Dependence Control at Large

12/29/2018 ∙ by Fengyou Sun, et al. ∙ 0

We study the dependence control theory, with a focus on the tail property and dependence transformability of wireless channel capacity, respectively, from the perspective of an information theoretic model of the wireless channel and from the perspective of a functional of controllable and uncontrollable random parameter processes. We find that the light-tailed behavior is an intrinsic property of the wireless channel capacity, which is due to the passive nature of the wireless propagation environment and the power limitation in the practical systems. We observe that the manipulation of the marginal distributions has a bias in favor of positive dependence and against negative dependence, e.g., when a parameter process bears negative dependence, the increases of the means of marginals can not leads effectively to a better system performance. On the other hand, the dependence bias indicates that the dependence is a tradable resource, i.e., when the dependence resource is utilized another resource can be saved. For example, the negative dependence can be traded for transmission power in terms of the performance measures.

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I Introduction

In mathematics, the stochastic dependence is a property of the dependent elements, specified by the probability measure, and independence is a special case with a product measure of probability. The dependence scenario, which is probably uncertain or is intractable to get an explicit mathematical expression, raises additional analytical issues that differ from the independence scenario. In real world, the dependence corresponds to the interrelationship of the system states through time and space, and different forms of dependence result in different system performances

[1]. In other words, the stochastic dependence is not only a mathematical property but also a physical resource. Considering the diverse characteristics and distinguishing effects of the stochastic dependence, it is intriguing to study how to control the dependence in a system in order to obtain an improved performance. Particularly, a theory of dependence control is built in [2], and it treats the system as a functional of controllable and uncontrollable random parameter processes and it proves that a manipulation of the dependence in a controllable random parameter process has a consequence on the overall system performance.

In this paper, we further study the dependence control theory, with respect to both the marginal distributions and the dependence structures [3], in the context of the wireless channel capacity, and we obtain two sets of results, namely tail domination and dependence bias, which are respectively about the light-tail property of the wireless channel capacity and the dependence influence on the dependence control mechanism.

I-a Tail Domination

The wireless signals are electromagnetic radiations and the signal propagation environment is a passive medium with dissipation that is the loss of field energy due to absorption, and dispersion that is the variation of the refractive index in the medium [4][5][6]. The dissipation causes the energy loss of the signals on the path from the transmitter to the receiver [5]. This effect is termed as the large-scale fading [7]. The dispersion causes the reflection, diffraction, and scattering of the transmitted signals [5], which result in the multipath interference and the Doppler shift of the received signals. This effect is termed as the small-scale fading [7]. As a characterization of the propagation channel, the channel gain is defined by the ratio of of the receiver-to-transmitter power, of which the reciprocal is defined as the channel loss. As a result of the energy conservation law, the channel gain is less than one or the channel loss is greater than one.

Consider the multiple-input-multiple-output channel model that is expressed as [8]

(1)

where , , , and is the channel gain matrix. For simplification, we omit the time index. The instantaneous capacity is defined by the mutual information, which is a function of the product of the transmission power and the channel matrix , i.e.,

(2)

where we treat the instantaneous power as a random variable. Specifically, if the tail distribution function satisfies

[9] , equivalently, , then the distribution is light-tailed; otherwise, it is heavy-tailed. The heavy-tailed distribution indicates that extreme values occur with a relatively high probability [10]

. Particularly, if the tail is super-heavy, it has no finite moments

[11], e.g., the distributions with slowly varying tails. The class of slowly varying functions includes constants, logarithms, iterated logarithms, powers of logarithms [12].

We obtain that the sufficient condition for the light-tail wireless channel capacity is the existence of the mean value of the power law of the product of the random power and the maximum eigenvalue of the channel matrix, i.e.,

(3)

where the right hand side is equivalent to , in terms of the tail behavior, they are equivalently expressed as , and . Specifically, corresponds to the deterministic power scenario. In addition, for the broadband channel scenario, the channel matrix is the diagonal matrix of each sub-channel matrices, i.e., .

We provide the following observations, which largely explain the light-tail property of the wireless channel capacity.

