INTERNATIONAL JOURNAL OF COMPUTERS COMMUNICATIONS & CONTROL
ISSN 1841-9836, 11(6):845-859, December 2016.

Stochastic Stability Analysis of Power Control
in Wireless Networks via a Norm-inequality-based Approach

R. Qian, Y. Qi

Rongrong Qian, Yuan Qi*
Beijing Univ. of Posts and Telecommunications (BUPT)
Beijing, 100876, China
rongrongqian@bupt.edu.cn
*Corresponding author: qiyuan@bupt.edu.cn

Abstract: Owing to the requirements from realistic wireless networks, the stochas-
tic stability analysis for discrete-time power control, which concerns the randomness
brought by the fading channels and noise of wireless systems, is of practical signifi-
cance. By developing a norm-inequality-based framework of analyzing the stochastic
stability of linear systems with random parameters, we show that a typical power-
control law with linear system model is stable in the sense of the pth-moment sta-
bility. Several conditions of achieving the pth-moment stability for the considered
power-control law are obtained, which can easily applied to realistic wireless net-
works. Besides, within this study, the stability analysis of power control for the first
time takes into account the effect of multiple-access methods.
Keywords: wireless networks, power control, stochastic stability.

1 Introduction

There has been a great deal of research over the past several decades on the power control of
wireless networks. These studies span multiple disciplines and include information theory [1,8],
communication [2, 6, 7, 9, 13], and control theory [3–5, 21]. It has already been recognized that
the stability analysis of power-control laws can efficiently investigate the intrinsic properties of
various power-control algorithms [3–5,21].

The power-control laws with two-sided scalable iterative (interference) functions are conver-
gent under the condition that an equilibrium exists [1], and the laws with contractive interference
functions guarantee the existence and uniqueness of equilibriums along with linear convergence
of iterates [2]. The global asymptotic stability of power control laws involving two-sided scalable
interference functions and the exponentially scalability of laws with contractive interference func-
tions are seen even under bounded time-varying delays [3]. A general class of power-control laws
whose interference functions are monotonic and scalable are considered in [4, 5]. By employing
appropriately constructed Lyapunov functions, [5] shows that any bounded power distribution ob-
tained from these laws is uniformly asymptotically stable. Further, in [5] Lyapunov-Razumikhin
functions are used to show that, even when the system incorporates time-varying delays, any
solution along which the generalized system nonlinearity is bounded must also be uniformly
asymptotically stable. In both of above cases the stability is shown to be global. Most of cur-
rent wireless networks are digital communication systems, in which the link gains are random
(stochastic) variables fluctuating as wireless channels of the underlying networks are experienc-
ing fading at all time and the noises are also random variables. It is therefore important for the
power-control laws to be designed and verified when considering the impact of randomness in
discrete-time wireless systems with fading channels and random noise.

Stochastic power-control algorithm that uses noisy interference estimates (observations) is
first studied in [6]. With conventional matched filter receivers, the stochastic power control is
shown by [6] to converge to the optimal power vector in the mean square error sense. These

Copyright © 2006-2016 by CCC Publications



846 R. Qian, Y. Qi

results are later extended to the cases when a linear receiver or a decision feedback receiver is
used [7, 22]. In [9], a stochastic-approximation based power-control algorithm is proposed to
handle both measurement errors and randomness in the channel gain matrix, which is proved to
converge to the optimal solution in the mean-squared sense.

Treading the elegant footsteps of recent works [3]- [5], one can gain a deep insight into the
stability theory of typical power-control laws in wireless networks. Rather than being concerned
with the stability analysis of power control, the studies [6]- [9] focus on developing extra tech-
niques of reducing the impact of randomness encountered by power control in wireless networks,
where these techniques includes matched filter receiver, decision feedback receiver, and stochastic
approximation. To the best of our knowledge, it still lacks of systemic study on stability analysis
of typical power-control laws taking into account the randomness existed in practical wireless
networks while the power control does not use extra randomness reducing techniques proposed
by [6]- [9]. We shall emphasize here that, this kind of study is indispensable, because, on the
one hand, it could attentively reveal the inherent attributes of the typical power-control laws
when the randomness exists (but no randomness reducing technique is involved) so as to make
the stability theory of typical power-control laws complete, on the other hand, extra technique of
reducing randomness may not necessarily be available in practical systems due to the objective
factors such as realtime processing demands so that engineers have to be aware of the stabil-
ity of power control in randomness environments without any randomness reducing technique.
Hence, the aim of our work is to perform this study; specifically, we will develop a framework of
stochastic stability analysis for discrete-time power control, which takes the randomness brought
by the fading channels and noise of wireless systems into consideration.

Our main works are: (i) developing a norm-inequality-based framework of analyzing the
stochastic stability (to be specific, the pth-moment stability) for linear systems with random
parameters so as to investigate the stochastic stability of the power control in consideration
of the randomness caused by the fading channels and noise; (ii) clarifying the conditions of
achieving the stochastic stability for the considered linear systems and power-control law; and
(iii) investigating the effect of multiple-access methods to stochastic stability of power control.

2 Notation and preliminaries

2.1 Notation

Throughout, the interval [0 + ∞) is denoted by R+, and the set of positive integers by Z+.
The non-negative orthant of the N-dimensional real space is represented by RN+ . The vectors are
written in bold lower case letters and matrices in bold capital letters, e.g., a and A. The ith
component of a vector a is denoted by ai, and the ijth entry of a matrix A is denoted by Aij
such that Aij = [A]ij and A = [Aij]. The notation A ≥ 0 means that all of the components of A
are greater than or equal to zero. The inequality A ≥ B implies that aij ≥ bij for all components
ij. We let (·)T denote the transpose of a vector or a matrix. If a is a vector with components
a1,a2, · · · ,aN, then its p-norm is defined by ‖a‖p =

(∑N
i=1 |ai|p

)1/p
and its Euclidean norm by

‖a‖2 =
(∑N

i=1 |ai|2
)1/2

that is actually p-norm with p = 2. For a square matrix A, the induced
norm corresponding to the p-norm of vectors is defined as

‖A‖p = max
‖x‖p 6=0

‖Ax‖p
‖x‖p

= max
‖x‖p=1

‖Ax‖p
‖x‖p

, (1)

where ‖A‖1 = maxj
∑N

i=1 |Aij| is also known as the maximum column sum matrix norm, and
‖A‖∞ = maxi

∑N
j=1 |Aij| is the maximum row sum matrix norm.



