TY - JOUR
T1 - Hole probability for entire functions represented by Gaussian Taylor series
AU - Nishry, Alon
N1 - Funding Information:
∗Research supported by the Israel Science Foundation of the Israel Academy of Sciences and Humanities, grants 171/07, 166/11 and by grant No 2006136 of the United States-Israel Binational Science Foundation.
PY - 2012/11
Y1 - 2012/11
N2 - Consider the Gaussian entire function f(z) = Σn=0∞ζnanzn, where {ζn} is a sequence of independent and identically distributed standard complex Gaussians and {an} is some sequence of non-negative coefficients, with a0 > 0. We study the asymptotics (for large values of r) of the hole probability for f (z), that is, the probability PH(r) that f(z) hcisk {{pipe}z{pipe} < r}. We prove that log PH(r) = -S(r) + o(S(r)), where S(r) = 2·Σn≥0log+(anrn) as r tends to ∞ outside a deterministic exceptional set of finite logarithmic measure.
AB - Consider the Gaussian entire function f(z) = Σn=0∞ζnanzn, where {ζn} is a sequence of independent and identically distributed standard complex Gaussians and {an} is some sequence of non-negative coefficients, with a0 > 0. We study the asymptotics (for large values of r) of the hole probability for f (z), that is, the probability PH(r) that f(z) hcisk {{pipe}z{pipe} < r}. We prove that log PH(r) = -S(r) + o(S(r)), where S(r) = 2·Σn≥0log+(anrn) as r tends to ∞ outside a deterministic exceptional set of finite logarithmic measure.
UR - http://www.scopus.com/inward/record.url?scp=84870685894&partnerID=8YFLogxK
U2 - 10.1007/s11854-012-0042-2
DO - 10.1007/s11854-012-0042-2
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AN - SCOPUS:84870685894
SN - 0021-7670
VL - 118
SP - 493
EP - 507
JO - Journal d'Analyse Mathematique
JF - Journal d'Analyse Mathematique
IS - 2
ER -