  • [wide, labelwidth=!, labelindent=0pt]

  • It is interesting to note that the typical large-scale fading distribution is heavy-tailed, e.g., the Lognormal distribution, while the typical small-scale fading distribution is light-tailed, e.g., the Rayleigh, Rice, and Nakagami distributions. Specifically, if a random variable is lognormal, then its reciprocal is also lognormal. The tail property indicates that the large-scale fading effects, like path loss and shadowing, are more likely to cause large values of both channel loss and gain, which may be due to the large shadow dynamics in the propagation environment; while the small-scale fading effects, like the multipath interference and Doopler shift, are less likely to cause large values of channel gain or the random values are more likely to be concentrated around the mean. Since both light-tailed and heavy-tailed distributions with finite mean are used to model the channel gain, the parametric distributions that can model both heavy-tailed and light-tailed distributions are of interest, e.g., the Weibull distribution [13][14]. These theoretical insights on the stochastic models match the empirical results [6]. The restriction that the passive channel gain is less than one exclude the existence of fading models with super-heavy tails. In addition, since the random values of the stochastic models, whether the light-tailed distribution or the heavy-tailed distribution, are unbounded, the stochastic models of the wireless channels are strictly not passive systems [4][15], because of the violation of the energy conservation law.

  • Though the wireless system can be energy unlimited [16], the transmission power is unlikely to have an infinite mean, thus, the tail of the power distribution is lighter than the super-heavy distribution. When there are active relays in the wireless channels, the whole channel gain is the product of each individual channel gain. However, the tail of the product distribution can be asymptotically bounded above and below by the tail of a dominating random variable of the product for both independence and dependence scenarios [17][18][19][20]. In addition, the gain saturation also exclude the possibility of unlimited gain in active medium [21]. Thus, the whole channel gain is more likely to have a tail behavior lighter than the super-heavy tail. On the other hand, when the power in the capacity formula is set to be deterministic, e.g., the mean value of power, normalization is usually considered for the channel matrix. Specifically, if the channel description is based on the average transmitter power [22], then, the channel matrix is non-normalized; and if the description uses the average receiver power , then the channel matrix is normalized [23][24]. Mathematically, it is expressed as [23] . For example, the normalized channel gain of the Rayleigh fading channel is [23][8] and . The normalization indicates that the mean values of the matrix identities exist, which excludes the existence of the fading models with super-heavy tails.

In all, for the typical stochastic channel models and the power supply systems in practice, the distribution of the capacity, which is a logarithm transform of the product of the fading effects and random power, is light-tailed, because the logarithm function transforms a less than super-heavy distribution to a light-tailed distribution.

I-B Dependence Bias

We treat the wireless channel capacity as a functional of random parameters [1][2], which are either uncontrollable or controllable, the uncontrollable parameters represent the property of the environment that can not be interfered, e.g., fading, and the controllable parameters represent the configurable property of the wireless system, e.g., power. We specify that the cardinality of the parameter set is time-invariant and the function is time-variant, i.e.,

(4)

We study how to transform the dependence in the functional process , by manipulating the dependence in parameter processes , . There are two ways to implement this dependence transform, i.e., one by transforming the dependence structure from the positive dependence to the negative dependence, and the other by transforming the marginal distributions.

This functional specification is extensible to the general stochastic process on the Polish space, i.e., the stochastic process as a function of a set of random parameters, each of which is itself a stochastic process, in other words, we treat the stochastic process as a functional of a multivariate stochastic process and the functional maps the multivariate stochastic process to a univariate stochastic process. For example, this functional perspective is useful for studying the dependence impact of an individual arrival process on the aggregation of a set of multiplexed arrival processes.

We highlight the following results, which provide guidelines for dependence control.

  • [wide, labelwidth=!, labelindent=0pt]

  • The dependence is a resource that can be traded off, i.e., when the dependence is utilized, another form of resource can be saved, e.g., more amounts of negative dependence can exchange for less amounts of transmission power. The chain relation, , means the supermodular order of the dependence structures implies the convex order of the variability of the partial sum with equal mean. To take into account both the mean and the variability, we use the increasing convex order for further elaboration. Specifically, the mean and the variability are exchangeable for each other, i.e., if the variability is relatively small, then a relatively greater mean can be tolerated while satisfying the increasing convex order, vice versa. The mathematical expressions are as follows, if and , then it is possible that , because we have [25]; and if , then such that , because we have [25]. Complementary results hold in the sense of the increasing concave order [25].

  • When the backlog and the delay are used as the performance measures [2], the arrival process and the service process are consistent in the manipulation of the dependence strength and are different in the manipulation of the marginals. Specifically, for the manipulation of the dependence, the objective is the convex ordering , where represents the instantaneous arrival amount minus the instantaneous service amount, while for the manipulation of the marginals, the objective for the arrival process is the increasing convex ordering and the objective for the service process is the increasing concave ordering . This is coherent with the intuition that a smaller and less variable arrival process or a greater and less variable service process leads to a better system performance in terms of the backlog and delay [2].

