Uniform Asymptotics by Alexei

Empirical specification
$$log(Vol)={\alpha }_0+[{\beta }_1+{\beta }_{11}|M|+{\beta }_{12}log]$$
tensor in days
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$$N,T\to \infty$$

$$N,T\to \infty$$

$$T\ge N^{1-2\alpha }$$

$$\alpha \in (0,1/2)$$

$${\rho }^2=co{s}^2\angle (\hat{F},F)$$

$$\widehat{F_t}$$

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$$FN(0,1_T)$$

$$ϵN$$

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$$\hat{F}=F+small$$

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$$A=A_0+small$$

$$R=R_0(1+small{R}_0{)}^{-1}=R_0-R_0(small{R}_0)+R_0(small{R}_0{)}^2-...$$

$$P=P_0+small{P}^{(1)}+(small{)}^2{P}^{(2)}+...$$

$$P=\frac{1}{2\pi i}\oint R(z)dz$$

$$R(z)=\frac{1}{{\lambda }_1-z}P+\frac{1}{{\lambda }_2-z}{w}_2{v}_2^{\prime }+...$$

$$A=\frac{X{X}^{\prime }}{Tn}$$

$$A_0=F{F}^{\prime }/T$$

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$${\rho }^2=tr[P{P}_0]=1+\frac{1}{n}tr[P^{(1)}{P}_0]+\frac{1}{n^2}tr[P^{(2)}{P}_0]$$

$$tr[P^{(j)}]$$

$$\sqrt{T}(n({\rho }^2-1)-{\mu }_1)=\sqrt{T}(tr[P^{(1)}{P}_0]-{\mu }_1)+\frac{\sqrt{T}}{1}$$

$$\sqrt{T}<<n^j*\alpha >\frac{1}{2j}$$

$$X{X}^{\prime }$$

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$$X{X}^{\prime }$$

$$ϵ{ϵ}^{\prime }$$

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$$X{X}^{\prime }=lowrank+ϵ{ϵ}^{\prime }=g$$

$$(A+f{f}^{\prime }-z{)}^{-1}=R(z)-\frac{R(z)f{f}^{\prime }R(z)}{1+f^{\prime }R(z)f}$$

$$1+f^{\prime }R(\hat{\lambda })f=0$$

$$(f^{\prime }\hat{f}{)}^2=1/f^{\prime }R(\hat{\lambda })$$

$$\frac{X{X}^{\prime }}{\sqrt{NT}}=\frac{n+1}{\sqrt{NT}}F{F}^{\prime }+\frac{u{u}^{\prime }}{\sqrt{NT}}$$

$$A=u{u}^{\prime }/\sqrt{NT}$$

$$f=\frac{\sqrt{n+1}}{(NT{)}^{\frac{1}{4}}}$$

$$F^{\prime }\frac{R(\hat{\lambda })}{T}F$$

$$=-\frac{1}{\sqrt{c}(c+1)}$$

$$-\frac{1}{}$$

$${\lambda }_0=\frac{1}{\sqrt{c}}+\sqrt{c}+\frac{1}{\sqrt{c}n}+\sqrt{c}n$$

$$\frac{n}{n+1}\sqrt{\frac{T/2}{}}$$

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$$\sqrt{c}n-1>K^1$$

$$\sqrt{c}n-1>1$$

$$\sqrt{c}n-1<<1$$

$$\sqrt{c}n-1>>K^{-1/3}$$

$$\sqrt{c}n-1\sim K^{-1/3}$$

Three factors

critical

critical a weak factor

critical point

sorted
$${\rho }^2=\frac{(F^{\prime }R(\hat{\lambda })F{)}^2}{(F^{\prime }F)(F^{\prime }{R}^2(\hat{\lambda })F)}$$

$$c{n}^2-1>K^{-1/3+\omega }$$

$${\rho }^2/{\rho }_0^2\to 1$$

$${\rho }_0^2=\frac{1}{2}$$

$$1+\sum _{j=1}^{\infty }{\mu }_j/n^j={\rho }_0^2$$

$$\sqrt{T}n({\rho }^2-{\rho }_0^2)$$

$$T=250,N=1000$$

$$

$$

$$tr\frac{1}{T}R(z)=$$

$$local-semicircle-law$$

$$\alpha \in (0,1)$$

$$\alpha =-1/3$$

$$\sum \in \mathbb{R}$$

$$\alpha =-1/3$$

$$\sum \in \mathbb{R}$$

$$Convergence$$