Denote by NF(T) the number of zeros in the interval [0, T] of a real stationary Gaussian process F whose spectral measure is supported on [−A, −B] ∪ [B, A], with 0 ≤ B < A. We study linear deviations events for NF(T), namely η-overcrowding events <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$(\{N_{F}(T)>\eta T\}\,\text{for}\,\eta>\mathbb{E}N_{F}(1))$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mo stretchy="false">(</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mi>N</mi> <mrow> <mi>F</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>></mo> <mi>η</mi> <mi>T</mi> <mo fence="false" stretchy="false">}</mo> <mspace width="thinmathspace" /> <mtext>for</mtext> <mspace width="thinmathspace" /> <mi>η</mi> <mo>></mo> <mrow> <mi mathvariant="double-struck">E</mi> </mrow> <msub> <mi>N</mi> <mrow> <mi>F</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </math></EquationSource> </InlineEquation> and η-undercrowding events <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$(\{N_{F}(T)<\eta T\}\,\text{for}\,\eta<\mathbb{E}N_{F}(1))$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mo stretchy="false">(</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mi>N</mi> <mrow> <mi>F</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo><</mo> <mi>η</mi> <mi>T</mi> <mo fence="false" stretchy="false">}</mo> <mspace width="thinmathspace" /> <mtext>for</mtext> <mspace width="thinmathspace" /> <mi>η</mi> <mo><</mo> <mrow> <mi mathvariant="double-struck">E</mi> </mrow> <msub> <mi>N</mi> <mrow> <mi>F</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </math></EquationSource> </InlineEquation>. We show that, as T tends to infinity, η-overcrowding probability undergoes a transition from exponential decay to Gaussian decay at <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\eta = {A \over \pi}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mi>η</mi> <mo>=</mo> <mrow> <mfrac> <mi>A</mi> <mi>π</mi> </mfrac> </mrow> </math></EquationSource> </InlineEquation>, while η-undercrowding probability undergoes the reverse transition at <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\eta = {B \over \pi}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mi>η</mi> <mo>=</mo> <mrow> <mfrac> <mi>B</mi> <mi>π</mi> </mfrac> </mrow> </math></EquationSource> </InlineEquation>.

A sharp transition in zero overcrowding and undercrowding probabilities for stationary Gaussian processes

期刊论文
发表日期：

2025年12月3日

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Naomi Dvora Feldheim,
Ohad Noy Feldheim,
Lakshmi Priya M. E.

摘要

Denote by N_F(T) the number of zeros in the interval [0, T] of a real stationary Gaussian process F whose spectral measure is supported on [−A, −B] ∪ [B, A], with 0 ≤ B < A. We study linear deviations events for N_F(T), namely η-overcrowding events $(\{N_{F}(T)>\eta T\}\,\text{for}\,\eta>\mathbb{E}N_{F}(1))$ $({N_{F} (T) > η T} for η > E N_{F} (1))$ and η-undercrowding events $(\{N_{F}(T)<\eta T\}\,\text{for}\,\eta<\mathbb{E}N_{F}(1))$ $({N_{F} (T) < η T} for η < E N_{F} (1))$ . We show that, as T tends to infinity, η-overcrowding probability undergoes a transition from exponential decay to Gaussian decay at $\eta = {A \over \pi}$ $η = \frac{A}{π}$ , while η-undercrowding probability undergoes the reverse transition at $\eta = {B \over \pi}$ $η = \frac{B}{π}$ .