) 2 (1 e y ) 2 dy π δ (1 e y ) 2 dy ( ) e 1 e + log(1 e ) 0 (37) from Eq. (36) and (37) we onlude that I 1 + I 2 0 thus, Y n is uniformly integrable hene Y n onverges in mean, namely, E[log X (n) log 0. On the other hand, by Lemma 2 we have log(e[x (n) ) log 0 thus we have E[log X (n) log(e[x (n) ) 0.

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