Stochastic Stability Analysis of Power Control
in Wireless Networks via a Norm-inequality-based Approach 847

A probability space is a triple (Ω,F,P) where Ω is a set of "outcomes", F is a set of "events",
and P : F → [0 1] is a function that assigns probabilities to events. If x is a random variable on
(Ω,F,P) then we define the expected value operator to be Ex =

∫
xdP . If Ex2 < +∞ then the

variance of x is defined to be var(x) = E(x−Ex)2. We let {x[k],k ∈ Z+} denote a stochastic
process with random values in a set of vectors, and {X[k],k ∈ Z+} denote a stochastic process
with random values in a set of matrices, by writing x[k] and X[k] in italic and bold letters. If
x is a N ×1 random vector then we define its expected value as x̄ = Ex , [Ex1 Ex2 · · ·ExN ]T .
Analogously, for a N ×N random matrix X, we define its expected value as X̄ = EX , [EXij]
and its variance as var(X) , [var(Xij)].

Let Lp(Ω,F,P) be the set of measurable function f on Ω such that
∫

Ω
|f|pdµ < +∞, we

introduce an operator ELp as

ELpf , (E|f|p)1/p . (2)

From [15, 2.2.5 Example], one shall find that Lp(Ω,F,P) is a linear space and ELp is a semi-norm.

2.2 Preliminaries

In this part, we collect basic properties and definitions of matrix theory, algebra theory,
probability theory, and stochastic stability theory, which will be used in the following analysis.
For more details, see, e.g., [10,11,23–25].

Basic norm inequalities [23,25]: The p-norm of vectors and the corresponding induced norm
of square matrices are nonnegative numbers have the properties that

1. ‖x + y‖p ≤‖x‖p + ‖y‖p and ‖A + B‖p ≤‖A‖p + ‖B‖p;

2. ‖Ax‖p ≤‖A‖p‖x‖p, which is derived from the definition of ‖A‖p;

3. ‖AB‖p ≤‖A‖p‖B‖p, since ‖ABx‖p ≤‖A‖p‖Bx‖p ≤‖A‖p‖B‖p‖x‖p and

max
‖x‖p=1

‖ABx‖p = ‖AB‖p.

Upper bound of induced matrix norm [17]: For any N × N matrix A, the induced norm
‖A‖p has no explicit representation unless p = 1, 2 or ∞. However, one can have the below
inequalities [17, (1.8),(1.11)]

‖A‖p ≤ N1−1/p‖A‖1, (3)

and

‖A‖p ≤‖A‖1/p1 ‖A‖
1−1/p
∞ , (4)

provide two closed-form upper bounds of ‖A‖p with p other than 1, 2 and ∞.
Cauchy-Schwarz-Buniakowsky inequality involving real numbers [25]: Let a1,a2, · · · ,aN and

b1,b2, · · · ,bN be any two arbitrary sets of real numbers, then(
N∑
i=1

aibi

)2
≤
(

N∑
i=1

a2i

)(
N∑
i=1

b2i

)
. (5)

This inequality can be expressed in the vector form as aT b ≤‖a‖2‖b‖2, where a = [a1 a2 · · · aN ]T
and b = [b1 b2 · · · bN ]T .



848 R. Qian, Y. Qi

Definition 1 [24]: Let (S,S) be a measurable space. A stochastic process {Φ[k],k ∈ Z+}
taking values in S is said to be a Markov chain with respect to a filtration Fk, if Φ[k] ∈Fk and
for all B ∈ S, P(Φ[k + 1] ∈ B|Fk) = P(Φ[k + 1] ∈ B|Φ[k]). In words, given the present, the
rest of the past is irrelevant for predicting the value of Φ[k + 1].

Stability properties of stochastic systems need to be established in the context of stochastic
stability, in which a variety of inter-related definitions exist [10, 11]. This study concerns the
so-called pth-moment stability which is borrowed from [11] with trivial differences and defined
as follows.

Definition 2 [11]: The pth-moment stability can be stated as, for each initial distribution,
there exists limk→+∞E (‖x[k]‖ pp ) < +∞, where p ∈ Z+, and it shall hold that (E (‖x[k]‖ pp ))1/p =
ELp‖x[k]‖p.

This study will seek for an analytical framework of deriving the upper bound of E (‖x[k]‖ pp )
so as to prove the pth-moment stability for linear systems with random parameters.

3 System model and problem statement

3.1 System model of power control

We consider a wireless network with N wireless nodes, which employs a discrete-time power
control algorithm given by x[k + 1] = I(x[k]), where x[k] = [x1[k] x2[k] · · · xN [k]]T and xj[k] ∈
R+ is the transmitted power of node j at the kth iteration, I(x) = [I1(x) I2(x) · · · IN (x)]T
and Ij : RN+ → R+ is the interference function modeling the interference together with noise
measured at the receivers for node j that mainly comes from other nodes and local noise source.
Denote the link gain between the transmitter of node j and the receiver for node i by Gij.

To perform the study in a systematic fashion, we proceed from a simple but considerably
typical law of power control that has a linear system model, helping us avoid any entanglement
due to nonlinear effects. This power-control law is given by

x[k + 1] = I(x[k]), (6)

where I(x[k]) = D[k] (C[k]x[k] + n[k]) = D[k]C[k]x[k] + D[k]n[k], D[k] is a N ×N diagonal
matrix whose diagonal elements are

{
γ1[k]
G11[k]

,
γ2[k]
G22[k]

, · · · , γN [k]
GNN [k]

}
, in which γj[k] is the target

Signal-to-Interference-and-Noise Ratio (SINR) of node j at the kth iteration, and Gij[k] is the
link gain Gij at the kth iteration. In this study, we set γj[k] = γj where γj is fixed target SINR
value for node j. C[k] = [Cij[k]] is a N × N matrix whose entries are either zero or positive
depending on whether the entry is diagonal or off-diagonal, i.e.,

C[k] =




0 G12[k] · · · G1N [k]
G21[k] 0 · · · G2N [k]

...
...

...
...

GN1[k] GN2[k] · · · 0


 . (7)

n[k] = [n1[k] n1[k] · · · nN [k]]T denotes the vector of noise power at the receivers for all N
nodes. Here, note that x[k],n[k] ≥ 0 and D[k],C[k] ≥ 0 because the powers and link gains are
all positive values.

The model (6) is thought to be typical because it covers the well-known Foschini-Miljanic
algorithm [12] and can be extended (in future) to describe the power-control algorithms of op-
portunistic communications e.g., the utility-based power control (UBPC) algorithm [13].



