Proof of Theorem 1

6.1.1 Reduction to pure random walk data.

Let G ( ) and ~G ( ) be distribution functions that may depend on p and T and are possibly random. We shall call them asymptotically equivalent if the a.s. weak convergence G ( ) ) F ( ) to some non-random d.f. F ( ) implies similar a.s. weak convergence for ~G( ); and vice versa. Let Si and ~S_i with i = 0; 1; 2 be, possibly random, matrices that may depend on p and T such that Si and ~S_iare a.s. positive de…nite for i = 0; 1: Below, we shall often refer to the following auxiliary lemma.

Lemma 10 If, almost surely, ¹_prank S_i S~_i ! 0 as p; T !^c1 for i = 0; 1; 2;

then G ( ) and ~G( ) are asymptoically equivalent, where G ( ) and ~G ( ) are the empirical d.f. of eigenvalues of S2S₁¹S₂⁰S₀¹ and ~S2S~₁¹S~₂⁰S~₀¹; respectively.

Proof of Lemma 10. Let R = rank S2S₁¹S₂⁰S₀¹ S~₂S~₁¹S~₂⁰S~₀¹ : The a.s.

convergence ¹_p rank S_i S~_i ! 0 implies the a.s. convergence R=p ! 0: On the other hand, by the rank inequality (Theorem A43 in Bai and Silverstein (2010)), L G; ~G R=p; where L G; ~G is the Lévy distance between G ( ) and ~G( ):

Recall that the Lévy distance metrizes the weak convergence. Therefore, the almost sure convergence L G; ~G ! 0 yields the asymptotic equivalence of G ( ) and G( ):~ respectively, without loss of generality. Furthermore, scaling Sij by T does not change the product S01S₁₁¹S₀₁⁰ S₀₀¹;and thus, in the rest of the proof, we shall work

Next, we show that, still without loss of generality, we may replace the data generated process (11) by pure random walk with zero initial value. Indeed, let X = [X _k+1; :::; X_T]; where X k+1; :::; X₀ are arbitrary and Xt with t 1 are generated by (11). Further, let ~X _k+1; :::; ~X₀ be zero vectors, ~X_t = X^t

s=1"_t for t 1; and ~X = [ ~X k+1; :::; ~XT]:

Lemma 11 rank X X~ 2 (r + rank + k + d_F) :

A proof of this lemma is given in the Supplementary Appendix. It is based on the representation of Xt as a function of the initial values, " and F (see Theorem 2.1 in Johansen (1995)), and requires only elementary algebraic manipulations.

Lemmas 11 and 10 together with the assumption (12) imply that replacing X_t and Xt 1 in (28) by X~_t and ~X_{t 1}; respectively, does not change the weak limit of Fp;T( ):Hence, in the rest of the proof of Theorem 1, without loss of generality, we shall assume that the data are generated by

X_t= "_t; t = 1; :::; T; with X0 = 0: (29) 6.1.2 Block-diagonalization

Assuming that ’s are the eigenvalues of S01S₁₁¹S₀₁⁰ S₀₀¹ with Sij satisfying (28) and (29), we can interpret them as the squared sample canonical correlations between lagged values of a random walk Xt 1 and its current innovations "t. Since the sample canonical correlations are invariant with respect to the multiplication of the data by any invertible matrix, we assume without loss of generality that the variance of "t equals = I_p=T: Further, we assume that T is even. The case of odd T can be analyzed similarly, and we omit it to save space.

