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❇❛❤❛❞✉r ❡✣❝✐❡♥❝② ♦❢ ❛ ❣♦♦❞♥❡ss✲♦❢✲✜t t❡sts ❜❛s❡❞ ♦♥ ❞✐str✐❜✉t✐♦♥ ❝❤❛r❛❝t❡r✐③❛t✐♦♥s✳

❱♦❧❦♦✈❛ ❑s❡♥✐❛ ❙❛✐♥t✲P❡t❡rs❜✉r❣ ❯♥✐✈❡rs✐t②✱ ❘✉ss✐❛ ▼❛r❝❤ ✶✺✱ ✷✵✶✵

❇❛❤❛❞✉r ❡✣❝✐❡♥❝② ♦❢ ❛ ❣♦♦❞♥❡ss✲♦❢✲✜t t❡sts ❜❛s❡❞ ♦♥ ❞✐str✐❜✉t✐♦♥ ❝❤❛r❛❝t❡r✐③❛t✐♦♥s✳ ■♥tr♦❞✉❝t✐♦♥✳

■♥tr♦❞✉❝t✐♦♥

■♥ t❤✐s t❛❧❦ ✇❡ ✇✐❧❧ ❝♦♥s✐❞❡r t✇♦ ❡①❛♠♣❧❡s ♦❢ ❣♦♦❞♥❡ss✲♦❢✲✜t t❡sts✿ t❡sts ♦❢ ♥♦r♠❛❧✐t② ❛♥❞ t❡sts ♦❢ ❡①♣♦♥❡♥t✐❛❧✐t②✳ ❋♦r ❡❛❝❤ ❡①❛♠♣❧❡ ✇❡ ✇✐❧❧ ❉❡s❝r✐❜❡ t❤❡ ❧✐♠✐t✐♥❣ ❞✐str✐❜✉t✐♦♥s ♦❢ st❛t✐st✐❝s ✉♥❞❡r ♥✉❧❧ ❤②♣♦t❤❡s✐s✳ ❋✐♥❞ t❤❡✐r ❧♦❣❛r✐t❤♠✐❝ ❧❛r❣❡ ❞❡✈✐❛t✐♦♥ ❛s②♠♣t♦t✐❝s ✉♥❞❡r ♥✉❧❧ ❤②♣♦t❤❡s✐s✳ ❈❛❧❝✉❧❛t❡ t❤❡ ❧♦❝❛❧ ❇❛❤❛❞✉r ❡✣❝✐❡♥❝② ♦❢ st❛t✐st✐❝s ✉♥❞❡r s♦♠❡ ❝♦♠♠♦♥ ♣❛r❛♠❡tr✐❝ ❛❧t❡r♥❛t✐✈❡s✳

❇❛❤❛❞✉r ❡✣❝✐❡♥❝② ♦❢ ❛ ❣♦♦❞♥❡ss✲♦❢✲✜t t❡sts ❜❛s❡❞ ♦♥ ❞✐str✐❜✉t✐♦♥ ❝❤❛r❛❝t❡r✐③❛t✐♦♥s✳ ❚❡sts ♦❢ ♥♦r♠❛❧✐t② ❜❛s❡❞ ♦♥ ❙❤❡♣♣ ♣r♦♣❡rt②✱ ❛♥❞ t❤❡✐r ❡✣❝✐❡♥❝✐❡s✳

❚❡sts ♦❢ ♥♦r♠❛❧✐t② ❜❛s❡❞ ♦♥ ❙❤❡♣♣ ♣r♦♣❡rt②✱ ❛♥❞ t❤❡✐r ❡✣❝✐❡♥❝✐❡s

■♥ ✶✾✻✹ ▲✳❙❤❡♣♣ ❞✐s❝♦✈❡r❡❞ t❤❛t ✐❢ X ❛♥❞ Y ❛r❡ t✇♦ ✐♥❞❡♣❡♥❞❡♥t ❝❡♥tr❡❞ ♥♦r♠❛❧ r✈✬s ✇✐t❤ s♦♠❡ ✈❛r✐❛♥❝❡ σ2 > 0, t❤❡♥ t❤❡ r✈ k(X, Y ) := 2XY/ ♣ X2 + Y 2 ❤❛s ❛❣❛✐♥ t❤❡ ♥♦r♠❛❧ ❞✐str✐❜✉t✐♦♥ N(0, σ2) ✭❙❤❡♣♣ ♣r♦♣❡rt②✳✮ ❏✳●❛❧❛♠❜♦s ❛♥❞ ■✳❙✐♠♦♥❡❧❧✐ ♣r♦✈❡❞ ✐♥ ✷✵✵✸✱ t❤❛t t❤❡ ❙❤❡♣♣ ♣r♦♣❡rt② ✐♠♣❧✐❡s t❤❡ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ t❤❡ ♥♦r♠❛❧ ❧❛✇ ✐♥ ❛ ❜r♦❛❞ ❝❧❛ss ♦❢ ❞✐str✐❜✉t✐♦♥s✳ ❈♦♥s✐❞❡r t❤❡ ❝❧❛ss F, ♦❢ ❞❢✬s F s❛t✐s❢②✐♥❣ t❤❡ ❝♦♥❞✐t✐♦♥s✿ ✐✮ 0 < F(0) < 1 ❛♥❞ ✐✐✮ F(x) − F(−x) ✐s r❡❣✉❧❛r❧② ✈❛r②✐♥❣ ✐♥ ③❡r♦ ✇✐t❤ t❤❡ ❡①♣♦♥❡♥t ✶✳ ❚❤❡♦r❡♠ ▲❡t X ❛♥❞ Y ❜❡ ✐♥❞❡♣❡♥❞❡♥t r✈✬s ✇✐t❤ ❝♦♠♠♦♥ ❞❢ F ❢r♦♠ t❤❡ ❝❧❛ss F. ❚❤❡♥ t❤❡ ❡q✉❛❧✐t② ✐♥ ❞✐str✐❜✉t✐♦♥ X

d

= k(X, Y ) ✐s ✈❛❧✐❞ ✐✛ X ∈ N(0, σ2) ❢♦r s♦♠❡ ✈❛r✐❛♥❝❡ σ2 > 0.

❇❛❤❛❞✉r ❡✣❝✐❡♥❝② ♦❢ ❛ ❣♦♦❞♥❡ss✲♦❢✲✜t t❡sts ❜❛s❡❞ ♦♥ ❞✐str✐❜✉t✐♦♥ ❝❤❛r❛❝t❡r✐③❛t✐♦♥s✳ ❚❡sts ♦❢ ♥♦r♠❛❧✐t② ❜❛s❡❞ ♦♥ ❙❤❡♣♣ ♣r♦♣❡rt②✱ ❛♥❞ t❤❡✐r ❡✣❝✐❡♥❝✐❡s✳

■♥tr♦❞✉❝t✐♦♥

▲❡t X1, . . . , Xn ❜❡ ✐♥❞❡♣❡♥❞❡♥t ♦❜s❡r✈❛t✐♦♥s ✇✐t❤ ③❡r♦ ♠❡❛♥ ❛♥❞ ❞❢ F, ❛♥❞ ❧❡t Fn ❜❡ t❤❡ ✉s✉❛❧ ❡♠♣✐r✐❝❛❧ ❞❢ ❜❛s❡❞ ♦♥ t❤✐s s❛♠♣❧❡✳ ❲❡ ❛r❡ ✐♥t❡r❡st❡❞ ✐♥ t❡st✐♥❣ t❤❡ ❝♦♠♣♦s✐t❡ ❤②♣♦t❤❡s✐s H0 : F ∈ N(0, σ2) ❢♦r s♦♠❡ ✉♥❦♥♦✇♥ ✈❛r✐❛♥❝❡ σ2 > 0 ❛❣❛✐♥st t❤❡ ❛❧t❡r♥❛t✐✈❡s H1, ✉♥❞❡r ✇❤✐❝❤ t❤❡ ❤②♣♦t❤❡s✐s H0 ✐s ❢❛❧s❡✳ ❲❡ ❜✉✐❧❞ t❤❡ U✲❡♠♣✐r✐❝❛❧ ❞❢ Hn ✉s✐♥❣ t❤❡ ❢♦r♠✉❧❛ Hn(t) = (C2

n)−1

❳

1≤i<j≤n

✶{k(Xi, Xj) < t}, t ∈ R1. ❈♦♥s✐❞❡r t✇♦ st❛t✐st✐❝s✿ In = ❩

R1(Hn(t) − Fn(t))dFn(t),

Dn = sup

t∈R1 |Hn(t) − Fn(t)|.

