The level crossings of random sums

12/19/2020

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by   Christopher Corley, et al.

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Let {a_j}_j = 0^N and {b_j}_j = 0^N be sequences of mutually independent and identically distributed, real, normal random variables with mean zero and variances {σ_a_j^2}_j = 0^N and {σ_b_j^2}_j = 0^N. Let {f_j}_j = 0^N be a sequence of basis functions that are entire and real-valued on ℝ. For studying the number of times a random sum crosses a complex level, we establish an exact formula for the expected intensity of the complex roots of ∑_j = 0^N (a_j + i b_j) f_j (z) = K_1 + i K_2, where K_1 and K_2 are constants independent of z, and apply this formula to a standard Brownian motion.