Let " = ["1; :::; "_T] and let U be the upper-triangular matrix with ones above the main diagonal and zeros on the diagonal. Then "U = [X0; :::; X_{T 1}] ; so that

S₀₀ = ""⁰; S₀₁ = "U⁰"⁰; and S11= "U U⁰"⁰: (30) We shall show that the empirical d.f. of the ’s, Fp;T ( ) ;is asymptotically equiv-alent to the empirical d.f. ^F_p;T ( ) of eigenvalues of CD ¹C⁰A ¹;where

C = " ⁰₂"⁰; D = " ₁"⁰; and A = ""⁰;

1 is a diagonal matrix,

1 = diagn

r₁¹I₂; :::; r_{T =2}¹ I₂o

; (31)

and 2 is a block-diagonal matrix,

2 = diagn

r₁¹(R₁ I₂) ; :::; r_{T =2}¹ R_{T =2} I₂ o

: (32)

Here I2 is the 2-dimensional identity matrix, and rj; R_j are de…ned as follows. Let

= 2 =T: Then for j = 1; 2; :::

r_j+1 = 2 2 cos j ; R_j+1= cos j sin j sin j cos j

!

;

whereas r1 = 4; R₁ = I₂:

Lemma 12 The distribution functions Fp;T ( ) and ^F_p;T ( ) are asymptotically equivalent.

Proof of Lemma 12. Let V be the circulant matrix (see Golub and Van Loan (1996, p.201)) with the …rst column v = ( 1; 1; 0; :::; 0)⁰: Direct calculations show that U V = IT le⁰_T and V U = IT e₁l⁰; where ej is the j-th column of IT; and l is the vector of ones. Using these identities, it is straightforward to verify that

U = (V + e₁e⁰₁) ¹ le⁰₁; and (33)

U U⁰ = V⁰V (e₁ e_T) (e₁ e_T)⁰+ e_Te⁰_T ¹ ll⁰: (34) Now, let us de…ne

C₁ = " (U + le⁰₁)⁰"⁰ and D1 = " (U U⁰ + ll⁰) "⁰:

Using identities (30) for Sij and Lemma 10, we conclude that Fp;T( )is asymptot-ically equivalent to F_p;T⁽¹⁾( ), where F_p;T⁽¹⁾( ) is the empirical d.f. of the eigenvalues of C1D₁¹C₁⁰A ¹:

Further, (33) and (34) yield

C₁ = " (V + e₁e⁰₁) ¹"⁰ and

D₁ = " V⁰V (e₁ e_T) (e₁ e_T)⁰ + e_Te⁰_T ¹"⁰:

Applying Lemma 10 one more time, we obtain the asymptotic equivalence of F_p;T⁽¹⁾( ) and F_p;T⁽²⁾( ) ; where F_p;T⁽²⁾( ) is the empirical d.f. of the eigenvalues of C₂D₂¹C₂⁰A ¹ with

C2 = "V ¹"⁰ and D2 = " (V⁰V ) ¹"⁰: (35) As is well known (see, for example, Golub and Van Loan (1996), chapter 4.7.7), T T circulant matrices can be expressed in terms of the discrete Fourier transform

matrices

F = fexp (i (s 1) (t 1))g^Ts;t=1

with = 2 =T: Precisely, V = 1

TF diag (Fv) F; and V⁰V = 1

TF diag (Fw) F;

where w = (2; 1; 0; :::; 0; 1)⁰ and the star superscript denotes transposition and complex conjugation. For the s-th diagonal elements of diag (Fv) and diag (Fw) ; we have

diag (Fv)s = 1 + expfi (s 1)g ; and diag (Fw)s= 2 2 cos (s 1) : Note that diag (Fw)s= diag (Fw)T +2 s for s = 2; 3; ::: If T is even, as we assumed above, then there are T =2 1 pairs (s; T + 2 s), and there is one pair (1; T =2 + 1) that correspond to

diag (Fw)1 = 0; diag (Fw)T =2+1= 4:

De…ne a permutation matrix P so that the equal diagonal elements of P⁰diag (Fw) P are grouped in adjacent pairs. Precisely, let P = fp^stg, where

pst = 8>

<

>:

1 if t = 2s 1 for s = 1; :::; T =2

1 if t = 2 (T s + 2) mod T for s = T =2 + 1; :::; T 0 otherwise

and let W be the unitary matrix

W = I2 0

0 I_{T =2} Z

!

with Z = 1 p2

1 1

i i

!