❇❛❤❛❞✉r ❡✣❝✐❡♥❝② ♦❢ ❛ ❣♦♦❞♥❡ss✲♦❢✲✜t t❡sts ❜❛s❡❞ ♦♥ ❞✐str✐❜✉t✐♦♥ ❝❤❛r❛❝t❡r✐③❛t✐♦♥s✳ ❚❡sts ♦❢ ♥♦r♠❛❧✐t② ❜❛s❡❞ ♦♥ ❙❤❡♣♣ ♣r♦♣❡rt②✱ ❛♥❞ t❤❡✐r ❡✣❝✐❡♥❝✐❡s✳

▲✐♠✐t✐♥❣ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ st❛t✐st✐❝ In

■t ✐s ✇❡❧❧✲❦♥♦✇♥ t❤❛t ♥♦♥✲❞❡❣❡♥❡r❛t❡ U✲ ❛♥❞ V ✲st❛t✐st✐❝s ❛r❡ ❛s②♠♣t♦t✐❝❛❧❧② ♥♦r♠❛❧ ✭❍♦❡✛❞✐♥❣✱ ✶✾✹✽✮✳ ▲❡t s❤♦✇ t❤❛t In ❜❡❧♦♥❣s t♦ t❤✐s ❝❧❛ss✳ ❚❤❡ st❛t✐st✐❝ In ✐s ❛s②♠♣t♦t✐❝❛❧❧② ❡q✉✐✈❛❧❡♥t t♦ t❤❡ U✲st❛t✐st✐❝ ♦❢ ❞❡❣r❡❡ ✸ ✇✐t❤ ❝❡♥t❡r❡❞ ❦❡r♥❡❧ Ψ(x, y, z) = 1 3 (✶{k(x, y) < z} + ✶{k(x, z) < y} + ✶{k(y, z) < x})−1/2. ✭✶✮ ▲❡t ❝❛❧❝✉❧❛t❡ t❤❡ ♣r♦❥❡❝t✐♦♥ ♦❢ t❤❡ ❦❡r♥❡❧ ✭✶✮ ψ(x) := E(Ψ(X, Y, Z)|X = x) = = 1 3 (P(k(x, Y ) < Z) + P(k(x, Z) < Y ) + P(k(Y, Z) < x)) − 1/2. ❆❢t❡r s♦♠❡ ❝❛❧❝✉❧❛t✐♦♥s ✇❡ ♦❜t❛✐♥ t❤❛t t❤❡ ♣r♦❥❡❝t✐♦♥ ♦❢ t❤❡ ❦❡r♥❡❧ Ψ ✐s ψ(x) = 1

3(Φ(x) − 1 2) ✇✐t❤ t❤❡ ✈❛r✐❛♥❝❡ σ2 =

❘

R1 ψ2(x)dΦ(x) = 1 108.

❍❡♥❝❡ t❤❡ ❦❡r♥❡❧ ✭✶✮ ✐s ♥♦♥✲❞❡❣❡♥❡r❛t❡✳ ❇② ❍♦❡✛❞✐♥❣✬s t❤❡♦r❡♠ √nIn

d

− → N(0, 1 12).

❇❛❤❛❞✉r ❡✣❝✐❡♥❝② ♦❢ ❛ ❣♦♦❞♥❡ss✲♦❢✲✜t t❡sts ❜❛s❡❞ ♦♥ ❞✐str✐❜✉t✐♦♥ ❝❤❛r❛❝t❡r✐③❛t✐♦♥s✳ ❚❡sts ♦❢ ♥♦r♠❛❧✐t② ❜❛s❡❞ ♦♥ ❙❤❡♣♣ ♣r♦♣❡rt②✱ ❛♥❞ t❤❡✐r ❡✣❝✐❡♥❝✐❡s✳

▲✐♠✐t✐♥❣ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ st❛t✐st✐❝ Dn

❚❤❡ r✈ Hn(t) − Fn(t) ❢♦r ✜①❡❞ t ✐s ❛s②♠♣t♦t✐❝❛❧❧② ❡q✉✐✈❛❧❡♥t t♦ U✲st❛t✐st✐❝ ✇✐t❤ t❤❡ ❦❡r♥❡❧ ❞❡♣❡♥❞✐♥❣ ♦♥ t ∈ R1 Ξ(x, y; t) = ✶{k(x, y) < t} − 1 2(✶{x < t} + ✶{y < t}). ❚❤❡ ♣r♦❥❡❝t✐♦♥ ♦❢ t❤✐s ❦❡r♥❡❧ ❢♦r ✜①❡❞ t ✐s ❡q✉❛❧ t♦ ξ(x; t) = ✽ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❁ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ✿ − 1

2(✶{x < t} + Φ(t)), ✐❢ t < −2|x|;

Φ ✒

t|x|

√

4x2−t2

✓ − 1

2(✶{x < t} + Φ(t)), ✐❢ − 2|x| ≤ t ≤ 2|x|;

1 − 1

2(✶{x < t} + Φ(t)), ✐❢ t > 2|x|.

❇❛❤❛❞✉r ❡✣❝✐❡♥❝② ♦❢ ❛ ❣♦♦❞♥❡ss✲♦❢✲✜t t❡sts ❜❛s❡❞ ♦♥ ❞✐str✐❜✉t✐♦♥ ❝❤❛r❛❝t❡r✐③❛t✐♦♥s✳ ❚❡sts ♦❢ ♥♦r♠❛❧✐t② ❜❛s❡❞ ♦♥ ❙❤❡♣♣ ♣r♦♣❡rt②✱ ❛♥❞ t❤❡✐r ❡✣❝✐❡♥❝✐❡s✳

▲✐♠✐t✐♥❣ ❞✐str✐❜✉t✐♦♥ ♦❢ st❛t✐st✐❝ Dn

❚❤❡ ✈❛r✐❛♥❝❡ ♦❢ t❤❡ ♣r♦❥❡❝t✐♦♥ ❤❛s t❤❡ ❢♦r♠ σ2

ξ(t) = Eξ2(X; t) =

✽ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❁ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ✿ 2 ❘ ∞

t/2 Φ2

✒

tx

√

4x2−t2

✓ dΦ(x) − ❘ t

t/2 Φ

✒

tx

√

4x2−t2

✓ dΦ(x)− − 1

4Φ(t) − 1 4Φ2(t) + Φ(t/2) − 1 2, ✐❢ t ≥ 0;

2 ❘ ∞

−t/2 Φ2

✒

tx

√

4x2−t2

✓ dΦ(x) − ❘ ∞

−t Φ

✒

tx

√

4x2−t2

✓ dΦ(x)+ + 1

4Φ(t) − 1 4Φ2(t), ✐❢ t < 0.

❚❛❦✐♥❣ ✐♥ ❝♦♥s✐❞❡r❛t✐♦♥ t❤❡ s②♠♠❡tr② ♦❢ t❤❡ ❢✉♥❝t✐♦♥ σ2

ξ(t)✱ ✇❡ ❝♦♥❝❧✉❞❡ t❤❛t

supt∈R σ2

ξ(t) = σ2 ξ(0) = 1

• 16. ❚❤✐s ✈❛❧✉❡ ✐s

✐♠♣♦rt❛♥t ✇❤❡♥ ❝❛❧❝✉❧❛t✐♥❣ ❧❛r❣❡ ❞❡✈✐❛t✐♦♥ ❛s②♠♣t♦t✐❝s✳

❇❛❤❛❞✉r ❡✣❝✐❡♥❝② ♦❢ ❛ ❣♦♦❞♥❡ss✲♦❢✲✜t t❡sts ❜❛s❡❞ ♦♥ ❞✐str✐❜✉t✐♦♥ ❝❤❛r❛❝t❡r✐③❛t✐♦♥s✳ ❚❡sts ♦❢ ♥♦r♠❛❧✐t② ❜❛s❡❞ ♦♥ ❙❤❡♣♣ ♣r♦♣❡rt②✱ ❛♥❞ t❤❡✐r ❡✣❝✐❡♥❝✐❡s✳

▲✐♠✐t✐♥❣ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ st❛t✐st✐❝ Dn

▲✐♠✐t✐♥❣ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ st❛t✐st✐❝ Dn ✐s ✉♥❦♥♦✇♥✳ ❯s✐♥❣ t❤❡ ♠❡t❤♦❞s ♦❢ ❙✐❧✈❡r♠❛♥ ✭✶✾✽✸✮✱ ♦♥❡ ❝❛♥ s❤♦✇ t❤❛t t❤❡ U✲❡♠♣✐r✐❝❛❧ ♣r♦❝❡ss ηn(t) = √n (Hn(t) − Fn(t)) , t ∈ R1, ❝♦♥✈❡r❣❡s ✇❡❛❦❧② ❛s n → ∞ t♦ s♦♠❡ ❝❡♥t❡r❡❞ ●❛✉ss✐❛♥ ♣r♦❝❡ss η(t) ✇✐t❤ ❝♦♠♣❧✐❝❛t❡❞ ❝♦✈❛r✐❛♥❝❡✳ ❈♦♥s❡q✉❡♥t❧② t❤❡ s❡q✉❡♥❝❡ ♦❢ st❛t✐st✐❝s √nDn ❝♦♥✈❡r❣❡s ✐♥ ❞✐str✐❜✉t✐♦♥ t♦ supt |η(t)|, ✇❤✐❝❤ ❤❛s ✈❡r② ❝♦♠♣❧✐❝❛t❡❞ ❞✐str✐❜✉t✐♦♥ ✭❝✉rr❡♥t❧② ✉♥❦♥♦✇♥✮✳ ❇✉t t❤❡ ❝r✐t✐❝❛❧ ✈❛❧✉❡s ❢♦r st❛t✐st✐❝s Dn ❝❛♥ ❜❡ ❢♦✉♥❞ ✈✐❛ s✐♠✉❧❛t✐♥❣ t❤❡✐r s❛♠♣❧❡ ❞✐str✐❜✉t✐♦♥✳

❇❛❤❛❞✉r ❡✣❝✐❡♥❝② ♦❢ ❛ ❣♦♦❞♥❡ss✲♦❢✲✜t t❡sts ❜❛s❡❞ ♦♥ ❞✐str✐❜✉t✐♦♥ ❝❤❛r❛❝t❡r✐③❛t✐♦♥s✳ ❚❡sts ♦❢ ♥♦r♠❛❧✐t② ❜❛s❡❞ ♦♥ ❙❤❡♣♣ ♣r♦♣❡rt②✱ ❛♥❞ t❤❡✐r ❡✣❝✐❡♥❝✐❡s✳

▲❛r❣❡ ❞❡✈✐❛t✐♦♥s ❛♥❞ ❧♦❝❛❧ ❡✣❝✐❡♥❝② ♦❢ st❛t✐st✐❝s In

❚❤❡ ❦❡r♥❡❧ Ψ ✐s ♥♦t ♦♥❧② ❝❡♥t❡r❡❞ ❜✉t ❜♦✉♥❞❡❞✳ ❚❤❡r❡❢♦r❡ ❢r♦♠ t❤❡♦r❡♠ ♦❢ ◆✐❦✐t✐♥ ❛♥❞ P♦♥✐❦❛r♦✈✭✶✾✾✾✮ ❞❡s❝r✐❜✐♥❣ ❧❛r❣❡ ❞❡✈✐❛t✐♦♥s ♦❢ ♥♦♥✲❞❡❣❡♥❡r❛t❡ U✲ ❛♥❞ V ✲st❛t✐st✐❝s ✇❡ ❤❛✈❡ ❢♦r a > 0 lim

n→∞ n−1 ln P(In > a) = −f(a),

✇❤❡r❡ t❤❡ ❢✉♥❝t✐♦♥ f ✐s ❝♦♥t✐♥✉♦✉s ❢♦r s✉✣❝✐❡♥t❧② s♠❛❧❧ a > 0, ❛♥❞✱ ♠♦r❡♦✈❡r✱ f(a) = 6a2(1 + o(1)), ❛s a → 0.

