❱Ò ♠ét ❞➵♥❣ ❤é✐ tô ❝ñ❛ ❞➲②
✈➭ ❝❤✉ç✐ ♥❤✐Ò✉ ❝❤Ø sè ❝➳❝ ➤➵✐ ❧➢î♥❣ ♥❣➱✉ ♥❤✐➟♥
◆❣✉②Ô♥ ❱➝♥ ◗✉➯♥❣
✭❛✮
➜➷♥❣ ❱➝♥ ❍➯✐
✭❜✮
✱ ◆❣✉②Ô♥ ❚❤Þ ❚❤Õ
✭❛✮
❚ã♠ t➽t✳
●✐➯ sö
N
❧➭ t❐♣ ❤î♣ ❝➳❝ sè ♥❣✉②➟♥ ❞➢➡♥❣✱
d
❧➭ ♠ét sè ♥❣✉②➟♥ ❞➢➡♥❣
Nd={n= (n1, n2, ..., nd) : ni∈N, i = 1,2, ..., d}.
✈➭
{X(n), n ∈Nd}
❧➭ ❞➲②
d
✲❝❤Ø sè ❝➳❝ ➤➵✐ ❧➢î♥❣ ♥❣➱✉ ♥❤✐➟♥✳ ▼ô❝ ➤Ý❝❤ ❝ñ❛ ❜➭✐ ❜➳♦
♥➭② ❧➭ t❤✐Õt ❧❐♣ ♠ét sè tÝ♥❤ ❝❤✃t ❝➡ ❜➯♥ ✈Ò sù ❤é✐ tô ❝ñ❛ ❞➲②
{X(n), n ∈Nd}
✈➭ ❝❤✉ç✐
Pn∈NdX(n)
❦❤✐
max
16i6dni=∨ni→ ∞
✳
✶✳
▼ë ➤➬✉
●✐➯ sö
d
❧➭ ♠ét sè ♥❣✉②➟♥ ❞➢➡♥❣✱ ❦ý ❤✐Ö✉
Nd={n= (n1, n2, ..., nd) : ni∈N, i = 1,2, ..., d}.
❚r♦♥❣
Nd
✱ q✉❛♥ ❤Ö t❤ø tù ➤➢î❝ ➤Þ♥❤ ♥❣❤Ü❛ ♥❤➢ s❛✉✿
❱í✐
m= (m1, m2, .., md), n = (n1, n2, ..., nd)
❧➭ ❤❛✐ ♣❤➬♥ tö ❝ñ❛
Nd
✱ ❦❤✐ ➤ã
m6n
➤✳♥
⇐⇒ mi6ni, i = 1, . . . , d.
●ä✐ t❐♣ ❝♦♥ ❝ñ❛
Nd
✱ ♠➭ ✈í✐ ♠ç✐ ♣❤➬♥ tö ❝ñ❛ ♥ã ❝➳❝ ❝❤Ø sè ➤Ò✉ ❜➺♥❣ ♥❤❛✉✱ ❧➭
Id={i=
(i, i, . . . , i) : i∈N}.
❱í✐ ♠ç✐
n= (n1, n2, . . . , nd)
✱ ➤➷t
∨ni= max
16i6dni;∧ni= min
16i6dni
✈➭
|n|=n1.n2. . . nd.