Stochastic Stability Analysis of Power Control
in Wireless Networks via a Norm-inequality-based Approach 849

In wireless channels, fading is deviation of the attenuation affecting a signal over certain
propagation media. The fading can vary with time or geographical position, and is often modeled
as a stochastic process. If let G[k] = [Gij[k]], now one should bear in mind that, {G[k],k ∈ Z+}
and {n[k],k ∈ Z+} are two stochastic processes when the fading channels and random noise
appear in the wireless networks. As a consequences, {D[k],k ∈ Z+} and {C[k],k ∈ Z+} will
also be stochastic processes. To define the randomness behavior of the wireless networks with
fading channels and noises, the below assumptions are always employed.

Assumption 1 (Additive White Gaussian Noise). The noises existed in the wireless networks
are i.i.d. additive white Gaussian noises with average power δ2n > 0 such that the noise power
vector n[k] satisfies E(‖n[k]‖ 11 ) = Nδ2n. The noises are independent with the link gains. In
words, if nq[k2] has distribution µnq,k2 and Gij[k1] has µ

G
i,j,k1

, then (nq[k2],Gij[k1]) has distribution
µnq,k2 ×µ

G
i,j,k1

[24], for 1 ≤ i,j,q ≤ N and k1,k2 ∈ Z+.
Assumption 2 (Temporal Independency of Link Gains). The link gains at different iterations

are independent. In essence this implies, if Gij[k1] has distribution µGi,j,k1 and Gpq[k2] has µ
G
p,q,k2

,
then (Gij[k1],Gpq[k2]) has distribution µGi,j,k1×µ

G
p,q,k2

[24], for 1 ≤ i,j,p,q ≤ N whenever k1 6= k2.
Assumption 3 (Stationarity). The distributions of Gij[k] and nq[k] are unrelated to k, i.e.,

whatever k is, Gij[k] and nq[k] has distribution µGi,j and µ
n
q , respectively, for 1 ≤ i,j,q ≤ N.

This assumption stipulates that {G[k],k ∈ Z+}, {D[k],k ∈ Z+}, {C[k],k ∈ Z+}, and
{n[k],k ∈ Z+} are stationary stochastic processes which do not change their statistical properties
with k.

Assumption 4 (Deployment of Multiple Access Methods). In wireless networks, multiple access
methods can suppress the leakage of signal power from one node to the receivers for other nodes
under certain level such that Gij[k] ≤ βijGjj[k] with constant values βij � 1 for any i 6= j.
Moreover, Gjj[k], 1 ≤ j ≤ N are N stationary and i.i.d. random variables with E(Gjj[k]) = µG,
E(1/Gjj[k]) = µ1/G, E(G2jj[k]) = µG2, and E(1/G

2
jj[k]) = µ1/G2.

This assumption holds in case of the power control of wireless networks with multiple access
methods, which implies

C[k] ≤




0 β12G22[k] · · · β1NGNN [k]
β21G11[k] 0 · · · β2NGNN [k]

...
...

...
...

βN1G11[k] βN2G22[k] · · · 0


 .

In summary, the power-control system (6) has a linear system model with random parameters.
This study will analyze the pth-moment stability for such a system.

3.2 Problem statement

In what follows, we begin by considering the linear systems with random parameters as

x[k + 1] = A[k]x[k] + B[k]n[k], (8)

where {A[k],k ∈ Z+} and {B[k],k ∈ Z+} be two stationary stochastic processes. {n[k],k ∈
Z+} is a stationary stochastic process of additive white Gaussian noise, and independent with
{A[k],k ∈ Z+} and {B[k],k ∈ Z+}. Clearly, the model of system (8) is a generalization of
the model (6), which is not same but closely related to the models appeared in several existed
works [10] [14] [16].

Main problem of this work. How can we estimate whether the linear system (8) is
pth-moment stable or not?



850 R. Qian, Y. Qi

Theorem. The stochastic process {x[k],k ∈ Z+} that corresponds to the state vector x[k]
in (8), is a Markov chain.

Proof The conclusion directly follows from the definition of Markov chain. 2
The result above is a trivial but fundamental understanding of the system (8). We then turn

to taking a closer look at this system. To state the further results, we need to rewrite (8) as

x[k + 1] =

(
k∏

u=1

A[u]

)
x[1]

︸ ︷︷ ︸
non-noise term

+

k∑
i=1

(
k∏

u=i+1

A[u]

)
B[i]n[i]

︸ ︷︷ ︸
noise term

. (9)

If the noise term does not exist in (9), the Furstenberg-Kesten theorem [14] and the analytical
framework developed by Feng et al. [10] would be available to study the stochastic stability
properties for the associated system. However now, due to the existence of the noise term, we
have to seek for a new proper framework to analyze such a system. In [16], Koning provided an
analytical framework which is usable to investigate (8), but such a framework can only reflect
the first- and second-order statistics of x[k]. In this work, we will develop a norm-inequality-
based framework that is capable of analyzing higher-order as well as the first- and second-order
statistics of x[k].

4 Results

In this section, the pth-moment stability of the linear system (8) is analyzed through a norm-
inequality-based approach, and then the analysis is applied to the power-control system (6). To
attain the main results of this study, we need to derive and use several lemmas; however, we
prefer not to introduce them in the main text but, rather, in Appendix, so as not to interrupt
the presentation. For details, please refer to Lemmas A.1 to A.5.

4.1 A Norm-inequality-based approach of the pth-moment stability analysis

We are now ready to perform the pth-moment stability analysis of the linear system (8) via
a norm-inequality-based approach.

Theorem 1. A sufficient condition for the first-moment stability of the system (8) is
limk→+∞ Ā

k = 0N×N, where 0N×N is the N ×N zero matrix.
Proof Since x[k + 1] ≥ 0,
E
(
‖x[k + 1])‖ 11

)
= ‖E(x[k + 1])‖ 11 = ‖x̄[k + 1]‖ 11 =

∥∥∥Ākx̄[1] + ∑ki=1 Āk−i B̄n̄∥∥∥ 1
1
. If the

matrix Ā has the property that limk→+∞ Āk = 0N×N, I−Ā will be nonsingular and its inverse
can be expressed by [23, Corollary 5.6.16]: (I − Ā)−1 =

∑+∞
k=0 Ā

k, and then we shall have
limk→+∞E

(
‖x[k + 1])‖ 11

)
=
∥∥(I − Ā)−1B̄n̄∥∥ 1

1
< +∞ as long as limk→+∞ Āk = 0N×N holds.