❇❛❤❛❞✉r ❡✣❝✐❡♥❝② ♦❢ ❛ ❣♦♦❞♥❡ss✲♦❢✲✜t t❡sts ❜❛s❡❞ ♦♥ ❞✐str✐❜✉t✐♦♥ ❝❤❛r❛❝t❡r✐③❛t✐♦♥s✳ ❚❡sts ♦❢ ♥♦r♠❛❧✐t② ❜❛s❡❞ ♦♥ ❙❤❡♣♣ ♣r♦♣❡rt②✱ ❛♥❞ t❤❡✐r ❡✣❝✐❡♥❝✐❡s✳

▲❛r❣❡ ❞❡✈✐❛t✐♦♥s ❛♥❞ ❧♦❝❛❧ ❡✣❝✐❡♥❝② ♦❢ st❛t✐st✐❝s In

▲❡t ❝❛❧❝✉❧❛t❡ ❧♦❝❛❧ ❇❛❤❛❞✉r ❡✣❝✐❡♥❝② ♦❢ st❛t✐st✐❝s In ❢♦r s♦♠❡ ❛❧t❡r♥❛t✐✈❡s✳ ▲♦❝❛t✐♦♥ ❛❧t❡r♥❛t✐✈❡✳ ❯♥❞❡r H1 t❤❡ ♦❜s❡r✈❛t✐♦♥s ❤❛✈❡ t❤❡ ❞❢ Φ(x − θ), θ ≥ 0. ❯s✐♥❣ t❤❡ ▲❛✇ ♦❢ ▲❛r❣❡ ◆✉♠❜❡rs ❢♦r U✲st❛t✐st✐❝s✱ ✇❡ s❡❡ t❤❛t t❤❡ ❧✐♠✐t ✐♥ ♣r♦❜❛❜✐❧✐t② ♦❢ In ✉♥❞❡r H1 ✐s ❡q✉❛❧ t♦ bI(θ) = Pθ(k(X, Y ) < Z) − 1 2. ❍❡♥❝❡✱ ❛s θ → 0 ✇❡ ❤❛✈❡ bI(θ) ∼ (2√π)−1θ. ❚❤❡r❡❢♦r❡ t❤❡ ❧♦❝❛❧ ❡①❛❝t s❧♦♣❡ ♦❢ t❤❡ s❡q✉❡♥❝❡ In ❛s θ → 0 ❛❞♠✐ts t❤❡ r❡♣r❡s❡♥t❛t✐♦♥ cI(θ) ∼ 12b2

I(θ) ∼ 3

π θ2 ∼ 0.955 θ2. ❚❤❡ t❤❡♦r❡t✐❝❛❧ ✉♣♣❡r ❜♦✉♥❞ ✐s 2K(θ) ∼ θ2 ❛s θ → 0 ✭▼✉❧✐❡r❡✱ ◆✐❦✐t✐♥✱ ✷✵✵✷✮✳ ■t ❢♦❧❧♦✇s t❤❛t t❤❡ ❧♦❝❛❧ ❇❛❤❛❞✉r ❡✣❝✐❡♥❝② ♦❢ ♦✉r t❡st ✐s r❛t❤❡r ❤✐❣❤ ❛♥❞ ✐s ❡q✉❛❧ t♦ eff B(I) := lim

θ→0{cI(θ)/2K(θ)} = 3/π ≈ 0.9549.

❇❛❤❛❞✉r ❡✣❝✐❡♥❝② ♦❢ ❛ ❣♦♦❞♥❡ss✲♦❢✲✜t t❡sts ❜❛s❡❞ ♦♥ ❞✐str✐❜✉t✐♦♥ ❝❤❛r❛❝t❡r✐③❛t✐♦♥s✳ ❚❡sts ♦❢ ♥♦r♠❛❧✐t② ❜❛s❡❞ ♦♥ ❙❤❡♣♣ ♣r♦♣❡rt②✱ ❛♥❞ t❤❡✐r ❡✣❝✐❡♥❝✐❡s✳

▲❛r❣❡ ❞❡✈✐❛t✐♦♥s ❛♥❞ ❧♦❝❛❧ ❡✣❝✐❡♥❝② ♦❢ st❛t✐st✐❝s In

▲❡t ❝❛❧❝✉❧❛t❡ ❧♦❝❛❧ ❇❛❤❛❞✉r ❡✣❝✐❡♥❝② ♦❢ st❛t✐st✐❝s In ❢♦r s♦♠❡ ❛❧t❡r♥❛t✐✈❡s✳ ❙❦❡✇ ❛❧t❡r♥❛t✐✈❡ ✐♥ ❆③③❛❧✐♥✐✬s s❡♥s❡✳ ❙✉♣♣♦s❡ t❤❛t t❤❡ ♦❜s❡r✈❛t✐♦♥s ❤❛✈❡ t❤❡ ❞❡♥s✐t② 2ϕ(x)Φ(θx), θ ≥ 0, x ∈ R1. ❚❤❡ ❝❛❧❝✉❧❛t✐♦♥s ❣✐✈❡ bI(θ) ∼ 1 π √ 2 θ, θ → 0. ❚❤❡ ❧♦❝❛❧ ❡①❛❝t s❧♦♣❡ ❤❛s t❤❡ r❡♣r❡s❡♥t❛t✐♦♥ cI(θ) ∼

6 π2 θ2, θ → 0. ❚❤❡

✉♣♣❡r ❜♦✉♥❞ 2K(θ) ❢♦r ❡①❛❝t s❧♦♣❡s ✉♥❞❡r s❦❡✇ ❛❧t❡r♥❛t✐✈❡ ✐s ❦♥♦✇♥ ❛s 2K(θ) ∼ 2

π θ2 ✭▼✉❧✐❡r❡ ❛♥❞ ◆✐❦✐t✐♥✱ ✷✵✵✷✮✳ ❈♦♥s❡q✉❡♥t❧②✱ t❤❡ ❧♦❝❛❧

❇❛❤❛❞✉r ❡✣❝✐❡♥❝② ♦❢ ♦✉r t❡st ❢♦r s❦❡✇ ❛❧t❡r♥❛t✐✈❡ ✐s ❛❣❛✐♥ ❡q✉❛❧ t♦ 3/π ≈ 0.9549.

❇❛❤❛❞✉r ❡✣❝✐❡♥❝② ♦❢ ❛ ❣♦♦❞♥❡ss✲♦❢✲✜t t❡sts ❜❛s❡❞ ♦♥ ❞✐str✐❜✉t✐♦♥ ❝❤❛r❛❝t❡r✐③❛t✐♦♥s✳ ❚❡sts ♦❢ ♥♦r♠❛❧✐t② ❜❛s❡❞ ♦♥ ❙❤❡♣♣ ♣r♦♣❡rt②✱ ❛♥❞ t❤❡✐r ❡✣❝✐❡♥❝✐❡s✳

▲❛r❣❡ ❞❡✈✐❛t✐♦♥s ❛♥❞ ❧♦❝❛❧ ❡✣❝✐❡♥❝② ♦❢ st❛t✐st✐❝s In

▲❡t ❝❛❧❝✉❧❛t❡ ❧♦❝❛❧ ❇❛❤❛❞✉r ❡✣❝✐❡♥❝② ♦❢ st❛t✐st✐❝s In ❢♦r s♦♠❡ ❛❧t❡r♥❛t✐✈❡s✳ ❈♦♥t❛♠✐♥❛t✐♦♥ ❛❧t❡r♥❛t✐✈❡✳ ❚❤❡ ♦❜s❡r✈❛t✐♦♥s ❤❛✈❡ t❤❡ ❞❢ G(x, θ) = (1 − θ)Φ(x) + θΦ2(x). ✭✷✮ ❲❡ ❣❡t bI(θ) ∼ 0.1667 θ, θ → 0. ❚❤❡ ❧♦❝❛❧ ❡①❛❝t s❧♦♣❡ ❛s θ → 0 ❛❞♠✐ts t❤❡ r❡♣r❡s❡♥t❛t✐♦♥ cI(θ) ∼ 0.3334 θ2. ■t ✐s ❦♥♦✇♥ t❤❛t 2K(θ) ∼ 4

5 θ2

✭▲✐t✈✐♥♦✈❛ ✷✵✵✹✮✳ ❚❤❡r❡❢♦r❡ t❤❡ ❧♦❝❛❧ ❇❛❤❛❞✉r ❡✣❝✐❡♥❝② ♦❢ ♦✉r t❡st ❢♦r t❤❡ ❝♦♥t❛♠✐♥❛t✐♦♥ ❛❧t❡r♥❛t✐✈❡ ✐s ❡q✉❛❧ t♦ ✵✳✹✶✻✼✳

❇❛❤❛❞✉r ❡✣❝✐❡♥❝② ♦❢ ❛ ❣♦♦❞♥❡ss✲♦❢✲✜t t❡sts ❜❛s❡❞ ♦♥ ❞✐str✐❜✉t✐♦♥ ❝❤❛r❛❝t❡r✐③❛t✐♦♥s✳ ❚❡sts ♦❢ ♥♦r♠❛❧✐t② ❜❛s❡❞ ♦♥ ❙❤❡♣♣ ♣r♦♣❡rt②✱ ❛♥❞ t❤❡✐r ❡✣❝✐❡♥❝✐❡s✳