❈➳❝ ❞➲② ✈➭ ❝❤✉ç✐ ♠➭ t❐♣ ❝❤Ø sè ❧➭
Nd
➤➢î❝ ❣ä✐ ❧➭ ❞➲② ✈➭ ❝❤✉ç✐
d
✲ ❝❤Ø sè✳
❚r♦♥❣ t❤ê✐ ❣✐❛♥ ❣➬♥ ➤➞②✱ ❝ã ♥❤✐Ò✉ ❜➭✐ ❜➳♦ ♥❣❤✐➟♥ ❝ø✉ ✈Ò ❣✐í✐ ❤➵♥ ❝ñ❛ ❞➲② ✈➭ ❝❤✉ç✐
d
✲ ❝❤Ø sè ❝➳❝ ➤➵✐ ❧➢î♥❣ ♥❣➱✉ ♥❤✐➟♥ ✭①❡♠ ❝❤➻♥❣ ❤➵♥ ❬✸❪✱ ❬✹❪✱ ❬✺❪ ❬✻❪✱ ❬✽❪✮✳ ❚✉② ♥❤✐➟♥✱
❝❤➢❛ ❝ã ❜➭✐ ❜➳♦ ♥➭♦ tr×♥❤ ❜➭② ✈➭ ❝❤ø♥❣ ♠✐♥❤ ♠ét ❝➳❝❤ ❝❤✐ t✐Õt ❝❤➷t ❝❤Ï ❝➳❝ tÝ♥❤ ❝❤✃t
❝➡ ❜➯♥ ❝ñ❛ ❧♦➵✐ ❣✐í✐ ❤➵♥ ♥➭②✳
➜è✐ ✈í✐ ❧♦➵✐ ❞➲② ✈➭ ❝❤✉ç✐
d
✲ ❝❤Ø sè✱ ♥❣➢ê✐ t❛ t❤➢ê♥❣ q✉❛♥ t➞♠ ♥❣❤✐➟♥ ❝ø✉ ❣✐í✐ ❤➵♥
❝ñ❛ ❝❤ó♥❣ ❦❤✐
∨ni→ ∞
❤♦➷❝ ❦❤✐
∧ni→ ∞
✳ ❈➯ ❤❛✐ ❞➵♥❣ ❣✐í✐ ❤➵♥ ♥➭② ➤Ò✉ ❧➭ ♠ë ré♥❣
❝ñ❛ ❣✐í✐ ❤➵♥ ❝ñ❛ ❞➲② ✈➭ ❝❤✉ç✐ ♠ét ❝❤Ø sè t❤➠♥❣ t❤➢ê♥❣✳ ❙ù ❤é✐ tô ❝ñ❛ ❞➲② ✈➭ ❝❤✉ç✐
d
✲ ❝❤Ø sè ❝➳❝ ➤➵✐ ❧➢î♥❣ ♥❣➱✉ ♥❤✐➟♥ ✭➜▲◆◆✮ ❦❤✐
∧ni→ ∞
➤➲ ➤➢î❝ ♥❣❤✐➟♥ ❝ø✉ tr♦♥❣ ❬✶❪✳
✶
◆❤❐♥ ❜➭✐ ♥❣➭② ✸✵✴✶✵✴✷✵✵✻✳ ❙ö❛ ❝❤÷❛ ①♦♥❣ ♥❣➭② ✷✾✴✶✷✴✷✵✵✻✳
▼ô❝ ➤Ý❝❤ ❝❤Ý♥❤ ❝ñ❛ ❜➭✐ ❜➳♦ ♥➭② ❧➭ ♥❣❤✐➟♥ ❝ø✉ sù ❤é✐ tô ❝ñ❛ ❞➲② ✈➭ ❝❤✉ç✐
d
✲ ❝❤Ø sè ❝➳❝
➜▲◆◆ ❦❤✐
∨ni→ ∞
✳ ❈❤ó♥❣ t➠✐ sÏ ❝❤Ø r❛ r➺♥❣ ♥❤✐Ò✉ tÝ♥❤ ❝❤✃t ❝ñ❛ ❞➲② ✈➭ ❝❤✉ç✐ ❝➳❝
➜▲◆◆ ✈➱♥ ❝ß♥ ➤ó♥❣ ➤è✐ ✈í✐ ❞➲② ✈➭ ❝❤✉ç✐ ♥❤✐Ò✉ ❝❤Ø sè✳ ▼➷t ❦❤➳❝✱ ❝ã ♠ét sè tÝ♥❤ ❝❤✃t
❦❤➠♥❣ ❝ß♥ ➤ó♥❣ ♥÷❛✳ ❈➳❝ ♣❤Ð♣ ❝❤ø♥❣ ♠✐♥❤ ❝➳❝ tÝ♥❤ ❝❤✃t ❝ò♥❣ ❝ã sù t❤❛② ➤æ✐ ♥❤✃t
➤Þ♥❤✱ ❞♦ q✉❛♥ ❤Ö t❤ø tù tr➟♥
Nd(d > 1)
❦❤➠♥❣ ♣❤➯✐ ❧➭ q✉❛♥ ❤Ö t❤ø tù t✉②Õ♥ tÝ♥❤✳
