Journal for Research| Volume 01| Issue 12 | February 2016 ISSN: 2395-7549
Expected Number of Level Crossings of a Random Trigonometric Polynomial Dr. P. K. Mishra Associate Professor Department of Mathematics CET, BPUT, BBSR, Odisha, India
Dipty Rani Dhal Research Scholar Department of Mathematics “SOA" University, Bhubaneswar
Abstract Let EN( T; Φ’ , Φ’’ ) denote the average number of real zeros of the random trigonometric polynomial T=Tn( Φ, ω )= n
a b K
K
cos k
. In the interval (Φ’, Φ’’). Assuming that ak(ω ) are independent random variables identically distributed
K 1
according to the normal law and that bk = kp (p ≥ 0) are positive constants, we show that EN( T : 0, 2π ) ~ Outside an exceptional set of measure at most (2/ n ) where 2 4 2 2 p 12 p 3 2 p 1 1
2
(1 n ) 2 p 3 2
2 n O log n
n
SS ' (log n )
2
β = constant S ~ 1, S’ ~ 1. 1991 Mathematics subject classification (amer. Math. Soc.): 60 B 99. Keywords: Independent, Identically Distributed Random Variables, Random Algebraic Polynomial, Random Algebraic Equation, Real Roots, Domain of Attraction of the Normal Law, Slowly Varying Function _______________________________________________________________________________________________________
I. INTRODUCTION Let N( T ; Φ’ , Φ’’ ) be the number of real zeros of trigonometric polynomial n
T = Tn (Φ, ω) = a K b K cos k
(1)
K 1
In the interval ( Φ’ , Φ’’ ) where the coefficients ak(ω) are mutually independent random variables identically distributed according to the normal law; bk=kp are positive constants and when multiple zeros are counted only once . Let EN (T; Φ’, Φ’’) denote the expectation of N (T; Φ’, Φ’’). Obviously, Tn (Φ, ω) can have at most 2n most zeros in the interval (0, 2π) Das [ 1 ] studied the class of polynomials n
k g p
2 K 1
cos k g 2 k sin k
(2)
K 1
where gk are independent normal random variables for fixed p > -1/2 and proved that in the interval (0 , 2π) the function (2) have number of real roots when n is large.
2 p 1 2 p 3
11
1
4q
1
2 n O ( n 13 13 ) Here, q= max(0, -p) and 1 2 13 1 2 q . The measure of exceptional set does not exceed n-2 1 Das [2 ] took the polynomial (1) where a K(ω ) are independent normal random variables identically distributed with mean zero and variance one . He proves that in the interval 0 ≤ θ ≤ 2π , the average number of real zeros of polynomials ( 1 ) is [(2p+1) / (2p+3) ]1/22n+O( n ) (3) p for bK = k ( p > -1/2 ) and of the order of np+3/2 if -3/2 ≤ p ≤ -1/2 for large n. In this paper we consider the polynomial (1) with conditions as in DAS [2] and use the Kac_Rice formula for the expectation of the number of real zeros and obtain that for p ≥ 0 2
EN (T; 0, 2π) ~ 2 p 1 1 2 n 2 p 3
1 2
2 n O (log n )
Where 2
n
4
2
2 p 12 p 3 SS ' (log n )
β = constant S ~ 1
S’ ~ 1
2
Our asymptotic estimate implies that Das’s estimate in [1] is approached from below. Also our term is smaller.
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37
Expected Number of Level Crossings of a Random Trigonometric Polynomial (J4R/ Volume 01 / Issue 12 / 007)
The particular case for p=0 has been considered by Dunnage [3] and Pratihari and Bhanja [4] . Dunnage has shown that in the interval 0 ≤ θ ≤ 2π all save a certain exceptional set of the functions T n( Φ, ω ) have zeros when n is large. 11 3 2n (4) O n 1 3 log n 1 3
3
The measure of the exceptional set does not exceed ( logn )-1. Using the kac_rice formula we tried to obtain in [4] that Professor Dunnage[5] comments that our result is incorrect. EN (T; 0, 2π) ~ 2 n O (log n ) 6
He is quite right when he says than an asymptotic estimate is unique and that both results (4) and (5) cannot be correct. But in his calculations given in paragraph 4 of [5] he seems to have imported a factor 2 and the correct calculation would give I’~ 2πn / √3. accepting his own statement in paragraph 3 that I < I’, our point is clear. However, since I’ ~ 2πn /√3 on direct integration, our estimation of EN as found in [4], contained in the statement (5) above, must be wrong. We are sorry about our mistake. In this paper we consider our original integral I and evaluate it directly instead of placing it between two integrals as in [4], the second one being possibly suspect . This rectification eventually raises our estimate for EN but, all the same, keeps it below Dunnage’s estimate stated in above. The purport of our result is that EN approaches the value 2n/√3 from below. This is something meaningful. We prove the following theorem. Theorem The average number of real zeros in the interval (0, 2π ) of the class of random trigonometric polynomials of the form n
a b K
K
cos k
K 1
where aK( ω ) are mutually independent random variables identically distributed according to normal law with mean zero and variance one and bK=kp ( p ≥ 0) are positive constants , is asymptotically equal to 2p 1 2 1 n 2 p 3
1 2
(5)
2 n O (log n )
Outside an exceptional set of measure at most (2/n) where 2
n
4
2
2 p 12 p 3 SS ' (log n )
β = constant S ~ 1
S’ ~ 1
2
II. THE APPROXIMATION FOR EN (T;0,2Π) Let L(n) be a positive-valued function of n such that L(n) and n/L(n) both approach infinity with n. We take =L(n)/n throughout. Outside a small exceptional set of ω, T n( θ ,ω ) has a negligible number of zeros in each of the intervals (0, ), (π- ,π+ ) and (2π- ,2π). By periodicity, of zeros in each of intervals (0, ) and (2π- ,2π) is the same as number in (- , ). We shall use the following lemma, which is due to Das [2] . Lemma The probability that T has more than 1 + (log n)-1(logn+logDn+4n ) Zeros in ω- ≤ θ ≤ ω+ does not exceed 2 exp(-n ), where Dn =
n
b
K
K 1
The steps in this section follow closely those in section 2 of [4]. therefore, we indicate only the modifications necessary. In this case we have n
T= a K b K cos k K 1
n
T’= ka K b K sin k K 1
1 y , z exp 2
p 0 ,
1
n
y cos K 1
1
2 2 zk sin k b K
n
dz exp i z exp y cos k 2 2 2
K 1
2 2 zk sin k b K dy
And finally
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38
Expected Number of Level Crossings of a Random Trigonometric Polynomial (J4R/ Volume 01 / Issue 12 / 007)
( for fixed non-zero real constants A and B to be chosen .