▲❛r❣❡ ❞❡✈✐❛t✐♦♥s ❛♥❞ ❧♦❝❛❧ ❡✣❝✐❡♥❝② ♦❢ st❛t✐st✐❝ Dn

❚❤❡ ❢❛♠✐❧② ♦❢ ❦❡r♥❡❧s {Ξ(x, y; t)}, t ∈ R1 ✐s ♥♦t ♦♥❧② ❝❡♥t❡r❡❞ ❜✉t ❜♦✉♥❞❡❞✳ ❍❡♥❝❡ ✉s✐♥❣ t❤❡ r❡s✉❧t ♦♥ ❧❛r❣❡ ❞❡✈✐❛t✐♦♥s ♦❢ t❤❡ s✉♣r❡♠✉♠ ♦❢ ❛ ❢❛♠✐❧② ♦❢ U✲st❛t✐st✐❝s ✭◆✐❦✐t✐♥✱ ✷✵✵✽✮✱ ✇❡ ♦❜t❛✐♥ lim

n→∞ n−1 ln P(Dn > a) = −f(a),

✇❤❡r❡ t❤❡ ❢✉♥❝t✐♦♥ f ✐s ❝♦♥t✐♥✉♦✉s ❢♦r s✉✣❝✐❡♥t❧② s♠❛❧❧ a > 0, ❛♥❞✱ ♠♦r❡♦✈❡r✱ f(a) = 2a2(1 + o(1)), ❛s a → 0.

❇❛❤❛❞✉r ❡✣❝✐❡♥❝② ♦❢ ❛ ❣♦♦❞♥❡ss✲♦❢✲✜t t❡sts ❜❛s❡❞ ♦♥ ❞✐str✐❜✉t✐♦♥ ❝❤❛r❛❝t❡r✐③❛t✐♦♥s✳ ❚❡sts ♦❢ ♥♦r♠❛❧✐t② ❜❛s❡❞ ♦♥ ❙❤❡♣♣ ♣r♦♣❡rt②✱ ❛♥❞ t❤❡✐r ❡✣❝✐❡♥❝✐❡s✳

▲❛r❣❡ ❞❡✈✐❛t✐♦♥s ❛♥❞ ❧♦❝❛❧ ❡✣❝✐❡♥❝② ♦❢ st❛t✐st✐❝ Dn

▲❡t ❝❛❧❝✉❧❛t❡ t❤❡ ❧♦❝❛❧ ❇❛❤❛❞✉r ❡✣❝✐❡♥❝② ♦❢ Dn ❢♦r s♦♠❡ ❛❧t❡r♥❛t✐✈❡s✳ ▲♦❝❛t✐♦♥ ❛❧t❡r♥❛t✐✈❡✳ ❆❝❝♦r❞✐♥❣ t♦ ●❧✐✈❡♥❦♦✲❈❛♥t❡❧❧✐ t❤❡♦r❡♠ ❢♦r U✲❡♠♣✐r✐❝❛❧ ❞❢✬s t❤❡ ❧✐♠✐t ✐♥ ♣r♦❜❛❜✐❧✐t② ♦❢ Dn ✐s bD(θ) = sup

t |Pθ(k(X, Y ) < t) − Φ(t − θ)|.

❍❡♥❝❡✱ ❛s θ → 0 ✇❡ ❤❛✈❡ bD(θ) ∼

θ √ 2π ✳ ❚❤❡r❡❢♦r❡ t❤❡ ❧♦❝❛❧ ❡①❛❝t s❧♦♣❡

❛❞♠✐ts t❤❡ r❡♣r❡s❡♥t❛t✐♦♥ cD(θ) ∼ 2 π θ2 ∼ 0.6366 · θ2. ❲❡ ❦♥♦✇ t❤❛t ✐♥ t❤✐s ❝❛s❡ 2K(θ) ∼ θ2 ❛s θ → 0. ■t ❢♦❧❧♦✇s t❤❛t t❤❡ ❧♦❝❛❧ ❇❛❤❛❞✉r ❡✣❝✐❡♥❝② ♦❢ ♦✉r t❡st ✐s 2/π ≈ 0.6366.

❇❛❤❛❞✉r ❡✣❝✐❡♥❝② ♦❢ ❛ ❣♦♦❞♥❡ss✲♦❢✲✜t t❡sts ❜❛s❡❞ ♦♥ ❞✐str✐❜✉t✐♦♥ ❝❤❛r❛❝t❡r✐③❛t✐♦♥s✳ ❚❡sts ♦❢ ♥♦r♠❛❧✐t② ❜❛s❡❞ ♦♥ ❙❤❡♣♣ ♣r♦♣❡rt②✱ ❛♥❞ t❤❡✐r ❡✣❝✐❡♥❝✐❡s✳

▲❛r❣❡ ❞❡✈✐❛t✐♦♥s ❛♥❞ ❧♦❝❛❧ ❡✣❝✐❡♥❝② ♦❢ st❛t✐st✐❝ Dn

▲❡t ❝❛❧❝✉❧❛t❡ t❤❡ ❧♦❝❛❧ ❇❛❤❛❞✉r ❡✣❝✐❡♥❝② ♦❢ Dn ❢♦r s♦♠❡ ❛❧t❡r♥❛t✐✈❡s✳ ❙❦❡✇ ❛❧t❡r♥❛t✐✈❡✳ ■t ✐s ❡❛s② t♦ s❡❡ t❤❛t ❢♦r ❛♥② t bD(θ) ∼ θ π , θ → 0. ❚❤❡r❡❢♦r❡ t❤❡ ❧♦❝❛❧ ❡①❛❝t s❧♦♣❡ ❛s θ → 0 ❧♦♦❦s ❧✐❦❡ cD(θ) ∼ 4 π2 θ2. ❲❡ ❤❛✈❡ ❛❧r❡❛❞② s❡❡♥ t❤❛t ❢♦r t❤❡ s❦❡✇ ❛❧t❡r♥❛t✐✈❡ 2K(θ) ∼ 2θ2

π

❛s θ → 0✳ ❈♦♥s❡q✉❡♥t❧②✱ t❤❡ ❧♦❝❛❧ ❇❛❤❛❞✉r ❡✣❝✐❡♥❝② ♦❢ ♦✉r t❡st ✐s ❛❣❛✐♥ ❡q✉❛❧ t♦ 2/π ≈ 0.6366.

❇❛❤❛❞✉r ❡✣❝✐❡♥❝② ♦❢ ❛ ❣♦♦❞♥❡ss✲♦❢✲✜t t❡sts ❜❛s❡❞ ♦♥ ❞✐str✐❜✉t✐♦♥ ❝❤❛r❛❝t❡r✐③❛t✐♦♥s✳ ❚❡sts ♦❢ ♥♦r♠❛❧✐t② ❜❛s❡❞ ♦♥ ❙❤❡♣♣ ♣r♦♣❡rt②✱ ❛♥❞ t❤❡✐r ❡✣❝✐❡♥❝✐❡s✳

▲❛r❣❡ ❞❡✈✐❛t✐♦♥s ❛♥❞ ❧♦❝❛❧ ❡✣❝✐❡♥❝② ♦❢ st❛t✐st✐❝ Dn

▲❡t ❝❛❧❝✉❧❛t❡ t❤❡ ❧♦❝❛❧ ❇❛❤❛❞✉r ❡✣❝✐❡♥❝② ♦❢ Dn ❢♦r s♦♠❡ ❛❧t❡r♥❛t✐✈❡s✳ ❈♦♥t❛♠✐♥❛t✐♦♥ ❛❧t❡r♥❛t✐✈❡✳ ❲❡ ❣❡t ❡❛s✐❧② t❤❛t bD(θ) ∼ θ/4, θ → 0. ❍❡♥❝❡ ✇❡ ❤❛✈❡ t❤❡ ❛s②♠♣t♦t✐❝s cD(θ) ∼ θ2/4. ❆s s❡❡♥ ❜❡❢♦r❡✱ ✐♥ t❤✐s ❝❛s❡ 2K(θ) ∼ 4

5θ2 ❛s θ → 0✳ ❍❡♥❝❡✱ t❤❡ ❧♦❝❛❧ ❇❛❤❛❞✉r ❡✣❝✐❡♥❝② ♦❢ ♦✉r

t❡st ✐s ✵✳✸✶✷✺✳

❇❛❤❛❞✉r ❡✣❝✐❡♥❝② ♦❢ ❛ ❣♦♦❞♥❡ss✲♦❢✲✜t t❡sts ❜❛s❡❞ ♦♥ ❞✐str✐❜✉t✐♦♥ ❝❤❛r❛❝t❡r✐③❛t✐♦♥s✳ ❚❡sts ♦❢ ♥♦r♠❛❧✐t② ❜❛s❡❞ ♦♥ ❙❤❡♣♣ ♣r♦♣❡rt②✱ ❛♥❞ t❤❡✐r ❡✣❝✐❡♥❝✐❡s✳