  • The manipulation of the marginal distributions has a dependence bias, while the manipulation of the dependence structure fixing the marginals has no such dependence bias. Specifically, the dependence bias means that, if a parameter process bears negative dependence, then the manipulation of each individual marginals with respect to the (increasing) convex order can not lead effectively to the (increasing) convex order of the partial sums, i.e., the (increasing) convex order of the marginals implies the (increasing) convex order of the partial sum holds for positive dependence and not for negative dependence [26]. The dependence bias of the marginals provides an opportunity for dependence control. Specifically, the dependence bias means that the increasing convex order of the partial sum is insensitive to the marginal manipulation of the parameter process with negative dependence, e.g., the increasing convex order still holds for a partial sum with smaller mean values of the marginals. In other words, a better system performance, in terms of backlog and delay, can be achieved in the scenario of negative dependence in the processes, even with a smaller mean value of the service process or a greater mean value of the arrival process.

I-C Related Work

There are some related work in the literature and the comparisons with this paper are as follows. (i) The light-tailed property of wireless channel capacity for the single-input-single-output channel is investigated in [1]. In this paper, we reason why the light-tailed behavior is an intrinsic property of the wireless channel capacity, and we extend the results to multiple-input-multiple-output channel, with an extensive study on the equivalent conditions and sufficient conditions regarding the statistical identities of the wireless channel. The statistical property of wireless channel capacity is studied in [27], where the focus is on the distribution functions and first and second order statistics of the capacity, the detailed fading distributions are used, e.g., Rayleigh, Rice, and Nakagami. In this paper, instead of the exact fading distributions, we use the light-tailed and heavy-tailed distribution classes and show that the distribution of the capacity based on these typical fading distributions is intrinsically light-tailed. Thus, the results in this paper is more general than [27] and indicate more possibilities of wireless channel modeling, i.e., more distributions as alternative to the typical fading models. (ii) The tail asymptotic is investigated in [28] for the product and sum of random variables in terms of the asymptotic equality . In this paper, we extend the analysis to specific heavy-tailed and light-tailed distribution classes, e.g., the long-tail distribution, the regular varying distribution, and the light-tailed distribution, specifically, we find that the slowly varying distribution can dominate the tail behavior for the sum and product distribution, moreover, we extend the analysis beyond the asymptotic equality to more asymptotic notations, e.g., , , , and . Since the capacity is a logarithm transform of the product of the power and the fading random variable, the less strict asymptotic bound provides more flexibility than the asymptotic equality, i.e., it has less restriction and can capture more distribution scenarios, most importantly, it is sufficiently enough to investigate the light-tail behavior that is defined by the asymptotic bound . Another related work is [29], which provides conditions for the product of a light-tailed random variable and a heavy-tailed random variable to be heavy-tailed. In contrast to the result with asymptotic precision up to some distribution classes in [29], we show results of the exact tail domination with respect to a certain distribution function in this paper. (iii) The dependence control theory is studied in [2], where the stochastic dependence is not only treated as a mathematical property but also as a physical resource [1], the benefits of utilizing the stochastic dependence resource are elaborated, and the dependence transformability is proved with respect to the supermodular order. In this paper, we extend the results to the (increasing) directionally convex order and the usual stochastic order, particularly, the (increasing) directionally convex order takes into account the impacts of both the marginals and dependence structures in comparison, while the supermodular order requires identical marginals of the compared stochastic processes. Such extension provides additional insights into the dependence control theory as regards the dependence manipulation and marginal manipulation.

The rest of this paper is structured as follows. The tail property of the MIMO (multiple-input-multiple-output) channel is studied in Sec. II. The dependence transform of stochastic processes is studied in Sec. III. Finally, this paper is concluded and future work are discussed in Sec. IV.

Ii Tail Property

Let be a probability space and be measurable with respect to and the Borel -algebra on . Denote , where represents the conjugate transpose. Denote the cone [30] , which introduces a partial order in , i.e., is equivalent to that all the eigenvalues of are nonnegative. Similarly, .

Ii-a Deterministic Power Fluctuation

Consider the flat fading MIMO channel , . The capacity, in bits per second, under total average transmit power constraint, is expressed as [22]

(5)

where is the bandwidth, , is the total average transmit power, is the noise power spectral density, is the covariance matrix for the transmitted signal .

