A generalization of some random variables involving in certain compressive sensing problems
In this paper we give a generalization of the discrete complex-valued random variable defined and investigated in \cite{ssa} and \cite{m8}. We prove the statements concerning the expressions for the excepted value and the variance of this random vari…
Authors: Romeo Mev{s}trovic
A GENERALIZA TION OF SOME RANDOM V ARIABLES INV OL VING IN CER T AI N COMPRESSIVE SENSING PR OBLEMS R OMEO ME ˇ STR O VI ´ C A B S T R AC T . In this paper we gi ve a generalization of the discrete comp le x-valued random variable defined and in vestigated in [23] and [10]. W e prove the statemen ts concern ing the exp r ess ions for the excepted value and the variance of this random variable. In partucular, such a random variable h e re is de fin ed for each of m rows of any m × N complex or real matrix A with 1 ≤ m ≤ N . W e consider the arithmetic mean ¯ X ( m ) of the se m r andom variables and we d educe the expressions f or the expected value E [ ¯ X ( m )] an d the variance V ar[ ¯ X ( m )] of ¯ X ( m ) . Using the expression fo r V ar[ ¯ X ( m )] , we establish some equ alities and in equalities inv olving V ar[ ¯ X ( m )] , the Frobeniu s norm , the largest eig en value, the largest sin g ular value and the coherence of a matrix A . I t is showed that some of these estimates are closely related to the W e lc h bound of the co herence of a m × N complex or real matr ix A with 1 ≤ m ≤ N . T aking into account th at the value of coh erence of the m easurement matrix in the theory of c o mpressi ve sensing h as a significant role , we believe that our results should be usefu l for so m e top ics of this theo r y . 1. I N T RO D U C T I O N A N D P R E L I M I N A R I E S As usually , throu ghout our considerations we use the term “multis et” (often written as “set”) to mean “a t o tality having possi b le multipl icities”; so that two (multi)sets will be counted as equal if and only i f th e y have th e same elements wi th identical multipli cities. Let C and R denote the fields of complex and real numbers, respecti vely . For a g iven positive int e ger N , let M N denote the collection of all mul tisets of the form (1) Φ N = { z 1 , z 2 , . . . , z N } , where z 1 , z 2 , . . . , z N ∈ C are arbitrary (not necessarily distinct) complex numbers. Furthermore, denote by M the set consisting of all mul tisets of the form (1), i.e., M = ∞ [ N =1 M N . Follo wing Definitio n 1.2 from [10] (also see [23, Section 2], [24, Section II], [11, Definition 1.1] and [11, Definition 1.1]), the random variable X ( m, Φ N ) can be gener- alized as follows. Definition 1.1. Let N and m b e arbitrary n onne gativ e integers such that 1 ≤ m ≤ N . For given not necessarily distinct comp l e x numbers z 1 , z 2 , . . . , z N , let Φ N ∈ M N 2010 Math e matics Su bject Classification . 05A19 , 9 4A12, 60C05, 05A1 0. K ey wor ds a nd phrases. Complex-valued discr e te random variable, Comp ressi ve sensing, Coherence of the matrix, W elch bound, Froben ius norm , Rand om pa r tial Fourier matr ix. 1 2 R OMEO ME ˇ STRO VI ´ C be a mult iset defined by (1). Define the discrete complex-valued random var iable X ( m, Φ N ) as Prob X ( m, Φ N ) = m X i =1 z n i ! (2) = 1 N m · {{ t 1 , t 2 , . . . , t m } ⊂ { 1 , 2 , . . . , N } : m X i =1 z t i = m X i =1 z n i | = : q ( n 1 , n 2 , . . . , n m ) N m , where { n 1 , n 2 , . . . , n m } is an arbit ra ry fixed subs et of { 1 , 2 , . . . , N } such that 1 ≤ n 1 < n 2 < · · · < n m ≤ N ; moreover , q ( n 1 , n 2 , . . . , n m ) is the cardinality o f a collection of all subsets { t 1 , t 2 , . . . , t m } of th e set { 1 , 2 , . . . , N } s uch that P m i =1 z t i = P m i =1 z n i . Notice that the abov e definition is correct taking into account that there are N m index sets T ⊂ { 1 , 2 , . . . , N } with m elements . M o re over , a very short, but not strongly exact version of Definition 1.1 is giv en as follows (cf. [10, Definition 1.2 ’]). Definition 1.1’. Let N and m be arbitrary nonnegative integers such that 1 ≤ m ≤ N . For giv en not necessarily dis tinct complex numbers z 1 , z 2 , . . . , z N , let Φ N ∈ M N be a multiset defined by (1). Choose a random subset S o f size m (the so -c alled m -element subset) without repl acement from th e set { 1 , 2 , . . . , N } . Then the complex-v alued discrete random variable X ( m, Φ N ) is defined as a sum X ( m, Φ N ) = X n ∈ S z n . Accordingly , here we prove the following result (cf. proof of Theorem 2.4 in [10] as a parti c ular case). Theor em 1.2. Let N and m be positi ve inte ger s such t h at N ≥ 2 and 1 ≤ m ≤ N . Let Φ N = { z 1 , z 2 , . . . , z N } be any multis et with z 1 , z 2 , . . . , z N ∈ C . Then th e e xpected value a n d the variance of the random variable X ( m, Φ N ) fr om Definition 1 . 1 ar e r especti vely given by (3) E [ X ( m, Φ N )] = m N N X i =1 z i and (4) V ar[ X ( m, Φ N )] = m ( N − m ) N 2 ( N − 1 ) N N X i =1 | z i | 2 − N X i =1 z i 2 ! . Mor eover , the second moment of the random variable | X ( m, Φ N ) | i s given by (5) E [ | X ( m, Φ N ) | 2 ] = m N ( N − 1) ( N − m ) N X i =1 | z i | 2 + ( m − 1 ) N X i =1 z i 2 ! . A GENE RALIZA TION OF S OME RANDOM V ARIABLES . . . 3 As a particular case of Theorem 1.2, we imm ediately obt ai n the fol lo wing s t ra ight- forward result. Corollary 1.3. Under notations and the ass u mptions of Theor em 1 . 2 , for m = N we have E [ X ( N , Φ N )] = N X i =1 z i , V ar [ X ( N , Φ N )] = 0 and E [ | X ( N , Φ N ) | 2 ] = N X i =1 z i 2 . Another consequence of Theorem 1.2 is given as follows. Corollary 1.4. Under notatio n s a n d the assumpt ions of Theor em 1 . 2 , we have (6) V ar[ X ( m, Φ N )] = m ( N − m ) N 2 ( N − 1 ) X 1 ≤ i
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