❈♦♥❝❧✉❞✐♥❣ r❡♠❛r❦s

❲❡ ❝❛♥ r❡♠❛r❦ t❤❛t t❤❡ ❡✣❝✐❡♥❝✐❡s ♦❢ ♦✉r t✇♦ t❡sts ❢♦r t❤❡ ❝♦♠♣♦s✐t❡ ❤②♣♦t❤❡s✐s H0 ♦❢ ♥♦r♠❛❧✐t② ✉♥❞❡r ❧♦❝❛t✐♦♥ ❛❧t❡r♥❛t✐✈❡ ✭r❡s♣❡❝t✐✈❡❧② 3/π ❛♥❞ 2/π✮ ❝♦✐♥❝✐❞❡ ✇✐t❤ t❤❡ ❡✣❝✐❡♥❝✐❡s ✉♥❞❡r t❤❡ s❛♠❡ ❛❧t❡r♥❛t✐✈❡ ♦❢ ❝❧❛ss✐❝❛❧ t❡sts ❜❛s❡❞ ♦♥ t❤❡ ❈❤❛♣♠❛♥✲▼♦s❡s st❛t✐st✐❝ ω1

n ❛♥❞ ❑♦❧♠♦❣♦r♦✈ st❛t✐st✐❝ ✇❤❡♥

t❡st✐♥❣ t❤❡ s✐♠♣❧❡ ❤②♣♦t❤❡s✐s ♦❢ ♥♦r♠❛❧✐t② ✭t❤❡② ❛r❡ ❦♥♦✇♥ s✐♥❝❡ ✶✾✼✵✲s✮✳ ■t ✇♦✉❧❞ ❜❡ ✐♥t❡r❡st✐♥❣ t♦ ✜♥❞ t❤❡ t❤❡♦r❡t✐❝❛❧ ❡①♣❧❛♥❛t✐♦♥ ♦❢ t❤✐s ❡♠♣✐r✐❝❛❧ ♦❜s❡r✈❛t✐♦♥✳

❇❛❤❛❞✉r ❡✣❝✐❡♥❝② ♦❢ ❛ ❣♦♦❞♥❡ss✲♦❢✲✜t t❡sts ❜❛s❡❞ ♦♥ ❞✐str✐❜✉t✐♦♥ ❝❤❛r❛❝t❡r✐③❛t✐♦♥s✳ ❚❡sts ♦❢ ❡①♣♦♥❡♥t✐❛❧✐t② ❜❛s❡❞ ♦♥ ❘♦ss❜❡r❣✬s ❝❤❛r❛❝t❡r✐③❛t✐♦♥✱ ❛♥❞ t❤❡✐r ❡✣❝✐❡♥❝✐❡s✳

❚❡sts ♦❢ ❡①♣♦♥❡♥t✐❛❧✐t② ❜❛s❡❞ ♦♥ ❘♦ss❜❡r❣✬s ❝❤❛r❛❝t❡r✐③❛t✐♦♥✱ ❛♥❞ t❤❡✐r ❡✣❝✐❡♥❝✐❡s

❲❡ ♣r♦♣♦s❡ t✇♦ ♥❡✇ t❡sts ♦❢ ❡①♣♦♥❡♥t✐❛❧✐t② ❜❛s❡❞ ♦♥ ❘♦ss❜❡r❣✬s ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ ❡①♣♦♥❡♥t✐❛❧ ❞✐str✐❜✉t✐♦♥✿ ▲❡t X1, . . . , Xn ❜❡ ♥♦♥✲♥❡❣❛t✐✈❡ ✐✳✐✳❞✳ r✈✬s✳ ❚❤❡♥ ❢♦r ❛♥② j st❛t✐st✐❝s Xj+s,n − Xj,n ❛♥❞ Xs,n−j ❛r❡ ✐❞❡♥t✐❝❛❧❧② ❞✐str✐❜✉t❡❞ ✐✛ t❤❡ s❛♠♣❧❡ ❤❛s t❤❡ ❡①♣♦♥❡♥t✐❛❧ ❞✐str✐❜✉t✐♦♥✳ ■ts s✐♠♣❧❡st ❢♦r♠✉❧❛t✐♦♥ ✐s ❛s ❢♦❧❧♦✇s✿ ❚❤❡♦r❡♠ ▲❡t X1, X2 ❛♥❞ X3 ❜❡ ✐✳✐✳❞✳ r✈✬s ✇✐t❤ ❝♦♥t✐♥✉♦✉s ❞❢ F. ❚✇♦ st❛t✐st✐❝s X2,3 − X1,3 ❛♥❞ min(X1, X2) ❛r❡ ✐❞❡♥t✐❝❛❧❧② ❞✐str✐❜✉t❡❞ ✐✛ F ✐s t❤❡ ❡①♣♦♥❡♥t✐❛❧ ❞❢✳

❇❛❤❛❞✉r ❡✣❝✐❡♥❝② ♦❢ ❛ ❣♦♦❞♥❡ss✲♦❢✲✜t t❡sts ❜❛s❡❞ ♦♥ ❞✐str✐❜✉t✐♦♥ ❝❤❛r❛❝t❡r✐③❛t✐♦♥s✳ ❚❡sts ♦❢ ❡①♣♦♥❡♥t✐❛❧✐t② ❜❛s❡❞ ♦♥ ❘♦ss❜❡r❣✬s ❝❤❛r❛❝t❡r✐③❛t✐♦♥✱ ❛♥❞ t❤❡✐r ❡✣❝✐❡♥❝✐❡s✳

■♥tr♦❞✉❝t✐♦♥

❲❡ ❛r❡ t❡st✐♥❣ t❤❡ ❤②♣♦t❤❡s✐s H0 : F ✐s ❡①♣♦♥❡♥t✐❛❧ ✇✐t❤ t❤❡ ❞❡♥s✐t② λe−λx, x ≥ 0, λ > 0 ❛❣❛✐♥st t❤❡ ❛❧t❡r♥❛t✐✈❡ ❤②♣♦t❤❡s✐s H1 : F ✐s ♥♦♥✲❡①♣♦♥❡♥t✐❛❧ ❞❢. ❲❡ ❝♦♥str✉❝t t✇♦ U−st❛t✐st✐❝❛❧ ❞❢ Hn ❛♥❞ Gn ✉s✐♥❣ t❤❡ ❢♦r♠✉❧❛s Hn(t) = ✥ n 3 ✦−1 ❳

1≤i<j<k≤n

✶{X2,{Xi,Xj,Xk} − X1,{Xi,Xj,Xk} < t}, t ≥ 0, Gn(t) = ✥ n 2 ✦−1 ❳

1≤i<j≤n

✶{min(Xi, Xj) < t}, t ≥ 0. ❈♦♥s✐❞❡r t✇♦ st❛t✐st✐❝s ❢♦r t❡st✐♥❣ ♦❢ H0✿ Sn = ❩ ∞ (Hn(t) − Gn(t)) dFn(t), Rn = sup

t≥0

| Hn(t) − Gn(t) | .

❇❛❤❛❞✉r ❡✣❝✐❡♥❝② ♦❢ ❛ ❣♦♦❞♥❡ss✲♦❢✲✜t t❡sts ❜❛s❡❞ ♦♥ ❞✐str✐❜✉t✐♦♥ ❝❤❛r❛❝t❡r✐③❛t✐♦♥s✳ ❚❡sts ♦❢ ❡①♣♦♥❡♥t✐❛❧✐t② ❜❛s❡❞ ♦♥ ❘♦ss❜❡r❣✬s ❝❤❛r❛❝t❡r✐③❛t✐♦♥✱ ❛♥❞ t❤❡✐r ❡✣❝✐❡♥❝✐❡s✳

▲✐♠✐t✐♥❣ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ st❛t✐st✐❝ Sn

❚❤❡ st❛t✐st✐❝ Sn ✐s ❛s②♠♣t♦t✐❝❛❧❧② ❡q✉✐✈❛❧❡♥t t♦ t❤❡ U✲st❛t✐st✐❝ ♦❢ ❞❡❣r❡❡ ✹ ✇✐t❤ ❝❡♥t❡r❡❞ ❦❡r♥❡❧ Φ(Xi,Xj, Xk, Xl) = 1 4✶{X2,{i,j,k} − X1,{i,j,k} < Xl} + 1 4✶{X2,{j,k,l} − X1,{j,k,l} < Xi} + 1 4✶{X2,{k,l,i} − X1,{k,l,i} < Xj} + 1 4✶{X2,{l,i,j} − X1,{l,i,j} < Xk}) − 1 12 ❳

j1,j2,j3∈(i,j,k,l) j1=j2=j3

✶{min(Xj1, Xj2) < Xj3}. ✭✸✮

❇❛❤❛❞✉r ❡✣❝✐❡♥❝② ♦❢ ❛ ❣♦♦❞♥❡ss✲♦❢✲✜t t❡sts ❜❛s❡❞ ♦♥ ❞✐str✐❜✉t✐♦♥ ❝❤❛r❛❝t❡r✐③❛t✐♦♥s✳ ❚❡sts ♦❢ ❡①♣♦♥❡♥t✐❛❧✐t② ❜❛s❡❞ ♦♥ ❘♦ss❜❡r❣✬s ❝❤❛r❛❝t❡r✐③❛t✐♦♥✱ ❛♥❞ t❤❡✐r ❡✣❝✐❡♥❝✐❡s✳

▲✐♠✐t✐♥❣ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ st❛t✐st✐❝ Sn

▲❡t ❝❛❧❝✉❧❛t❡ t❤❡ ♣r♦❥❡❝t✐♦♥ ♦❢ t❤❡ ❦❡r♥❡❧ ✭✸✮ ϕ(x) := E(Φ(X, Y, Z, W)|X = x) = 3 4P(X2,{X,Y,Z}−X1,{X,Y,Z} < W | X = x) + 1 4P(X2,{Y,Z,W } − X1,{Y,Z,W } < x) − 3 12P(min(Y, Z) < x) − 3 12P(min(Y, Z) < W) − 6 12P(min(x, Y ) < Z). ❆❢t❡r s♦♠❡ ❝❛❧❝✉❧❛t✐♦♥s ✇❡ ♦❜t❛✐♥ t❤❛t t❤❡ ♣r♦❥❡❝t✐♦♥ ♦❢ t❤❡ ❦❡r♥❡❧ Φ ✐s ϕ(x) = 1 4 ✒ (3x − 1 2)e−2x − 1 6 ✓ ✇✐t❤ t❤❡ ✈❛r✐❛♥❝❡ σ2

ϕ =

❩ ∞ ϕ2(x)dΦ(x) = 13 4500. ❍❡♥❝❡ t❤❡ ❦❡r♥❡❧ ✭✸✮ ✐s ♥♦♥✲❞❡❣❡♥❡r❛t❡✳ ❇② ❍♦❡✛❞✐♥❣✬s t❤❡♦r❡♠ √nSn

d

− → N(0, 52 1125).