;

where denotes the Kronecker product. Further, let Q = ^p1_TW P⁰F. As is easy to check, Q is an orthogonal matrix. Furthermore,

V = Q⁰ ₂¹ + 2e₁e⁰₁ Q; and V V⁰ = Q⁰ ₁¹ 4e₁e⁰₁ Q;

where 1 and 2 are as de…ned in (31) and (32). Combining this with (35) and using Lemma 10 once again, we obtain the asymptotic equivalence of F_p;T⁽²⁾( ) and

F_p;T⁽³⁾( ) ; where F_p;T⁽³⁾( ) is the empirical d.f. of the eigenvalues of C3D₃¹C₃⁰A ¹ with

C₃ = "Q⁰ ₂Q"⁰ and D3 = "Q⁰ ₁Q"⁰:

Because of the rotational invariance of the Gaussian distribution, the distributions of "Q⁰ and " are the same. Hence, F_p;T⁽³⁾( )is asymptotically equivalent to ^F_p;T ( ) ; and thus, ^F_p;T ( ) is asymptotically equivalent to Fp;T( ).

6.1.3 A system of equations for the Stieltjes transform

Our proof of the almost sure weak convergence of ^F_p;T( ) to the Wachter distrib-ution consists of showing that the Stieltjes transform of ^F_p;T ( ),

^

mp;T(z) =

Z 1

z

F^p;T (d ) ; (36)

almost surely converges pointwise in z 2 C⁺ =f : I > 0g to the Stieltjes trans-form m(z) of the Wachter distribution. To establish such a convergence, we show that, if m is a limit of ^m_p;T(z) along any subsequence of p; T !^c 1; then it must satisfy a system of equations with unique solution given by m(z). The almost sure convergence of ^F_p;T ( ) (and thus, also of Fp;T( )) to the Wachter distribution follows then from the Continuity Theorem for the Stieltjes transforms (see, for example, Corollary 1 in Geronimo and Hill (2003)).

We shall write ^m for the Stieltjes transform ^m_p;T(z) to simplify notation. Let M = CD ¹C⁰ zA and ~M = C⁰A ¹C zD: (37) Then by de…nition (36), ^m must satisfy the following equations

^

m = 1

ptr AM ¹ ; (38)

^

m = 1

ptrh

D ~M ¹i

: (39)

Let us study the above traces in detail. De…ne

"_(j)= ["_{2j 1}; "_2j] ; j = 1; :::; T =2:

We now show that the traces in (38) and (39) can be expressed as functions of the terms having form "⁰_(j) j"_(j); where j is independent from "(j): Then, we argue

that

"⁰_{(j) j}"_(j) 1

T tr [ _j] I₂

a.s. converge to zero, and use this fact to derive equations that the limit of ^m; if it exists, must satisfy.

First, consider (38). Note that 1

ptr AM ¹ = 1 p

XT =2 j=1

tr "⁰_(j)M ¹"_(j) : (40)

Let us introduce new notation:

1j = r_j¹I₂; _2j = r_j¹(R_j I₂) ; C_j = C "_(j) ⁰_2j"⁰_(j); D_j = D "_{(j) 1j}"⁰_(j); A_j = A "_(j)"⁰_(j); and Mj = C_jD_j ¹C_j⁰ zA_j: In addition, let

s_j = "⁰_(j)D_j¹"_(j); u_j = "⁰_(j)D_j ¹C_j⁰M_j ¹"_(j); v_j = "⁰_(j)M_j ¹"_(j); and

w_j = "⁰_(j)D_j¹C_j⁰M_j ¹C_jD_j¹"_(j):

A straightforward algebra that involves multiple use of the Sherman-Morrison-Woodbury formula (see Golub and Van Loan (1996), p.50)

(V + XW Y ) ¹ = V ¹ V ¹X W ¹+ Y V ¹X ¹Y V ¹; (41) and the identity

2j 0

2j = ⁰_{2j 2j} = _1j; (42)

establishes the following equality

"⁰_(j)M ¹"_(j)= v_j [v_j; u⁰_j] _j[v_j; u⁰_j]⁰; (43)

where

j =

1

1 zI2+ vj 1 1 zrj 0

2j + u⁰_j

1

1 zr_{j 2j} + u_{j 1 z}^z r_jI₂ s_j + w_j

! 1

:

A derivation of (43) can be found in the Supplementary Appendix.