The frequency-selective fading channel formulation requires a block diagonal extension of the flat fading channel model. The capacity, in bits per second, under total average transmit power constraint, is expressed as [22]

(6)

where is the bandwidth, , is the total average transmit power, is the noise power spectral density, is the number of sub-channels, is the block diagonal matrix with as the block diagonal elements, and is the covariance matrix for the transmitted signal .

Remark 1.

The identity matrix

in the capacity formula implies that the capacity is non-negative, i.e., .

Remark 2.

The typical stochastic models of the channel gain are the Rayleigh, Rice, and Nakagami distributions [31]. The shadowing model is the Lognormal distribution [7][31], which is able to superimpose the path loss.

Remark 3.

If , then , where , thus, if is lognormal, then is also lognormal in general. This result explains the product form of the combined effect of the multiple path interference, shadowing, and path loss [32].

Remark 4.

Since the normal distribution with zero mean is symmetric, we have the equal lognormal distributions

, because of . This result implies that the quotient of an arbitrary random variable with a lognormal random variable , where and are independent, equals in distribution the product of the two random variable, i.e., . This relation does not hold for the general normal distribution.

Remark 5.

For a almost surely positive random variable, , the right tail behavior of , , and , corresponds to left tail behavior of [33][34], , and , i.e., , , and , . Letting , we obtain the complementary results. Considering the reciprocal relation between the channel loss and channel gain , both the right tail and the left tail matter for the stochastic channel models.

We present some equivalence results of the function of random variables. The proof is shown in Appendix A.

Lemma 1.

Consider a flat MIMO channel . The capacity is upper bounded by , where and is the maximum eigenvalue of .

  1. For the tail property, we have the equivalent results

    (7)
  2. For the power law function of the maximum eigenvalue, we have the equivalent expressions

    (8)

    Specifically, corresponds to . In addition, we have , .

  3. In addition, we have another pair of equivalent expressions for the exponential function of the eigenvalue, i.e.,

    (9)
Remark 6.

The parameters in the two equations, and , are not necessarily equal.

Remark 7.

The above equivalent results indicate that, for , . However, it is interesting to notice that, for a common and , we only have , and the reverse does not hold in general. Because it is shown in [28] that , where and is a nonnegative random variable, if and only if and .

Remark 8.

The mean identity , for and , is a special case of Mellin-Stieltjes transform [35] of the distribution function .

Remark 9.

Suppose is a regularly varying non-negative random variable with index . Then [36], , for ; and , for .

Ii-A1 Arbitrary Channel Side Information

We present the sufficient condition for the light-tailed capacity of the flat channel. We present the proof in Appendix B.

Theorem 1.

Consider a flat MIMO channel . If the mean identity exists, i.e.,

(10)

where is the maximum eigenvalue of , the distribution of the capacity of the MIMO channel (with or without full channel side information) is light-tailed.

We present the sufficient condition for the light-tailed capacity of the frequency-selective channel. The proof is shown in Appendix C.

Theorem 2.

Consider a frequency-selective MIMO channel with sub-channel , , and block diagonal matrix . For the scenarios where channel side information known or unknown at the transmitter, if the mean identity exists for each sub-channel, i.e.,

(11)

where is the maximum eigenvalue of , or the equivalent condition is satisfied, i.e.,

(12)

where is the maximum eigenvalue of , the distribution of the capacity is light-tailed.

Remark 10.

The equivalent expression of the sufficient condition means that it is equivalent to consider the block diagonal matrix of the frequency-selective channel as a whole or to consider the matrix of each sub-channel individually.

Remark 11.

If the trace identity exists, i.e., , then the maximum eigenvalue distribution and the capacity distribution are light-tailed. Because the maximum eigenvalue distribution is exponentially bounded [37]

(13)

where . Since the matrix is block diagonal [38], , and

(14)

Thus, entails , , , which is non-negative.

Ii-A2 Without Channel Side Information at Transmitter

We present the sufficient and necessary condition for the flat channel capacity distribution to be light-tailed, when the channel side information is not known at the transmitter. We present the proof in Appendix D.

Theorem 3.

Consider that the channel side information is only known at the receiver. The capacity distribution of the flat channel is light tailed, if and only if the tail of the determinant term is expressed as

(15)

where , , and , . Equivalently, the condition is expressed as

(16)

where and is the rank of .

Remark 12.

The conditions are equivalently expressed as . The proof is similar to that of Lemma 1.

Remark 13.

Considering the inequality , where , the sufficient and necessary condition relaxes to a sufficient condition, i.e.,

(17)

which is equivalent to .

Remark 14.

Considering the Fredholm determinant [39], for small enough, , the condition is alternatively expressed as

(18)

where or . Specifically, for , we have .