❇❛❤❛❞✉r ❡✣❝✐❡♥❝② ♦❢ ❛ ❣♦♦❞♥❡ss✲♦❢✲✜t t❡sts ❜❛s❡❞ ♦♥ ❞✐str✐❜✉t✐♦♥ ❝❤❛r❛❝t❡r✐③❛t✐♦♥s✳ ❚❡sts ♦❢ ❡①♣♦♥❡♥t✐❛❧✐t② ❜❛s❡❞ ♦♥ ❘♦ss❜❡r❣✬s ❝❤❛r❛❝t❡r✐③❛t✐♦♥✱ ❛♥❞ t❤❡✐r ❡✣❝✐❡♥❝✐❡s✳

▲✐♠✐t✐♥❣ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ st❛t✐st✐❝ Rn

❚❤❡ r✈ Hn(t) − Gn(t) ❢♦r ✜①❡❞ t ✐s ❛s②♠♣t♦t✐❝❛❧❧② ❡q✉✐✈❛❧❡♥t t♦ U✲st❛t✐st✐❝ ✇✐t❤ t❤❡ ❦❡r♥❡❧ ❞❡♣❡♥❞✐♥❣ ♦♥ t ≥ 0 Ω(X, Y, Z; t) = ✶{X2,{X,Y,Z} − X1,{X,Y,Z} < t} − 1 3✶{min(X, Y ) < t}− −1 3✶{min(Y, Z) < t} − 1 3✶{min(X, Z) < t}. ❚❤❡ ♣r♦❥❡❝t✐♦♥ ♦❢ t❤✐s ❦❡r♥❡❧ ❢♦r ✜①❡❞ t ✐s ❡q✉❛❧ t♦ ω(x; t) = ✶{x > t}(1 − 1 3e−t + e−2x(et − 1)) + ✶{x < t}(1 − e−2x)+ +e−2x(1 − e−2t) − 1 + 1 3e−2t.

❇❛❤❛❞✉r ❡✣❝✐❡♥❝② ♦❢ ❛ ❣♦♦❞♥❡ss✲♦❢✲✜t t❡sts ❜❛s❡❞ ♦♥ ❞✐str✐❜✉t✐♦♥ ❝❤❛r❛❝t❡r✐③❛t✐♦♥s✳ ❚❡sts ♦❢ ❡①♣♦♥❡♥t✐❛❧✐t② ❜❛s❡❞ ♦♥ ❘♦ss❜❡r❣✬s ❝❤❛r❛❝t❡r✐③❛t✐♦♥✱ ❛♥❞ t❤❡✐r ❡✣❝✐❡♥❝✐❡s✳

▲✐♠✐t✐♥❣ ❞✐str✐❜✉t✐♦♥ ♦❢ st❛t✐st✐❝ Rn

❚❤❡ ✈❛r✐❛♥❝❡ ♦❢ t❤❡ ♣r♦❥❡❝t✐♦♥ ❤❛s t❤❡ ❢♦r♠ σ2

ω(t) = Eω2(X; t) = 4

45e−3t(−2e−3t+e−t+1). ❲❡ ❝♦♥❝❧✉❞❡ t❤❛t σ2

ω = supt≥0 σ2 ω(t) ≈ 0.03532.

❚❤✐s ✈❛❧✉❡ ✐s ✐♠♣♦rt❛♥t ✇❤❡♥ ❝❛❧❝✉❧❛t✐♥❣ ❧❛r❣❡ ❞❡✈✐❛t✐♦♥ ❛s②♠♣t♦t✐❝s✳

❇❛❤❛❞✉r ❡✣❝✐❡♥❝② ♦❢ ❛ ❣♦♦❞♥❡ss✲♦❢✲✜t t❡sts ❜❛s❡❞ ♦♥ ❞✐str✐❜✉t✐♦♥ ❝❤❛r❛❝t❡r✐③❛t✐♦♥s✳ ❚❡sts ♦❢ ❡①♣♦♥❡♥t✐❛❧✐t② ❜❛s❡❞ ♦♥ ❘♦ss❜❡r❣✬s ❝❤❛r❛❝t❡r✐③❛t✐♦♥✱ ❛♥❞ t❤❡✐r ❡✣❝✐❡♥❝✐❡s✳

▲✐♠✐t✐♥❣ ❞✐str✐❜✉t✐♦♥ ♦❢ st❛t✐st✐❝ Rn

▲✐♠✐t✐♥❣ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ st❛t✐st✐❝ Rn ✐s ✉♥❦♥♦✇♥✳ ❯s✐♥❣ t❤❡ ♠❡t❤♦❞s ♦❢ ❙✐❧✈❡r♠❛♥ ✭✶✾✽✸✮✱ ♦♥❡ ❝❛♥ s❤♦✇ t❤❛t t❤❡ U✲❡♠♣✐r✐❝❛❧ ♣r♦❝❡ss ηn(t) = √n (Hn(t) − Gn(t)) , t ≥ 0, ❝♦♥✈❡r❣❡s ✇❡❛❦❧② ❛s n → ∞ t♦ s♦♠❡ ❝❡♥t❡r❡❞ ●❛✉ss✐❛♥ ♣r♦❝❡ss η(t) ✇✐t❤ ❝♦♠♣❧✐❝❛t❡❞ ❝♦✈❛r✐❛♥❝❡✳ ❈♦♥s❡q✉❡♥t❧② t❤❡ s❡q✉❡♥❝❡ ♦❢ st❛t✐st✐❝s √nRn ❝♦♥✈❡r❣❡s ✐♥ ❞✐str✐❜✉t✐♦♥ t♦ supt≥0 |η(t)|, ✇❤✐❝❤ ❤❛s ✈❡r② ❝♦♠♣❧✐❝❛t❡❞ ❞✐str✐❜✉t✐♦♥ ✭❝✉rr❡♥t❧② ✉♥❦♥♦✇♥✮✳ ❇✉t t❤❡ ❝r✐t✐❝❛❧ ✈❛❧✉❡s ❢♦r st❛t✐st✐❝s Rn ❝❛♥ ❜❡ ❢♦✉♥❞ ✈✐❛ s✐♠✉❧❛t✐♥❣ t❤❡✐r s❛♠♣❧❡ ❞✐str✐❜✉t✐♦♥✳

❇❛❤❛❞✉r ❡✣❝✐❡♥❝② ♦❢ ❛ ❣♦♦❞♥❡ss✲♦❢✲✜t t❡sts ❜❛s❡❞ ♦♥ ❞✐str✐❜✉t✐♦♥ ❝❤❛r❛❝t❡r✐③❛t✐♦♥s✳ ❚❡sts ♦❢ ❡①♣♦♥❡♥t✐❛❧✐t② ❜❛s❡❞ ♦♥ ❘♦ss❜❡r❣✬s ❝❤❛r❛❝t❡r✐③❛t✐♦♥✱ ❛♥❞ t❤❡✐r ❡✣❝✐❡♥❝✐❡s✳

▲❛r❣❡ ❞❡✈✐❛t✐♦♥s ❛♥❞ ❧♦❝❛❧ ❡✣❝✐❡♥❝② ♦❢ st❛t✐st✐❝s Sn

❚❤❡ ❦❡r♥❡❧ Φ ✐s ♥♦t ♦♥❧② ❝❡♥t❡r❡❞ ❜✉t ❜♦✉♥❞❡❞✱ t❤❡r❡❢♦r❡ ❢r♦♠ t❤❡♦r❡♠ ♦❢ ◆✐❦✐t✐♥ ❛♥❞ P♦♥✐❦❛r♦✈ ✇❡ ❤❛✈❡ ❢♦r a > 0 lim

n→∞ n−1 ln P(Sn > a) = −f(a),

✇❤❡r❡ t❤❡ ❢✉♥❝t✐♦♥ f ✐s ❝♦♥t✐♥✉♦✉s ❢♦r s✉✣❝✐❡♥t❧② s♠❛❧❧ a > 0, ❛♥❞✱ ♠♦r❡♦✈❡r✱ f(a) = 1125 104 a2(1 + o(1)), ïðè a → 0.

❇❛❤❛❞✉r ❡✣❝✐❡♥❝② ♦❢ ❛ ❣♦♦❞♥❡ss✲♦❢✲✜t t❡sts ❜❛s❡❞ ♦♥ ❞✐str✐❜✉t✐♦♥ ❝❤❛r❛❝t❡r✐③❛t✐♦♥s✳ ❚❡sts ♦❢ ❡①♣♦♥❡♥t✐❛❧✐t② ❜❛s❡❞ ♦♥ ❘♦ss❜❡r❣✬s ❝❤❛r❛❝t❡r✐③❛t✐♦♥✱ ❛♥❞ t❤❡✐r ❡✣❝✐❡♥❝✐❡s✳

▲❛r❣❡ ❞❡✈✐❛t✐♦♥s ❛♥❞ ❧♦❝❛❧ ❡✣❝✐❡♥❝② ♦❢ st❛t✐st✐❝s Sn

▲❡t ❝❛❧❝✉❧❛t❡ ❧♦❝❛❧ ❇❛❤❛❞✉r ❡✣❝✐❡♥❝② ♦❢ st❛t✐st✐❝s Sn ❢♦r s♦♠❡ ❛❧t❡r♥❛t✐✈❡s✳ ❲❡✐❜✉❧❧ ❛❧t❡r♥❛t✐✈❡✳ ❯♥❞❡r H1 t❤❡ ♦❜s❡r✈❛t✐♦♥s ❤❛✈❡ t❤❡ ❞❡♥s✐t② (1 + θ)xθ exp(−x1+θ), x ≥ 0. ❯s✐♥❣ t❤❡ ▲❛✇ ♦❢ ▲❛r❣❡ ◆✉♠❜❡rs ❢♦r U✲st❛t✐st✐❝s✱ ✇❡ s❡❡ t❤❛t t❤❡ ❧✐♠✐t ✐♥ ♣r♦❜❛❜✐❧✐t② ♦❢ Sn ✉♥❞❡r H1 ✐s ❡q✉❛❧ t♦ bS(θ) = Pθ(X2,3 − X1,3 < X4) − Pθ(X1,2 < X3). ❍❡♥❝❡✱ ❛s θ → 0 ✇❡ ❤❛✈❡ bS(θ) ∼ 2