2

The model (8) has an alternative formulation as

x[k + 1] = sx[k] +
k∑
i=1

skn[i], (10)

where, for notational simplicity, sx[k] =
(∏k

u=1 A[u]
)
x[1], skn[i] =

(∏k
u=i+1 A[u]

)
B[i]n[i].

The forthcoming analysis will involve applying the operator ELp to p-norm of some random
vector x, i.e., substituting f = ‖x‖p into ELpf, or to induced matrix norm of some random



Stochastic Stability Analysis of Power Control
in Wireless Networks via a Norm-inequality-based Approach 851

matrix X, i.e., substituting f = ‖X‖p into ELpf. There is one important issue herein that must
be mentioned:

Remark 2: Computing the induced matrix norm is a nonlinear optimization problem1, and
the induced norm ‖X‖p has no explicit representation unless p = 1, 2 or ∞. If p 6= 1, 2, or ∞,
the integral

∫
Ω
‖X‖ pp dµ < +∞ might not exist, in which case we can not take ELp to ‖X‖p.

Therefore, in this study when it is needed to apply the operator ELp to ‖X‖p, we will seek for
an integrable upper-bound ϕ(X) of ‖X‖p and use ELpϕ(X) for theoretical analysis.

Theorem 2. Assume that ‖X‖p has an upper-bound ϕ(X) > 0, i.e., ‖X‖p ≤ ϕ(X), where∫
Ω
|ϕ(X)|pdµ is integrable. A sufficient condition for the pth-moment stability of the system (8)

is ELpϕ(A) < 1, ELpϕ(B) < +∞, and ELp‖n‖p < +∞.2
Proof By (10), we get

ELp ‖x[k + 1])‖p = ELp
∥∥∥∥∥sx[k] +

k∑
i=1

skn[i]

∥∥∥∥∥
p

≤ ELp ‖sx[k]‖p +
k∑
i=1

ELp
∥∥∥skn[i]∥∥∥

p

≤
k∏

u=1

ELpϕ(A[u])ELp‖x[1]‖p +
k∑
i=1

k∏
u=i+1

ELpϕ(A[u])ELpϕ(B[i])ELp‖n[i]‖p,

where the first inequality follows from Lemma A.4 and the second one is from Lemma A.5.
Under the assumption of stationarity, ELpϕ(A[k]), ELpϕ(B[k]), and ELp‖n[k]‖p shall not

change with k such that we can drop k for notational simplicity. Thus,

ELp ‖x[k + 1])‖p ≤ [ELpϕ(A)]
k
ELp‖x[1]‖p +

k∑
i=1

[ELpϕ(A)]
k−i

ELpϕ(B)ELp‖n‖p

= [ELpϕ(A)]
k
ELp‖x[1]‖p +

1 − [ELpϕ(A)]k
1 −ELpϕ(A)

ELpϕ(B)ELp‖n‖p.

If ELpϕ(A) < 1, ELp ‖B‖p < +∞, and ELp‖n‖p < +∞, then

lim
k→+∞

E
(
‖x[k + 1]‖ pp

)
≤
(
ELpϕ(B)ELp‖n‖p

1 −ELpϕ(A)

)p
< +∞.

Therefore, Theorem 2 is justified. 2
Theorem 2 holds for all p ∈ Z+, thus it can reveal any pth-order statistics of x[k]. It’s a

progress made by the proposed analytical framework of this study, compared to the framework
developed by [16] that can only investigate the first- and second-order statistics of x[k]. Since
the derivation methods of Theorems 1 and 2 are not exactly same, the sufficient condition of the
first-moment stability obtained by Theorem 1 is not necessarily identical as that by Theorem 2
with p = 1.

Theorem 3. If ELpϕ(A) < 1, ELp ‖B‖p < +∞, and ELp‖n‖p < +∞, there exists α < +∞,
such that limk→+∞E

(
‖x[k])‖ pp

)
= α.

Proof Let us begin by assuming that k > j > J. We get ELp ‖x[j + 1])‖p ≥ ELp
∥∥∥∑ji=1 sjn[i]∥∥∥

p
,

1An already known approach is to make use of the algorithm of estimating the induced matrix norm as well
as the Matlab routines provided by Higham [17].

2Note that A, B, and n are short for A[k], B[k], and n[k] by dropping the index k, since the statistics of
A[k], B[k], and n[k] are irrelevant with k under the stationarity assumption.



852 R. Qian, Y. Qi

because sx[j] and s
j
n[i] (i,j ∈ Z+, 1 ≤ i ≤ j) are all positive vectors. By (10), we can have

ELp ‖x[k + 1])‖p ≤ ELp ‖sx[k]‖p + ELp

∥∥∥∥∥∥
k∑

i=k−j+1
skn[i]

∥∥∥∥∥∥
p

+ ELp

∥∥∥∥∥
k−j∑
i=1

skn[i]

∥∥∥∥∥
p

. (11)

Due to the stationarity property, it holds true that ELp
∥∥∥∑ji=1 sjn[i]∥∥∥

p
= ELp

∥∥∥∑ki=k−j+1 skn[i]∥∥∥
p
.

Then, with ELpϕ(A) < 1, ELp ‖B‖p < +∞, and ELp‖n‖p < +∞, it implies that

ELp ‖x[k + 1])‖p −ELp ‖x[j + 1])‖p ≤ ELp ‖sx[k]‖p + ELp
∥∥∥∥∥
k−j∑
i=1

skn[i]

∥∥∥∥∥
p

≤ [ELpϕ(A)]k ELp‖x[1]‖p +
[ELpϕ(A)]

j

1 −ELpϕ(A)
ELpϕ(B)ELp‖n‖p. (12)

Now we can conclude that, ∀ε > 0, ∃J > 0, such that ELp ‖x[k + 1])‖p−ELp ‖x[j + 1])‖p < ε
as long as k,j ≥ J. It states that ELp ‖x[j + 1])‖p has a limit value as k → +∞, and thus finishes
the proof. 2

Theorem 4. For the system (8), if let ϕ(A) = N1−1/p‖A‖1 or ϕ(A) = ‖A‖
1/p
1 ‖A‖

1−1/p
∞ ,

where A is short for A[k],
∫

Ω
|ϕ(A)|pdµ would exist which means ELpϕ(A) exists.