We have the following lemma, where k k denotes the spectral norm. Its proof is given in the Supplementary Appendix.

The lemma yields an approximation to the right hand side of (43), which we use in (40) and (38) to obtain the following result.

Proposition 14 There exists > 0 such that, for any z with zero real part, Rz = 0, and the imaginary part satisfying Iz > ; we have

^

Proof of Proposition 14. Consider a 2 2 matrix ^S_j that is obtained from

"⁰_(j)M ¹"_(j) by replacing sj; vj; uj and wj in (43) with ^sI2; ^vI2; ^uI2; and ^wI2;

A simple algebra and the identity 2j + ⁰_2j = I₂ yield

where Rj is an orthogonal matrix, so that kr^{j 2j}k and r^{j 0}2j are clearly bounded uniformly in j: Therefore, the norm of matrix ~j almost surely remains bounded as p; T !^c1, uniformly in j: Regarding ^j; which appear in the denominator on the right hand side of (45), the Supplementary Appendix establishes the following result. surely remains bounded as p; T !^c 1, uniformly in j: Therefore, by Lemma 13,

"⁰_(j)M ¹"_(j) = ^S_j + o(1); (47) where o(1)^a:s:! 0 as p; T !^c1; uniformly in j:

A straightforward algebra reveals that

S^_j = ( ^w s^ r_j) ^v u^²

j

:

Using this in equations (47) and (40), we obtain

^

where, in the latter expression, the term corresponding to j = 0 is included in the o(1) term to take into account the special de…nition of r1:

As follows from Lemma 15 and the boundedness of ^s; ^u; ^v;and ^w;the derivative d

d'

f₁(')

(1 z) f₁(') + f₂(')

almost surely remains bounded by absolute value as p; T !^c 1; uniformly in '2 [0; 2 ] : Therefore

The statement of Proposition 14 now follows by noting that the latter integral is one quarter of the integral over [0; 2 ] :

A similar analysis of equation (39) gives us another proposition, describing ^m as function of ~s; ~u; ~v; and ~w; where

We omit the proof because it is very similar to that of Proposition 14.

Proposition 16 There exists > 0 such that, for any z with Rz = 0 and Iz > ;

we have

^ m = 1

2 c Z 2

0

g₁

(1 z) g₁+ g₂(')d' + o(1); where (48)

g₁ = ( ~w ~s 1) ~v u~²;

g₂(') = ~v 4 sin²' (~s + 1 u~ w) ;~ and o(1)! 0, as p; T !^{a:s c}1:

Although we now have two asymptotic equations for ^m; (44) and (48), they contain many unknowns: ^s; ^u; ^v; ^w; and the corresponding variables with tildes.

The following result establishes simple relationships between the unknowns with hats and tildes.

Lemma 17 We have the following three identities

^

u = ~u; z~v + ^s = ^w; and z^v + ~s = ~w: (49) Proof of Lemma 17. The identity ^u = ~uis established by the following sequence of equalities

T ^u = tr D ¹C⁰M ¹ = tr D ¹C⁰ CD ¹C⁰ zA ¹

= tr C zA (C⁰) ¹D

1

= tr C⁰ zD (C) ¹A ¹

= tr A ¹C C⁰A ¹C zD ¹ = tr A ¹C ~M ¹ = T ~u:

The relationship z~v + ^s = ^w is obtained as follows

T (z~v + ^s) = tr D ¹ zI_p C⁰A ¹CD ¹ zI_p ¹+ I_p

= tr D ¹ I_p+ C⁰A ¹CD ¹ C⁰A ¹CD ¹ zI_p ¹+ I_p

= tr D ¹ I_p DC ¹A (C⁰) ¹z

1

= tr D ¹C⁰ CD ¹C⁰ Az ¹CD ¹ = T ^w:

The identity z^v + ~s = ~w is obtained similarly to z~v + ^s = ^w by interchanging the roles of D; C and A; C⁰.