9θ. ❚❤❡r❡❢♦r❡ t❤❡ ❧♦❝❛❧ ❡①❛❝t s❧♦♣❡ ♦❢

t❤❡ s❡q✉❡♥❝❡ Sn ❛s θ → 0 ❛❞♠✐ts t❤❡ r❡♣r❡s❡♥t❛t✐♦♥ cS(θ) = bS(θ)2/(16σ2

ϕ) = 125

117θ2 ∼ 1.068θ2. ❚❤❡ t❤❡♦r❡t✐❝❛❧ ✉♣♣❡r ❜♦✉♥❞ ✐s 2K(θ) = π2θ2/6 ❛s θ → 0 ✭▲✐t✈✐♥♦✈❛✱ ✷✵✵✹✮✳ ■t ❢♦❧❧♦✇s t❤❛t t❤❡ ❧♦❝❛❧ ❇❛❤❛❞✉r ❡✣❝✐❡♥❝② ♦❢ ♦✉r t❡st ✐s ❡q✉❛❧ t♦ eff B(S) := lim

θ→0{cS(θ)/2K(θ)} ≈ 0.6495.

❇❛❤❛❞✉r ❡✣❝✐❡♥❝② ♦❢ ❛ ❣♦♦❞♥❡ss✲♦❢✲✜t t❡sts ❜❛s❡❞ ♦♥ ❞✐str✐❜✉t✐♦♥ ❝❤❛r❛❝t❡r✐③❛t✐♦♥s✳ ❚❡sts ♦❢ ❡①♣♦♥❡♥t✐❛❧✐t② ❜❛s❡❞ ♦♥ ❘♦ss❜❡r❣✬s ❝❤❛r❛❝t❡r✐③❛t✐♦♥✱ ❛♥❞ t❤❡✐r ❡✣❝✐❡♥❝✐❡s✳

▲❛r❣❡ ❞❡✈✐❛t✐♦♥s ❛♥❞ ❧♦❝❛❧ ❡✣❝✐❡♥❝② ♦❢ st❛t✐st✐❝s Sn

▲❡t ❝❛❧❝✉❧❛t❡ ❧♦❝❛❧ ❇❛❤❛❞✉r ❡✣❝✐❡♥❝② ♦❢ st❛t✐st✐❝s Sn ❢♦r s♦♠❡ ❛❧t❡r♥❛t✐✈❡s✳ ▼❛❦❡❤❛♠ ❛❧t❡r♥❛t✐✈❡✳ ❙✉♣♣♦s❡ t❤❛t t❤❡ ♦❜s❡r✈❛t✐♦♥s ❤❛✈❡ t❤❡ ❞❡♥s✐t② (1 + θ(1 − e−x)) exp(−x − θ(e−x − 1 + x)), x ≥ 0. ❚❤❡ ❝❛❧❝✉❧❛t✐♦♥s ❣✐✈❡ bS(θ) ∼ 1 24θ, θ → 0. ❚❤❡ ❧♦❝❛❧ ❡①❛❝t s❧♦♣❡ ❤❛s t❤❡ r❡♣r❡s❡♥t❛t✐♦♥ cS(θ) ∼ 0.0376θ2, θ → 0. ❚❤❡ ✉♣♣❡r ❜♦✉♥❞ 2K(θ) ❢♦r ❡①❛❝t s❧♦♣❡s ✉♥❞❡r ▼❛❦❡❤❛♠ ❛❧t❡r♥❛t✐✈❡ ✐s ❦♥♦✇♥ ❛s 2K(θ) ∼ θ2/12 ✭▲✐t✈✐♥♦✈❛✱ ✷✵✵✹✮✳ ❈♦♥s❡q✉❡♥t❧②✱ t❤❡ ❧♦❝❛❧ ❇❛❤❛❞✉r ❡✣❝✐❡♥❝② ♦❢ ♦✉r t❡st ❢♦r ▼❛❦❡❤❛♠ ❛❧t❡r♥❛t✐✈❡ ✐s ❡q✉❛❧ t♦ ✵✳✹✺✵✼✳

❇❛❤❛❞✉r ❡✣❝✐❡♥❝② ♦❢ ❛ ❣♦♦❞♥❡ss✲♦❢✲✜t t❡sts ❜❛s❡❞ ♦♥ ❞✐str✐❜✉t✐♦♥ ❝❤❛r❛❝t❡r✐③❛t✐♦♥s✳ ❚❡sts ♦❢ ❡①♣♦♥❡♥t✐❛❧✐t② ❜❛s❡❞ ♦♥ ❘♦ss❜❡r❣✬s ❝❤❛r❛❝t❡r✐③❛t✐♦♥✱ ❛♥❞ t❤❡✐r ❡✣❝✐❡♥❝✐❡s✳

▲❛r❣❡ ❞❡✈✐❛t✐♦♥s ❛♥❞ ❧♦❝❛❧ ❡✣❝✐❡♥❝② ♦❢ st❛t✐st✐❝s Sn

▲❡t ❝❛❧❝✉❧❛t❡ ❧♦❝❛❧ ❇❛❤❛❞✉r ❡✣❝✐❡♥❝② ♦❢ st❛t✐st✐❝s Sn ❢♦r s♦♠❡ ❛❧t❡r♥❛t✐✈❡s✳ ▲✐♥❡❛r ❢❛✐❧✉r❡ r❛t❡ ❛❧t❡r♥❛t✐✈❡✳ ❚❤❡ ♦❜s❡r✈❛t✐♦♥s ❤❛✈❡ t❤❡ ❞❡♥s✐t② (1 + θx)e−x− 1

2 θx2, x ≥ 0. ❲❡ ❣❡t

bS(θ) ∼ 2 27θ, θ → 0. ❚❤❡ ❧♦❝❛❧ ❡①❛❝t s❧♦♣❡ ❛s θ → 0 ❛❞♠✐ts t❤❡ r❡♣r❡s❡♥t❛t✐♦♥ cS(θ) ∼ 0.1187θ2. ■t ✐s ❦♥♦✇♥ t❤❛t 2K(θ) ∼ θ2 ✭▲✐t✈✐♥♦✈❛✱ ✷✵✵✹✮✳ ❚❤❡r❡❢♦r❡ t❤❡ ❧♦❝❛❧ ❇❛❤❛❞✉r ❡✣❝✐❡♥❝② ♦❢ ♦✉r t❡st ❢♦r t❤❡ ❝♦♥t❛♠✐♥❛t✐♦♥ ❛❧t❡r♥❛t✐✈❡ ✐s ❡q✉❛❧ t♦ ✵✳✶✶✽✼✳

❇❛❤❛❞✉r ❡✣❝✐❡♥❝② ♦❢ ❛ ❣♦♦❞♥❡ss✲♦❢✲✜t t❡sts ❜❛s❡❞ ♦♥ ❞✐str✐❜✉t✐♦♥ ❝❤❛r❛❝t❡r✐③❛t✐♦♥s✳ ❚❡sts ♦❢ ❡①♣♦♥❡♥t✐❛❧✐t② ❜❛s❡❞ ♦♥ ❘♦ss❜❡r❣✬s ❝❤❛r❛❝t❡r✐③❛t✐♦♥✱ ❛♥❞ t❤❡✐r ❡✣❝✐❡♥❝✐❡s✳

▲❛r❣❡ ❞❡✈✐❛t✐♦♥s ❛♥❞ ❧♦❝❛❧ ❡✣❝✐❡♥❝② ♦❢ st❛t✐st✐❝ Rn.

❚❤❡ ❢❛♠✐❧② ♦❢ ❦❡r♥❡❧s {Ω(X, Y, Z; t)}, t ≥ 0 ✐s ♥♦t ♦♥❧② ❝❡♥t❡r❡❞ ❜✉t ❜♦✉♥❞❡❞✳ ❍❡♥❝❡ ✉s✐♥❣ t❤❡ r❡s✉❧t ♦♥ ❧❛r❣❡ ❞❡✈✐❛t✐♦♥s ♦❢ t❤❡ s✉♣r❡♠✉♠ ♦❢ ❛ ❢❛♠✐❧② ♦❢ U✲st❛t✐st✐❝s ✭◆✐❦✐t✐♥✱ ✷✵✵✽✮✱ ✇❡ ♦❜t❛✐♥ lim

n→∞ n−1 ln P(Rn > a) = −f(a),

✇❤❡r❡ t❤❡ ❢✉♥❝t✐♦♥ f ✐s ❝♦♥t✐♥✉♦✉s ❢♦r s✉✣❝✐❡♥t❧② s♠❛❧❧ a > 0, ❛♥❞✱ ♠♦r❡♦✈❡r✱ f(a) = 1.5729a2(1 + o(1)), ❛s a → 0.