Proof Both ‖A‖1 and ‖A‖∞ are continues measurable functions. Then
∫

Ω
Np−1 ‖A‖ p1 dµ

and
∫

Ω
‖A‖1 ‖A‖

p−1
∞ dµ exist. This leads to the results of Theorem 4. 2

Remark 3: Theorem 4 yields two sufficient conditions for the pth-moment stability of the
system (8), i.e., ELp(N1−1/p‖A‖1) < 1 and ELp(‖A‖

1/p
1 ‖A‖

1−1/p
∞ ) < 1. Although there might

exist certain conservation, these two conditions are convenient for practical operations, because
both ‖A‖1 and ‖A‖∞ have the explicit representations.

Remark 4: Taniguchi [18] provided stochastic stability theorems of the nonlinear difference
equations through using norm inequalities; however, the theorems obtained in [18] can not assist
us to achieve the results with practical significance for the system (8). While by employing the
norm-inequality-based framework, this study dedicates to derive the results for the system (8).
One could also find the moment stability studies attract many interests recently, e.g., the pth-
moment exponential ultimate boundedness is investigated for impulsive stochastic differential
systems [19], and the pth-moment asymptotic stability is analyzed for stochastic delayed hybrid
systems with Levy noise [20].

4.2 The pth-moment stability of power control

Going back to the power-control system (6), we can obtain many useful results without too
much efforts based on the previous analysis.

Remark 5: By letting A[k] = D[k]C[k] and B[k] = D[k], one can directly apply Theorems
1 to 4 to the power-control system (6).

One important novelty of this study is not only to assess the stability of power-control system
(6) but also to acquire more knowledge of relations between the stochastic stability and power
control together with other wireless communication technologies. We will show that the proposed
norm-inequality-based approach allows us to recognize the effect of multiple-access methods to
the pth-moment stability of power control.

The sufficient conditions for the pth-moment stability given by Theorems 1, 2, 3, and 4 are
only related to A[k] (= D[k]C[k]), while D[k]C[k] is partly determined by the target SINRs



Stochastic Stability Analysis of Power Control
in Wireless Networks via a Norm-inequality-based Approach 853

and link gains according to (6)-(7). This fact inspires us to investigate the pth-moment stability
by thinking over the power control together with the effect of multiple access technique.

Consider the power-control system (6) with a multiple access method, under Assumption 4,
we have an upper bound of A[k] as

A[k] = D[k]C[k] ≤




0 γ1β12
G22[k]
G11[k]

· · · γ1β1N GNN [k]G11[k]
γ2β21

G11[k]
G22[k]

0 · · · γ2β2N GNN [k]G22[k]
...

...
...

...
γNβN1

G11[k]
GNN [k]

γNβN2
G22[k]
GNN [k]

· · · 0


 .

In the reminder of this section, the index k will be dropped from A[k], D[k], C[k], and Gii[k]
such that A, D, C, and Gii are used.

From above, we get the following results.
Theorem 5. If the power-control system (6) employs a multiple access method so that

Assumptions 1 to 4 are satisfied, it will hold that Ā = EA ≤ µG ·µ1/G · Θγ̄,β, where

Θγ,β =




0 γ1β12 · · · γ1β1N
γ2β21 0 · · · γ2β2N

...
...

...
...

γNβN1 γNβN2 · · · 0


 .

Then if the values of γi (1 ≤ i ≤ N) and βij (1 ≤ i,j ≤ N) are properly chosen such that

min


maxi

N∑
j 6=i

γiβij, max
j

N∑
i 6=j

γiβij


 < 1µG ·µ1/G , (13)

the system will be first-moment stable.
Proof The upper bound of Ā can be obtained by taking expectation to the upper bound of

A given above.
Let ρ(·) denote the spectral radius. Using [23, Theorem 8.1.22] to show that

max


mini

N∑
j=1

Āij, min
j

N∑
i=1

Āij


 ≤ ρ(Ā) ≤ min


maxi

N∑
j=1

Āij, max
j

N∑
i=1

Āij


 .

i.e., the smallest row sum of a nonnegative matrix is a lower bound on its spectral radius, and
the largest row sum is an upper bound. Then, by applying Āij ≤ µG ·µ1/G ·γiβij, we have

ρ
(
Ā
)
≤ µG ·µ1/G · min


maxi

N∑
j 6=i

γiβij, max
j

N∑
i 6=j

γiβij


 . (14)

Combing this result with [23, Theorem 5.6.12] which says limk→+∞ Āk = 0 if and only if
ρ(Ā) < 1, we see that letting the right-hand side of (14) be less than 1 is a sufficient condition
for limk→+∞ Āk = 0, which therefore makes the power-control system to be first-moment stable
(see Theorem 1 for reference). 2



854 R. Qian, Y. Qi

(a)
5 10 15 20 25 30 35 40

10
−2

10
0

10
2

10
4

10
6

10
8

10
10

10
12

k

E
(|

|x
[k

]|
| 1  

1

 

 

γ=0.1, ρ=0.3

γ=0.2, ρ=0.6

γ=0.3, ρ=0.9

γ=0.4, ρ=1.2

γ=0.5, ρ=1.5

γ=0.6, ρ=1.8

ρ<1

ρ>1

(b)
5 10 15 20 25 30 35 40

10
0

10
5

10
10

10
15

10
20

10
25

k

E
(|

|x
[k

]|
| 2  

2
)

 

 
γ=0.1,   E

L
2 ||A||

2
=0.28

γ=0.2,   E
L

2 ||A||
2
=0.56

γ=0.3,   E
L

2 ||A||
2
=0.84

γ=0.4,   E
L

2 ||A||
2
=1.12

γ=0.5,   E
L

2 ||A||
2
=1.40

γ=0.6,   E
L

2 ||A||
2
=1.68

E
L

2 ||A||
2
<1

E
L

2 ||A||
2
>1

Figure 1: (a): E(‖x[k]‖ 11 ) versus k, where ρ is short for ρ
(
Ā
)
. The curves with ρ < 1 tend

to finite values, while those with ρ > 1 increase ceaselessly; (b): E(‖x[k]‖ 22 ) versus k. The
curves with EL2 ‖A‖2 < 1 tend to finite values, while those with EL2 ‖A‖2 > 1 are progressively
growing

Theorem 6. Consider the power-control system (6) with a multiple access method such
that Assumptions 1 to 4 are satisfied, if the values of γi (1 ≤ i ≤ N) and βij (1 ≤ i,j ≤ N) are
properly chosen such that

N∑
j=1

N∑
i 6=j

γ2j β
2
ij <

1

µG2 ·µ1/G2
, (15)

the system will be second-moment stable.
Proof Since ‖A‖2 =

√
tr(AT A) [23], where tr(·) is the trace operation, by setting ϕ(A) =√

µ1/G2 · tr
(

ΘTγ,βΘγ,β

)
, it follows that EL2ϕ(A) = µG2 · µ1/G2 · tr

(
ΘTγ,βΘγ,β

)
= µG2 · µ1/G2 ·(∑N

j=1

∑N
i 6=j γ

2
j β

2
ij

)
. Observe that EL2ϕ(A) < 1 as long as (15) holds. Then, recalling Theorem

2 completes the proof. 2
Remark 6: The Cauchy-Schwarz inequality [24] leads to 1

µG·µ1/G < 1 and
1

µ
G2
·µ

1/G2
< 1. So we

can have more conservative but simpler conditions than (13) and (15) to achieve the first- and
second-moment stability, respectively, which are min

{
maxi

∑N
j 6=i γiβij, maxj

∑N
i 6=j γiβij

}
< 1,

and
∑N

j=1

∑N
i 6=j γ

2
j β

2
ij < 1.

