Real zeros of a random sum of orthogonal polynomials

Source abstract

Let c 0 , c 1 , c 2 , ⋯ {c_0},{c_1},{c_2}, \cdots be a sequence of normally distributed independent random variables with mathematical expectation zero and variance unity. Let P k ∗ ( x ) ( k = 0 , 1 , 2 , ⋯ ) P_k^ \ast (x)(k = 0,1,2, \cdots ) be the normalised Legendre polynomials orthogonal with respect to the interval ( − 1 , 1 ) ( - 1,1) . It is proved that the average number of the zeros of c 0 P 0 ∗ ( x ) + c 1 P 1 ∗ ( x ) + ⋯ + c n P n ∗ ( x ) {c_0}P_0^ \ast (x) + {c_1}P_1^ \ast (x) + \cdots + {c_n}P_n^ \ast (x) in the same interval is asymptotically equal to ( 3 ) − 1 / 2 n {(3)^{ - 1/2}}n when n n is large.