❇❛❤❛❞✉r ❡✣❝✐❡♥❝② ♦❢ ❛ ❣♦♦❞♥❡ss✲♦❢✲✜t t❡sts ❜❛s❡❞ ♦♥ ❞✐str✐❜✉t✐♦♥ ❝❤❛r❛❝t❡r✐③❛t✐♦♥s✳ ❚❡sts ♦❢ ❡①♣♦♥❡♥t✐❛❧✐t② ❜❛s❡❞ ♦♥ ❘♦ss❜❡r❣✬s ❝❤❛r❛❝t❡r✐③❛t✐♦♥✱ ❛♥❞ t❤❡✐r ❡✣❝✐❡♥❝✐❡s✳

▲❛r❣❡ ❞❡✈✐❛t✐♦♥s ❛♥❞ ❧♦❝❛❧ ❡✣❝✐❡♥❝② ♦❢ st❛t✐st✐❝ Rn

▲❡t ❝❛❧❝✉❧❛t❡ t❤❡ ❧♦❝❛❧ ❇❛❤❛❞✉r ❡✣❝✐❡♥❝② ♦❢ Rn ❢♦r s♦♠❡ ❛❧t❡r♥❛t✐✈❡s✳ ❲❡✐❜✉❧❧ ❛❧t❡r♥❛t✐✈❡✳ ▲✐♠✐t ✐♥ ♣r♦❜❛❜✐❧✐t② ♦❢ Rn ✐s bR(θ) = sup

t≥0

|Pθ(X2,3 − X1,3 < t) − Pθ(X1,2 < t)|. ❍❡♥❝❡✱ ❛s θ → 0 ✇❡ ❤❛✈❡ bR(t, θ) ∼ ✒2 3e−2t(ln t + γ + ln 3) + 2 3Ei(1, 3t)et ✓ θ, ✇❤❡r❡ γ ≈ 0.5772157✱ ❛♥❞ Ei(k, z) = ❘ ∞

1

e−ztt−kdt. ❲❡ ❝♦♥❝❧✉❞❡ t❤❛t bR(θ) = supt≥0 bR(t, θ) ∼ 0.4088θ, θ → 0. ❚❤❡r❡❢♦r❡ t❤❡ ❧♦❝❛❧ ❡①❛❝t s❧♦♣❡ ❛❞♠✐ts t❤❡ r❡♣r❡s❡♥t❛t✐♦♥ ❛s θ → 0 cR(θ) ∼ 0.5258θ2. ❲❡ ❦♥♦✇ t❤❛t ✐♥ t❤✐s ❝❛s❡ 2K(θ) ∼ π2θ2/6 ❛s θ → 0. ■t ❢♦❧❧♦✇s t❤❛t t❤❡ ❧♦❝❛❧ ❇❛❤❛❞✉r ❡✣❝✐❡♥❝② ♦❢ ♦✉r t❡st ✐s 0.3196.

❇❛❤❛❞✉r ❡✣❝✐❡♥❝② ♦❢ ❛ ❣♦♦❞♥❡ss✲♦❢✲✜t t❡sts ❜❛s❡❞ ♦♥ ❞✐str✐❜✉t✐♦♥ ❝❤❛r❛❝t❡r✐③❛t✐♦♥s✳ ❚❡sts ♦❢ ❡①♣♦♥❡♥t✐❛❧✐t② ❜❛s❡❞ ♦♥ ❘♦ss❜❡r❣✬s ❝❤❛r❛❝t❡r✐③❛t✐♦♥✱ ❛♥❞ t❤❡✐r ❡✣❝✐❡♥❝✐❡s✳

▲❛r❣❡ ❞❡✈✐❛t✐♦♥s ❛♥❞ ❧♦❝❛❧ ❡✣❝✐❡♥❝② ♦❢ st❛t✐st✐❝ Rn

▲❡t ❝❛❧❝✉❧❛t❡ t❤❡ ❧♦❝❛❧ ❇❛❤❛❞✉r ❡✣❝✐❡♥❝② ♦❢ Rn ❢♦r s♦♠❡ ❛❧t❡r♥❛t✐✈❡s✳ ▼❛❦❡❤❛♠ ❛❧t❡r♥❛t✐✈❡✳ ■t ✐s ❡❛s② t♦ s❡❡ t❤❛t ❢♦r ❛♥② t bR(t, θ) ∼ 1 2e−2t(1 − e−t)θ, θ → 0. ❆❢t❡r s✐♠♣❧❡ ❝♦♠♣✉t❛t✐♦♥ ✇❡ ❤❛✈❡ bR(θ) = supt≥0 bR(t, θ) = bR(ln 3 − ln 2, θ) ∼ 0.(074)θ, θ → 0.❚❤❡r❡❢♦r❡ t❤❡ ❧♦❝❛❧ ❡①❛❝t s❧♦♣❡ ❛s θ → 0 ❧♦♦❦s ❧✐❦❡ cR(θ) ∼ 0.01735θ2. ❲❡ ❤❛✈❡ ❛❧r❡❛❞② s❡❡♥ t❤❛t ❢♦r t❤❡ ▼❛❦❡❤❛♠ ❛❧t❡r♥❛t✐✈❡ 2K(θ) ∼ θ2/12 ❛s θ → 0✳ ❈♦♥s❡q✉❡♥t❧②✱ t❤❡ ❧♦❝❛❧ ❇❛❤❛❞✉r ❡✣❝✐❡♥❝② ♦❢ ♦✉r t❡st ✐s ❡q✉❛❧ t♦ 0.2071.

❇❛❤❛❞✉r ❡✣❝✐❡♥❝② ♦❢ ❛ ❣♦♦❞♥❡ss✲♦❢✲✜t t❡sts ❜❛s❡❞ ♦♥ ❞✐str✐❜✉t✐♦♥ ❝❤❛r❛❝t❡r✐③❛t✐♦♥s✳ ❚❡sts ♦❢ ❡①♣♦♥❡♥t✐❛❧✐t② ❜❛s❡❞ ♦♥ ❘♦ss❜❡r❣✬s ❝❤❛r❛❝t❡r✐③❛t✐♦♥✱ ❛♥❞ t❤❡✐r ❡✣❝✐❡♥❝✐❡s✳

▲❛r❣❡ ❞❡✈✐❛t✐♦♥s ❛♥❞ ❧♦❝❛❧ ❡✣❝✐❡♥❝② ♦❢ st❛t✐st✐❝ Rn

▲❡t ❝❛❧❝✉❧❛t❡ t❤❡ ❧♦❝❛❧ ❇❛❤❛❞✉r ❡✣❝✐❡♥❝② ♦❢ Rn ❢♦r s♦♠❡ ❛❧t❡r♥❛t✐✈❡s✳ ▲✐♥❡❛r ❢❛✐❧✉r❡ r❛t❡ ❛❧t❡r♥❛t✐✈❡✳ ❲❡ ❣❡t ❡❛s✐❧② t❤❛t bR(t, θ) ∼ 2 3te−2tθ, bR(θ) = sup

t≥0

bR(t, θ) = bR(1 2, θ) ∼ θ 3e. ❍❡♥❝❡ ✇❡ ❤❛✈❡ t❤❡ ❛s②♠♣t♦t✐❝s cR(θ) ∼ 0.0473θ2. ❆s s❡❡♥ ❜❡❢♦r❡✱ ✐♥ t❤✐s ❝❛s❡ 2K(θ) ∼ θ2 ❛s θ → 0✳ ❍❡♥❝❡✱ t❤❡ ❧♦❝❛❧ ❇❛❤❛❞✉r ❡✣❝✐❡♥❝② ♦❢ ♦✉r t❡st ✐s ✵✳✵✹✼✸✳

❇❛❤❛❞✉r ❡✣❝✐❡♥❝② ♦❢ ❛ ❣♦♦❞♥❡ss✲♦❢✲✜t t❡sts ❜❛s❡❞ ♦♥ ❞✐str✐❜✉t✐♦♥ ❝❤❛r❛❝t❡r✐③❛t✐♦♥s✳ ❚❡sts ♦❢ ❡①♣♦♥❡♥t✐❛❧✐t② ❜❛s❡❞ ♦♥ ❘♦ss❜❡r❣✬s ❝❤❛r❛❝t❡r✐③❛t✐♦♥✱ ❛♥❞ t❤❡✐r ❡✣❝✐❡♥❝✐❡s✳

❑✳❨✉✳❱♦❧❦♦✈❛✱ ❨❛✳❨✉✳◆✐❦✐t✐♥✳ ❖♥ t❤❡ ❛s②♠♣t♦t✐❝ ❡✣❝✐❡♥❝② ♦❢ ♥♦r♠❛❧✐t② t❡sts ❜❛s❡❞ ♦♥ t❤❡ ❙❤❡♣♣ ♣r♦♣❡rt②✳✴✴ ❱❡st♥✐❦ ❙t✳ P❡t❡rs❜✉r❣ ❯♥✐✈❡rs✐t②✳ ▼❡t❤❡♠❛t✐❝s✱ ✷✵✵✾✱ ✈✳✹✷✱ ◆♦✳✹✱ ✷✺✻✲✷✻✶✳ ❑✳❨✉✳❱♦❧❦♦✈❛✳ ❖♥ ❛s②♠♣t♦t✐❝ ❡✣❝✐❡♥❝② ♦❢ ❡①♣♦♥❡♥t✐❛❧✐t② t❡sts ❜❛s❡❞ ♦♥ ❘♦ss❜❡r❣✬s ❝❤❛r❛❝t❡r✐③❛t✐♦♥✳✴✴ ❩❛♣✐s❦✐ P❖▼■✱ ✷✵✵✾✱ ✈✳✸✻✽✱ ✾✺✲✶✵✾✳

❇❛❤❛❞✉r ❡✣❝✐❡♥❝② ♦❢ ❛ ❣♦♦❞♥❡ss✲♦❢✲✜t t❡sts ❜❛s❡❞ ♦♥ ❞✐str✐❜✉t✐♦♥ ❝❤❛r❛❝t❡r✐③❛t✐♦♥s✳ ❚❡sts ♦❢ ❡①♣♦♥❡♥t✐❛❧✐t② ❜❛s❡❞ ♦♥ ❘♦ss❜❡r❣✬s ❝❤❛r❛❝t❡r✐③❛t✐♦♥✱ ❛♥❞ t❤❡✐r ❡✣❝✐❡♥❝✐❡s✳

❚❤❛♥❦ ②♦✉ ❢♦r ②♦✉r ❛tt❡♥t✐♦♥✳