Furthermore, let us extend Theorems 5 and 6 to a generalized case, i.e., the pth-moment
stability with any p ∈ Z+.

Theorem 7. Suppose that the power-control system (6) employs a multiple access method
so that Assumptions 1 to 4 are established, the system will be pth-moment stable if ELpϕ(A) < 1
with ϕ(A) = N1−1/p

(
maxj

∑N
i 6=j γiβij

Gjj
Gii

)
, or

ϕ(A) =


max

j

N∑
i 6=j

γiβij
Gjj
Gii


1/p


max

i

N∑
j 6=i

γiβij
Gjj
Gii


1−1/p .

Proof Through replacing ‖A‖1 and ‖A‖∞ in Theorem 4 with the maximum column sum
and maximum row sum of A[k], respectively, Theorem 7 can be validated. 2

The importance of Theorems 5, 6, 7, and Remark 6 lies in that they can guide system
designers to assess and select suitable target SINR schemes and multiple access methods for
wireless-network systems and also to pick out the proper system parameters for them, from the
perspective of power-control stability.



Stochastic Stability Analysis of Power Control
in Wireless Networks via a Norm-inequality-based Approach 855

Table 1: Numerical Values of Examples 2 and 3.

γ 73.7 86.0 98.3 110.6

min
{

maxi
∑N

j 6=i γβij, maxj
∑N

i 6=j γβij
}

0.90 1.05 1.20 1.35

ρ
(
Ā
)

0.66 0.77 0.87 1.01∑N
j=1

∑N
i 6=j γ

2β2ij 0.85 1.16 1.51 1.91

EL2 ‖A‖2 0.53 0.72 0.94 1.19(∫
Ω
‖A‖ 55 dµ

) 1/5
2.05 2.40 2.70 3.13

EL5
(
‖A‖ 1/51 ‖A‖

4/5
∞
)

2.26 2.64 2.98 3.45

5 Numerical examples

Example 1: We consider the power-control system (8) in main text with i.i.d. Rayleigh fading
link gains (that is, all Gij are Rayleigh distributed with unit variance) and fixed target SINRs
γ1[k] = γ2[k] = · · · = γN [k] = γ. There are four nodes in the network.

We let n[k] be the power vector of Gaussian noise with unit variance, and initially set
x[1] = [1 0 0 0]T . In case of γ = 0.1, 0.2, · · · , 0.6, Figs. 1(a) and 1(b) illustrate how E(‖x[k]‖ 11 )
and E(‖x[k]‖ 22 ) grow with k, where ρ

(
Ā
)
and EL2 ‖A‖2 are estimated during the simulations.

Fig. 1(a) shows that the curves of E(‖x[k]‖ 11 ) with γ = 0.3, 0.6, 0.9 tend to finite values (in
other words, the system is first-moment stable), while others increase ceaselessly. This result is
in accordance with Theorem 1 because limk→+∞ Āk = 0 if and only if ρ(Ā) < 1 [23, Theorem
5.6.12]. From Fig. 1(b), it is observed that the curves of E(‖x[k]‖ 22 ) with γ = 0.28, 0.56, 0.84
tend to finite values (or, equivalently, the system is second-moment stable), while others are
progressively growing. This observation is a consequence of Theorems 2, 3.

Example 2: We consider the power-control system (8) in main text with multiple access
method and fixed target SINRs γ1[k] = γ2[k] = · · · = γN [k] = γ. All Gjj = 1 + G′j where G′j
is the Rayleigh distributed with unit variance. Let Gij[k] = βijGjj[k] without loss of generality.
It can be obtained that µG · µ1/G = 1.20 and µG2 · µ1/G2 = 2.03. There are four nodes in the
network which employs a multiple access method such that




0 β12 β13 β14
β21 0 β23 β24
β31 β32 0 β34
β41 β42 β43 0


 =




0 1
100

1
200

1
300

1
400

0 1
500

1
600

1
700

1
800

0 1
900

1
1000

1
1100

1
1200

0


 .

For above,

min


maxi

N∑
j 6=i

βij, max
j

N∑
i 6=j

βij


 = 1.22 × 10−2,

N∑
j=1

N∑
i 6=j

β2ij = 1.565 × 10−4. (16)

Again, let n[k] be the power vector of Gaussian noise with unit variance, and initially
set x[1] = [1 0 0 0]T . In case of γ = 73.7, 86.0, 98.3, 110.6, we build Table 1 by recalling
two previous equalities, i.e., (16), and performing simulations. From Table 1, it is clear that
min

{
maxi

∑N
j 6=i γβij, maxj

∑N
i 6=j γβij

}
is less than ρ

(
Ā
)
, and

∑N
j=1

∑N
i 6=j γ

2β2ij is less than
EL2 ‖A‖2, as we could expect from the relations between Theorems 1, 2, 5, and 6. The nu-
merical results of Figs. 2(a) and 2(b) are in agreement with Theorems 1, 2, 5, and 6.



856 R. Qian, Y. Qi

(a)
0 10 20 30 40 50 60 70

10
2

10
3

10
4

k

E
(|

|x
[k

]|
| 1  

1

 

 

γ=73.7
γ=86.0
γ=98.3
γ=110.6

(b)
0 10 20 30 40 50 60 70

10
3

10
4

10
5

10
6

10
7

10
8

k

E
(|

|x
[k

]|
| 2  

2

 

 

γ=73.7
γ=86.0
γ=98.3
γ=110.6

Figure 2: (a): E(‖x[k]‖ 11 ) versus k; (b): E(‖x[k]‖
2

2 ) versus k.