The identities (49) imply the following equality

(1 z) f1(') + f2(') = (1 z) g1+ g2(') :

We denote the reciprocal of the common value of the right and left hand sides of this equality as ^h (z; ') :A direct calculation shows that

^h (z; ') = (1 z) z~v^v u^² + z~v + 4 sin²' (z^v + ^u 1) ¹; (50)

and the asymptotic relationships (44) and (48) can be written in the following form ( m =^ _{2 c}¹ R2

0 ^h (z; ') z~v 4 sin²' ^v u^² d' + o(1)

^

m = _{2 c}¹ R2

0 ^h (z; ') ((z^v 1) ~v u^²) d' + o(1) : (51) This can be viewed as an asymptotic system of two equations with four unknowns:

^

m; ~v; ^v; and ^u: We shall now complete the system by establishing the other two asymptotic relationships connecting these unknowns.

Multiplying both sides of the identity

M A ¹ = CD ¹C⁰A ¹ zI_p (52)

by AM ¹; taking trace, dividing by p, and rearranging terms, we obtain 1 + z ^m = 1

ptr CD ¹C⁰M ¹ : (53)

Next, we analyze (53) similarly to the above analysis of (38). That is, …rst, we note that

1

ptr CD ¹C⁰M ¹ = 1 p

XT =2 j=1

tr ⁰_2j"⁰_(j)D ¹C⁰M ¹"_(j) : (54)

Then elementary algebra, based on the Sherman-Morrison-Woodbury formula (41), yields

"⁰_(j)D ¹C⁰M ¹"_(j) = r_j(r_jI₂+ s_j) ¹s_{j 2j} v_j v_j; u⁰_{j j} v_j; u⁰_j ⁰ (55) +r_j(r_jI₂+ s_j) ¹ u_j [u_j; w_j] _j v_j; u⁰_j ⁰ :

Multiplying both sides of (55) by ⁰_2jand replacing sj; u_j; v_j;and wj by ^sI₂; ^uI₂; ^vI₂; and ^wI₂;respectively, yields an asymptotic approximation to ⁰_2j"⁰_(j)D ¹C⁰M ¹"_(j); which can be used in (54) and (53) to produce the following result. Its proof, as well as the proof of (55), are given in the Supplementary Appendix.

Proposition 18 There exists > 0 such that, for any z with Rz = 0 and Iz > ;

One might think that the remaining asymptotic relationship can be obtained by using the identity

M D~ ¹ = C⁰A ¹CD ¹ zI_p; (57)

which parallels (52). Unfortunately, following this idea delivers a relationship equivalent to (56). Therefore, instead of using (57), we consider the identity

1

Then, we proceed as in the above analysis of (54) and (40) to obtain the remaining asymptotic relationship. The proof of the following proposition is given in the Supplementary Appendix.

Summing up the results in Propositions 14, 16, 18, and 19, the unknowns

^

m; ^v; ~v; and ^u must satisfy the following system of asymptotic equations 8>

6.1.4 Solving the system

Recall that the unknowns ^m; ^v; ~v;and ^uin the asymptotic relationships (61) depend on p; T: The de…nition (36) of ^mimplies that j ^mj is bounded by (Iz) ¹:Further, as shown in the proof of Proposition 14, ^u and ^v are a.s. bounded by absolute value, and it can be similarly shown that ~v is a.s. bounded by absolute value. Therefore, there exist a subsequence of p; T along which ^m; ^v; ~v; and ^u a.s. converge to some limits m; v; y; and u:

These limits must satisfy a non-asymptotic system of equations 8>

Let us consider, until further notice, only such z that have zero real part,Rz = 0, and the imaginary part satisfying Iz > ; for some > 0. Let us solve system (62) for m: Adding two times the last equation to the …rst one, and subtracting the second equation we obtain we have m = 0; which cannot be true because ^m is the Stieltjes transform of the empirical distribution of the squared canonical correlations, all of which lie between zero and one. Indeed, clearly, for any 0 1 and z with Rz = 0;

with y 6= 0 and u 6= 0 (if one of them equals zero, the other equals zero too, and m = 0 by the second equation of (62), which is impossible). Since u 6= 0; the last equation implies that v 6= 0 as well.