6)
III. ESTIMATION OF THE INTEGRAL OF EQUATION (6) Consider the integral n u cos k k sin k I log K 1 Au B 2
2 b K 2
du
Which exists in general as a principal value if n
A2 =
bK2 cos2kθ
K 1
Let B2 =
k2bK2 sin2kθ
n
K 1
And C2 =
n
2 kbk coskθ sinkθ K 1
As in Das[2], letting bk = kp(p ≥ 0) we get
n 2 p2 L (n)
2 p2
C
n O L (n)
2
Say (β = constant) outside the set {0, π, 2π, … } of the values of θ, AB > C2. We have n u cos k k sin k I log K 1 Au B 2
2 b K 2
u 2 2c 2u b 2 log u 2 2 bu b 2 du
du
Where c=(C/A) and b=(B/A). Now by integration by parts, X
log u
2
2c u b 2
2
du u log u
2
2c u b 2
0
2
x
X
u
0
0
2u 2
2
c ub
X
2c u 2
2c u b 2
2
u
2
c ub
2
2 du = 2 X log X 2 c 2 X 2 2
0
2
1 du O 2c u b X 2
2
And 0
log u
2
2c u b 2
2
du
X
log u
X
2
2c u b 2
2
du
X
2
1 du O u 2c u b X 0
= 2 X log X 2 c 2 2 X 2
0
2
2
2
Therefore X
log u 2 c u b 2
2
2
du
X
4 X log X 4 X 2
X
X
c ub 2
2
u 2c u b 2
2
2
1 du O X
Again X
X
log u 2 bu b 2
2
du
X
4 X log X 4 X 2
X
bu b
2
1 du O u 2 bu b X 2
2
Hence the integral
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39
Expected Number of Level Crossings of a Random Trigonometric Polynomial (J4R/ Volume 01 / Issue 12 / 007) X
X
u 2 2c 2u b 2 log 2 2 u 2 bu b
c ub
X
2
X
2
2
X
2
2
X 2 c log X
du c log u 2 c u b 2 2
2
2
u 2c u b 2
X 2 2 bu b 1 c ub O du 2 du 2 2 2 2 2 2 u 2 bu b u 2 c u b X X X
2
2
2
1 u c 2 2 4 2 b c tan 2 4 x b c x
2 2 2 4 2c X b b c 2 4 1 2 X 2 b c tan 2 2 b2 X 2 2 c X b
x x
When X→∞, we have
c ub 2
2 2
u 2c u b 2
du b c 2
2
u
4
bu b
2
2 bu b
2
2
du 0
and O
1 X
=0
Therefore X
lim
I=
x
2 b
...=
X
2
c
4
u 2 2c 2 u b 2 du 2 2 u 2 bu b
log
1 2
A2B2 C 2 4 A
S ' 2 p 1 2 2p 3 1 n S
4
1 2
1 2
2 n
Where 2
n
4
2
2 p
1 2 p 3
SS ' ( L ( n ))
S~1
S’ ~ 1
2
Thus I ~
2 p 1 2 p 3
1 2
2 n 1 n
2
1 2
Now the result follows from the section 4 of [4] , choosing L(n) =log n. the cases where -1/2 < p < 0 and p= -1/2 can be similarly dealt with and results can be obtained to show that Das’s estimates are approached from below. We express our reverence and deep gratitude to Sir A.C.offord for his approval of the thesis [6] which presents this result, and for his valuable suggestions .
REFERENCES [1] [2] [3] [4]
Das, M.K. Real zeros of a random sum of orthogonal polynomials, Proc. Amer. Math. Soc. 27, 1 (1971), 147-153. Das, M.K. Real zeros of a class of random algebraic Polynomials, Journal of Indian Math. Soc.. 36 (1972), 53-63. Dunnage, J.E.A. The number of real zeros of a class of random algebraic polynomials (I), Proc. London Math. Soc. (3) 18 (1968), 439-460. Pratihari and Bhanja .On the number of real zeros of random trigonometric polynomial, Trans. Of Amer. Math. Soc. 238 (1978) 57-70.
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