(a)
0 5 10 15 20 25 30 35 40

0

2

4

6

8

10

12

p

 

 

( ∫
Ω

 ||A||
p
  p dµ)1/p

E
L

p (N1−1/p ||A||
1
)

E
L

p (||A||
1
  1/p ||A||

∞
  1−1/p)

(b)
0 5 10 15 20 25 30 35 40

10
10

10
15

10
20

10
25

k

E
(|

|x
[k

]|
| 5  

5

 

 

γ=73.7
γ=86.0
γ=98.3
γ=110.6

Figure 3: (a): Comparison among
(∫

Ω
‖A‖ pp dµ

)1/p
, ELp(N1−1/p‖A‖1), and

ELp
(
‖A‖ 1/p1 ‖A‖

1−1/p
∞

)
for different p. Note that, the Matlab routines provided by

Higham [17] which can directly estimate ‖A‖p is used for computing
(∫

Ω
‖A‖ pp dµ

)1/p
; (b):

E(‖x[k]‖ 55 ) versus k

Example 3: The same system model and parameters as Example 2 are used.

In Fig. 3(a), we have compared
(∫

Ω
‖A‖ pp dµ

)1/p
with its two upper bounds as given by Re-

mark 3, i.e., ELp(N1−1/p‖A‖1) and ELp
(
‖A‖ 1/p1 ‖A‖

1−1/p
∞

)
, which can also be refereed to The-

orem 4. It is seen that, ELp
(
‖A‖ 1/p1 ‖A‖

1−1/p
∞

)
stays quite close to

(∫
Ω
‖A‖ pp dµ

)1/p
, however,

there is an evident gap between ELp(N1−1/p‖A‖1) and
(∫

Ω
‖A‖ pp dµ

)1/p
, which is amplified as

p increases. Therefore, being the upper bound of
(∫

Ω
‖A‖ pp dµ

)1/p
, ELp

(
‖A‖ 1/p1 ‖A‖

1−1/p
∞

)
is

more tight than ELp(N1−1/p‖A‖1). As a consequence, we propose to use ELp
(
‖A‖ 1/p1 ‖A‖

1−1/p
∞

)
when the upper bound of

(∫
Ω
‖A‖ pp dµ

)1/p
is needed. For γ = 73.7, 86.0, 98.3, 110.6, Fig. 3(b)

illustrates how E(‖x[k]‖ 55 ) evolves with k, and Table 1 presents the data of
(∫

Ω
‖A‖ 55 dµ

)1/5
and EL5

(
‖A‖ 1/51 ‖A‖

4/5
∞
)
. When γ = 73.7, 86.0, E(‖x[k]‖ 55 ) tends to finite values as long as

k is sufficiently large. Then if γ = 98.3, 110.6, E(‖x[k]‖ 55 ) will be found to grow infinitely. The
numerical result is in accordance with Theorem 3.



Stochastic Stability Analysis of Power Control
in Wireless Networks via a Norm-inequality-based Approach 857

Conclusion

This study develops a norm-inequality-based framework of analyzing the pth-moment stabil-
ity of linear systems with random parameters, so as to show that a typical power control law
with linear system model is stable in the sense of the pth-moment stability. It is the first time
to recognize the effect of multiple-access methods to stability analysis of power control.

Acknowledgment

The authors are grateful to the anonymous reviewers for many valuable comments concerning
the presentation. This work was supported by the National Natural Science Foundation of China
under Grant 61501043. We would like to thank Dr. Miao Diao and Prof. Duan Zhisheng for all
their help and support while writing this paper.

Bibliography

[1] C.W. Sung, K.K. Leung (2005); A Generalized framework for distributed power control in
wireless networks, IEEE Trans. Inf. Theory, 51(7): 2625-2635.

[2] H. Feyzmahdavian, M. Johansson,T. Charalambous (2012); Contractive interference func-
tions and rates of convergence of distributed power control laws, IEEE Trans. Wireless Com-
mun., 11(12):4494-4502.

[3] H. Feyzmahdavian, T. Charalambous, and M. Johansson (2014); Stability and perfor-
mance of continuous-time power control in wireless networks, IEEE Trans. Automat. Contr.,
59(8):2012-2023.

[4] I. Lestas (2012); Power control in wireless networks: Stability and delay independence for a
general class of distributed algorithms, IEEE Trans. Automat. Contr., 57(5):1253-1258.

[5] E. Devane and I. Lestas (2014); Stability of a general class of distributed algorithms for power
control in time-varying wireless networks, IEEE Trans. Automat. Contr., 59(8):1999-2011.

[6] S. Ulukus, R. Yates (1998); Stochastic power control for cellular radio systems, IEEE Trans.
Commun., 46(6):784-798.

[7] M. Varanasi, D. Das (2002); Fast stochastic power control algorithms for nonlinear multiuser
receivers, IEEE Trans. Commun., 50(11):1817-1827.

[8] J. Lu, S. Ulukus,A. Ephremides (2005); Standard and quasi-standard stochastic power control
algorithms, IEEE Trans. Inf. Theory, 51(7):2612-2624.

[9] H. Zhang, W. S. Wong, W. Ge, P. E. Caines (2007); A stochastic approximation approach
to the power-control problem, IEEE Trans. Commun., 55(8): 878-886.

[10] X. Feng, K A. Loparo, Y. Ji, H. J. Chizeck (1992); Stochastic stability properties of jump
linear systems, IEEE Trans. Automat. Contr., 37(1):38-53.

[11] J. Leth, H. Schioler, M. Gholami, V. Cocquempot (2013); Stochastic stability of Markovianly
switched systems, IEEE Trans. Automat. Contr., 58(8): 2048-2054.

[12] G. J. Foschini and Z. Miljanic (1993); A simple distributed autonomous power control
algorithm and its convergence, IEEE Trans. Vehic. Technol., 42: 641-646.



858 R. Qian, Y. Qi

[13] M. Xiao, N. Shroff, and E. Chong (2003); A utility-based power-control scheme in wireless
cellular systems, IEEE J. Sel. Areas Commun., 11(2):210-221.

[14] H. Furstenberg, H. Kesten (1960); Products of random matrices, Annals of mathematical
statistics, 31(2):457-469.

[15] A. Bobrowski (2005); Functional analysis for probability and stochastic processes, Cambridge
Universtity Press, 2005.

[16] W. L. D. Koning (1984); Optimal estimation of linear discrete-time systems with stochastic
parameters, Automatica, 20(1):113-115.