Further, subtracting from the third equation the sum of z times the second and u=v times the last equation, and using (64), we obtain

1 = 1 2 c

Z 2 0

h (z; ')u

v (2zv + u) (zv v 1) d': (65) This equation, together with the second equation of (62) yield

m = v (2zv + u 2)

(1 + v zv) (2zv + u): (66)

Next, for the integrand in the last equation of (62), we have h (z; ') 2v sin²' + u = 1

2 v

zv + u 1 (67)

+h (z; ')u 2

(1 z) v (2zv + u) + 2 (2zv + u 1)

zv + u 1 :

This assumes that

zv + u 16= 0: (68)

If not, then

h (z; ') = (1 z) zvy u² + zy ¹

would not depend on ' and the last equation of (62) would imply that u + v = 0: The latter equation and the equality zv + u 1 = 0 would yield v = (1 z) ¹;which when combined with the second equation of (62) would give us m = c ¹(1 z) ¹; which cannot be true because m; being a limit of ^m, must satisfy Im 0 forIz > 0:

Equations (65), (67), and the last equation of (62) imply that

u = 2c

2c 1 (1 z) v (1 c) 2zv: (69)

Combining this with (66) yields

m = v1 c

c : (70)

Finally, elementary calculations given in the Supplementary Appendix show

The latter inclusion holds because otherwise h (z; ') is not a bounded function of '; which would contradict Lemma 15.

Using (69) in (72), and simplifying, we …nd that there exist only three possi-bilities. Either Equation (73) cannot hold because otherwise, (70) would imply that Im < 0;

which is impossible as argued above. Equation (74) taken together with (69) implies that

u + zv 1 = 0;

which was ruled out above. This leaves us with (75), so that, using (70), we get

m = (z c cz) q

(z c cz)² 4c (1 z) z

2z (1 z) c : (76)

For z 2 C⁺ with Rz = 0; the imaginary part of the right hand side of (76) is

negative when ‘ ’is used in front of the square root. Here we choose the branch of the square root, with the cut along the positive real semi-axis, which has positive imaginary part. Since Im cannot be negative, we conclude that

m = (z c cz) + q

(z c cz)² 4c (1 z) z

2z (1 z) c : (77)

But the right hand side of the above equality is the value of the limit of the Stieltjes transforms of the eigenvalues of the multivariate beta matrix Bp(p; (T p) =2) as p; T !^c 1: This can be veri…ed directly by using the formula for such a limit, given for example in Theorem 1.6 of Bai, Hu, Pan and Zhou (2015). As follows from Wachter (1980), the weak limit of the empirical distribution of the eigen-values of the multivariate beta matrix Bp(p; (T p) =2) as p; T !^c 1 equals W ( ; c= (1 + c) ; 2c= (1 + c)).

Equation (77) shows that, for z with Rz = 0 and Iz > ; any converging subsequence of ^m converges to the same limit. Hence, ^m a.s. converges for all z withRz = 0 and Iz > : Note that ^mis a sequence of bounded analytic functions in the domain fz : Iz > g ; where is an arbitrary positive number. Therefore, by Vitaly’s convergence theorem (see Titchmarsh (1939), p.168) ^m a.s. converges to m; described by (77), for any z 2 C⁺: The almost sure convergence of ^F_p;T ( ) (and thus, also of Fp;T ( )) to the Wachter distribution follows from the Continuity Theorem for the Stieltjes transforms (see, for example, Corollary 1 in Geronimo and Hill (2003)).

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