scholarly journals Lower bound for the number of real roots of a random algebraic polynomial

1983 ◽  
Vol 35 (1) ◽  
pp. 18-27 ◽  
Author(s):  
M. N. Mishra ◽  
N. N. Nayak ◽  
S. Pattanayak
Keyword(s):  
Lower Bound ◽  
Algebraic Polynomial ◽  
Random Variables ◽  
Domain Of Attraction ◽  
Independent Random Variables ◽  
Real Numbers ◽  
Exceptional Set ◽  
Real Roots ◽  
Normal Law ◽  
Number Of Real Roots

AbstractLet X1, X2, …, Xn be identically distributed independent random variables belonging to the domain of attraction of the normal law, have zero means and Pr{Xr ≠ 0} > 0. Suppose a0, a1, …, an are non-zero real numbers and max and εn is such that as n → ∞, εn. If Nn be the number of real roots of the equation then for n > n0, Nn > εn log n outside an exceptional set of measure at most provided limn→∞ (kn/tn) is finite.

scholarly journals On the number of real roots of a random algebraic equation

1993 ◽  
Vol 54 (1) ◽  
pp. 86-96
Author(s):  
D. Pratihari ◽  
R. K. Panda ◽  
B. P. Pattanaik
Keyword(s):  
Algebraic Equation ◽  
Positive Constant ◽  
Random Variables ◽  
Domain Of Attraction ◽  
Real Numbers ◽  
Real Roots ◽  
Normal Law ◽  
Image Position ◽  
Number Of Real Roots ◽  
Independent Identically Distributed

AbstractLet Nn(ω) be the number of real roots of the random algebraic equation Σnv = 0 avξv (ω)xv = 0, where the ξv(ω)'s are independent, identically distributed random variables belonging to the domain of attraction of the normal law with mean zero and P{ξv(ω) ≠ 0} > 0; also the av 's are nonzero real numbers such that (kn/tn) = 0(log n) where kn = max0≤v≤n |av| and tn = min0≤v≤n |av|. It is shown that for any sequence of positive constants (εn, n ≥ 0) satisfying εn → 0 and ε2nlog n → ∞ there is a positive constant μ so that for all n0 sufficiently large.

On the Number of Real Roots of a Random Algebraic Equation

1962 ◽  
Vol 58 (3) ◽  
pp. 433-442 ◽  
Author(s):  
G. Samal
Keyword(s):  
Algebraic Equation ◽  
Random Variables ◽  
Independent Random Variables ◽  
Absolute Moment ◽  
Exceptional Set ◽  
Real Roots ◽  
Image Position ◽  
Number Of Real Roots

ABSTRACTLet Nn be the number of real roots of a random algebraic equation The coefficients ξν are independent random variables identically distributed with expectation zero; the variance and third absolute moment are finite and non-zero. It is proved thatwhere εν tends to zero, but εν log n tends to infinity. The measure of the exceptional set tends to zero as n tends to infinity.

scholarly journals On the Lower Bound for the Number of Real Roots of a Random Algebraic Equation

2007 ◽  
Vol 2007 ◽  
pp. 1-8
Author(s):  
Takashi Uno
Keyword(s):  
Lower Bound ◽  
Algebraic Equation ◽  
Random Variables ◽  
Real Roots ◽  
Number Of Real Roots

We estimate a lower bound for the number of real roots of a random alegebraic equation whose random coeffcients are dependent normal random variables.

scholarly journals Marcinkiewicz-type strong law of large numbers for double arrays of pairwise independent random variables

1999 ◽  
Vol 22 (1) ◽  
pp. 171-177 ◽  
Author(s):  
Dug Hun Hong ◽  
Seok Yoon Hwang
Keyword(s):  
Law Of Large Numbers ◽  
Double Sequence ◽  
Random Variables ◽  
Independent Random Variables ◽  
Real Numbers ◽  
Strong Law ◽  
One Dimensional ◽  
Large Numbers ◽  
Pairwise Independent Random Variables ◽  
The One

Let {Xij}be a double sequence of pairwise independent random variables. If P{|Xmn|≥t}≤P{|X|≥t}for all nonnegative real numbers tandE|X|p(log+|X|)3<∞, for1<p<2, then we prove that∑i=1m∑j=1n(Xij−EXij)(mn)1/p→0    a.s.   as  m∨n→∞.                                     (0.1)Under the weak condition ofE|X|plog+|X|<∞, it converges to 0inL1. And the results can be generalized to anr-dimensional array of random variables under the conditionsE|X|p(log+|X|)r+1<∞,E|X|p(log+|X|)r−1<∞, respectively, thus, extending Choi and Sung's result [1] of the one-dimensional case.

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