For arbitrary , according to the Plemelj-Smithies formulas [39], we have , thus the condition is expressed as

(19)

where

(20)

We present a sufficient condition for the light-tailed property of the frequency-selective channel capacity.

Theorem 4.

Consider that the channel side information is only known at the receiver. The capacity distribution of the frequency-selective channel is light tailed, if

(21)

where is the channel model of each sub-channel and is the corresponding eigenvalue of .

Proof.

The proof of the frequency-selective channel scenario follows that the light-tailed distribution of the capacity of each sub-channel implies the light-tailed distribution of the overall channel capacity. ∎

Remark 15.

The sufficient condition relaxes to , , where . Equivalently, it is expressed as , where .

Ii-B Random Power Fluctuation

We consider the channel scenario, where the channel knowledge is known at the receiver and is unknown at the transmitter, and the transmission power randomly fluctuates over the coherence periods and remains constant in each coherence period.

For the flat fading MIMO channel , when the transmit power is allocated evenly across the transmit antennas during each coherence period, the capacity, in bits per second, is expressed as

(22)

where is the bandwidth, is the noise power spectral density, and is the transmit power that is constant during each coherence period and randomly fluctuates over periods. Equivalently, the capacity is expressed as

(23)
(24)

where is the eigenvalue of the matrix and is the rank of .

Ii-B1 Sufficient Conditions for Light Tails

We present some preliminary results considering the random power.

Lemma 2.

Consider a flat MIMO channel . The capacity is upper bounded by , where , is the random power, and is the maximum eigenvalue of .

  1. For the tail property, we have the equivalent results

    (25)
  2. Alternatively, the tail property is expressed as , , and we have the equivalent expressions

    (26)

    In addition, if and are independent, then , and the condition relaxes to and .

Proof.

The proof is analog to the proof of the deterministic power scenario. ∎

We present the sufficient and necessary condition for the light-tailed property of the capacity.

Theorem 5.

The capacity distribution of the flat channel is light tailed, if and only if the tail of the determinant term is no heavier than the fat tail, i.e.,

(27)

where , , and , . Equivalently, the condition is expressed as

(28)

where and is the rank of .

Proof.

The proof is analog to the proof of the deterministic power scenario. ∎

Remark 16.

Considering the inequality , where , the sufficient and necessary condition relaxes to a sufficient condition, i.e.,

(29)

We present a set of sufficient conditions for the light-tailed capacity and their relationships. The proof is in Appendix E.

Theorem 6.

Consider a flat MIMO channel with random power fluctuation . We have the sufficient condition chain for the light-tailed property of the capacity.

(30)

(31)

(32)

(33)

(34)

(35)

(36)

(37)

Note we have the equivalent conditions

and

. Particularly, letting , we obtain the corresponding sufficient conditions for the deterministic power fluctuation scenario of arbitrary channel side information.

Remark 17.

Particularly, we have where . For example, and .

Theorem 7.

Consider the frequency-selective MIMO channel with random power fluctuation. If each sub-channel satisfies any one of the sufficient conditions in Theorem 6, then the distribution of the overall channel capacity is light-tailed.

Proof.

The proof follows that the light-tail property is preserved for the sum of random variables. ∎

Remark 18.

The channel model is the product formulation of the large-scale fading and small-scale fading effects.

Remark 19.

The tail property of the capacity is determined by the product of the random power and the random eigenvalues of the channel matrix. Thus, it is necessary to investigate the tail property of the product of two random variables. It is reasonable to assume independence between these two random variables, because the channel side information is not necessarily known at the transmitter. On the other hand, it is interesting to take into account the dependence for refinement.

Ii-B2 Tail Distribution of Random Variable Arithmetic

We study the tail property of the product distribution and sum distribution of random variables, or the impact of the tail property of one random variable on the overall product or sum distribution.

We consider the nonnegative functions, and , and define the asymptotic notations, , , , , , and .

We define a class of functions , , , such that and , i.e.,

(38)

For example, , , or . This class of functions are useful in decomposing the distribution function of the product or sum of random variables.

We study the asymptotic behavior of the composition of the function and some classes of tail distributions, e.g., the light-tail distribution, the regularly varying distribution , and the long-tail distribution (containing the subexponential distribution as a subset). We present the proof of the following results in Appendix F.

Lemma 3.

Consider the independent random variables , .

  1. If , i.e., , , then, and , . We specify , .

    1. If , , and , , then, .

    2. If , , and , , and , then, .

  2. If , i.e., , where