[17] N. J. Higham (1992); Estimating the matrix p-norm, Numer. Math, 62:511-538.

[18] T. Taniguchi (1990); Stability theorems of stochastic difference equations, Journal of Math-
ematical Analysis and Applications, 147(1):81-96.

[19] L. Xua, S. S. Geb (2015); The pth moment exponential ultimate boundedness of impulsive
stochastic differential systems, Applied Mathematics Letters, 42:22-29.

[20] J. Yang, W. Zhou, X. Yang, X. Hu, L. Xie (2015); pth moment asymptotic stability
of stochastic delayed hybrid systems with Levy noise, International Journal of Control,
88(9):1726-1734.

[21] T. Charalambous, I. Lestas, G. Vinnicombe (2008); On the stability of the Foschini-Miljanic
algorithm with time-delays, in Proc. 47th IEEE Conf. on Decision Control, 2991-2996.

[22] M. Varanasi (1999); Nonlinear multiuser receivers with distributed power control in cellular
radio networks, in Proc. 37th Annu. Allerton Conf. Communication, Control and Computers,
820-830.

[23] R. A. Horn and C. R. Johnson (1985); Matrix analysis, Cambridge University Press, Second
Edition, 1985.

[24] R. Durrett (2005); Probability: Theory and examples, Cengage Learning, Third Edition,
2005.

[25] I. S. Gradshteyn, I. M. Ryzhik (2007); Table of integrals, series, and products, 7th edition,
Academic, 2007.

Appendix: Several lemmas

This Appendix is devoted to present the lemmas (and their proofs) which are required to
derive the main results of our work.

Lemma A.1: Let the nonnegative numbers a1,a2, · · · ,aN and the positive p1,p2, · · · ,pN be
given. Set

∑N
j=1

1
pj

= 1 then the inequality
∏N
j=1 aj ≤

∑N
j=1

1
pj
a
pj
j holds with equality if and

only if all ak with pk > 0 are equal.
Proof:

∏N
j=1 aj ≤

∑N
j=1

1
pj
a
pj
j in Lemma A.1 is an inequality of the weighted arithmetic mean

and geometric mean, which can be proved by using the finite form of Jensen’s inequality [24] for
the natural logarithm. 2



Stochastic Stability Analysis of Power Control
in Wireless Networks via a Norm-inequality-based Approach 859

Lemma A.2: Let x1,x2, · · · ,xN be N random variables and p1,p2, · · · ,pN be nonnegative
numbers numbers. If

∑N
j=1

1
pj

= 1 and E|xj|pj < +∞ for 1 ≤ j ≤ N, then E
(∏N

j=1 |xj|
)
≤∏N

j=1 (E|xj|pj )
1
pj , where (E|xj|pj )

1
pj = ELp||xj|| with p = pj.

Proof: By using Lemma A.1, we get
∏N
j=1 |xj|∏N

j=1(E|xj|
pj )

1
pj

=
∏N
j=1

|xj|

(E|xj|pj )
1
pj

≤
∑N

j=1
|xj|pj

pjE|xj|pj
.

Then applying the expectation to above inequality,
E(

∏N
j=1 |xj|)∏N

j=1(E|xj|
rj )

1
pj

≤
∑N

j=1
E|xj|pj
pjE|xj|pj

=
∑N

j=1
1
pj

=

1. Thus, Lemma A.2 is verified. 2
Lemma A.3: Suppose X1,X2, · · · ,XK are random matrices with size S1 ×S2, S2 ×S3, · · · ,

SK × SK+1, where S1,S2, · · · ,SK,SK+1 are all simply positive integers and the subscripts are
labels corresponding to the matrices. If the entries of Xk are independent with those of Xl for
any k 6= l then E

(∏K
k=1 Xk

)
=
∏K
k=1 E(Xk).

Proof: The product of K matrices can be expressed in the index notation as[
K∏
k=1

Xk

]
ij

=

S1∑
i1=1

S2∑
i2=1

· · ·
SK−1∑
iK−1=1

[X1]ii1 [X2]i1i2 [X3]i2i3 · · · [Xn−1]in−2in−1 [Xn]in−1j.

This implies every entry of the resultant matrix after matrix product is a linear function of the
entries of all Ak matrices. The independence condition can further yield [24]

E


[ K∏

k=1

Xk

]
ij


 = S1∑

i1=1

S2∑
i2=1

· · ·
SK−1∑
iK−1=1

E([X1]ii1 )E([X2]i1i2 ) · · ·E([Xn−1]in−2in−1 )E([Xn]in−1j).

Therefore, Lemma A.3 is proved. 2
Lemma A.4: Suppose that x1,x2, · · · ,xI are random matrices with size S×1, one can have

ELp
∥∥∥∑Ii=1 xi∥∥∥

p
≤
∑I

i=1 ELp ‖xi‖p, where p ∈ Z+.
Proof: The norm inequalities and Lemma A.2 combine to provide

ELp
∥∥∥∥∥

I∑
i=1

xi

∥∥∥∥∥
p


p = E


∥∥∥∥∥

I∑
i=1

xi

∥∥∥∥∥
p

p


 ≤ E

((
I∑
i=1

‖xi‖p

)p)

=

I∑
i1=1

I∑
i2=1

· · ·
I∑

ip=1

E
(
‖xi1‖p‖xi2‖r · · ·

∥∥xip∥∥p)

≤
I∑

i1=1

· · ·
I∑

ip=1

(
E
(
‖xi1‖

p
p

)
· · ·E

(∥∥xip∥∥ pp ))1/p =
(

I∑
i=1

ELp ‖xi‖p

)p
.

This gives the desired result. 2
Lemma A.5: Let X1,X2, · · · ,XK be random matrices with size S1 ×S2, S2 ×S3, · · · , SK ×

SK+1 and y be a SK+1×1 random vector. If the entries of Xk are independent with those of Xl for
any k 6= l and y for any 1 ≤ k 6= K, then ELp

∥∥∥(∏Kk=1 Xk)y∥∥∥
p
≤
(∏K

k=1 ELpϕ(Xk)
)(

ELp ‖y‖p
)
,

where p ∈ Z+.
Proof: The norm inequalities implies

∥∥∥(∏Kk=1 Xk)y∥∥∥
p
≤
(∏K

k=1 ‖Xk‖p
)
‖y‖p, and thus we

get E
(∥∥∥(∏Kk=1 Xk)y∥∥∥ p

p

)
≤
(∏K

k=1 E(|ϕ(Xk)| p)
)
E(‖y‖ pp ). Such an inequality directly yields

the result. 2