➜Ó ❧➭♠ ❝➡ së✱ tr➢í❝ ❤Õt ❝❤ó♥❣ t❛ ❝➬♥ ♥❣❤✐➟♥ ❝ø✉ ✈Ò sù ❤é✐ tô ❝ñ❛ ❞➲② sè ✈➭ ❝❤✉ç✐ sè
d
✲ ❝❤Ø sè✳
✷✳
❙ù ❤é✐ tô ❝ñ❛ ❞➲② sè ✈➭ ❝❤✉ç✐ sè
d
✲ ❝❤Ø sè
➜Þ♥❤ ♥❣❤Ü❛
✷✳✶
✳
●✐➯ sö
{x(n), n ∈Nd}
❧➭ ❞➲② sè
d
✲ ❝❤Ø sè✳ ❚❛ ♥ã✐ ❞➲②
{x(n), n ∈Nd}
❤é✐ tô tí✐
x∈R
❦❤✐
∨ni→ ∞
✭t➢➡♥❣ ø♥❣ ❦❤✐
∧ni→ ∞)
✱ ♥Õ✉ ✈í✐ ♠ä✐
ε > 0
✱ tå♥
t➵✐
n0∈N
✭t➢➡♥❣ ø♥❣ tå♥ t➵✐
n0∈Nd
✮ s❛♦ ❝❤♦ ✈í✐ ♠ä✐
n= (n1, n2, . . . , nd)∈Nd
♠➭
∨ni≥n0
✭t➢➡♥❣ ø♥❣
n≥n0
✮ t❤×
|x(n)−x|< ε
✳ ❑ý ❤✐Ö✉
lim
∨ni→∞ x(n) = x
✭t➢➡♥❣ ø♥❣
lim
∧ni→∞ x(n) = x).
➜Þ♥❤ ♥❣❤Ü❛
✷✳✷
✳
●✐➯ sö
{x(n), n ∈Nd}
❧➭ ❞➲② sè
d
✲ ❝❤Ø sè✳ ❚❛ ♥ã✐ ❞➲②
{x(n), n ∈Nd}
❤é✐ tô tí✐
x∈R
❦❤✐
|n| → ∞
✱ ♥Õ✉ ✈í✐ ♠ä✐
ε > 0
✱ tå♥ t➵✐
n0∈N
s❛♦ ❝❤♦ ✈í✐ ♠ä✐
n= (n1, n2, . . . , nd)∈Nd
♠➭
|n| ≥ n0
t❤×
|x(n)−x|< ε.
❑ý ❤✐Ö✉
lim
|n|→∞ x(n) = x.
◆❤❐♥ ①Ðt✿ ❉Ô ♥❤❐♥ t❤✃② r➺♥❣ ♥Õ✉
n≥n0
t❤×
|n| ≥ |n0|
✈➭
∨ni→ ∞
❦❤✐ ✈➭ ❝❤Ø ❦❤✐
|n| → ∞
✳ ❉♦ ➤ã✱ ❣✐÷❛ ❤❛✐ ➤Þ♥❤ ♥❣❤Ü❛ tr➟♥ ❝ã ♠è✐ q✉❛♥ ❤Ö s❛✉✿
lim
|n|→∞ x(n) = x⇔lim
∨ni→∞ x(n) = x⇒lim
∧ni→∞ x(n) = x.
➜Þ♥❤ ♥❣❤Ü❛
✷✳✸
✳
●✐➯ sö
{x(n), n ∈Nd}
❧➭ ❞➲② sè
d
✲ ❝❤Ø sè✳ ❑❤✐ ➤ã
Pn∈Ndx(n) (1)
➤➢î❝ ❣ä✐ ❧➭ ❝❤✉ç✐ sè
d
✲ ❝❤Ø sè✳
➜➷t
S(n) = Pm6nx(m)
✱
S(n)
➤➢î❝ ❣ä✐ ❧➭ tæ♥❣ r✐➟♥❣ t❤ø
n
❝ñ❛ ❝❤✉ç✐ ✭✶✮✳ ❚❛ ♥ã✐
❝❤✉ç✐ ✭✶✮ ❤é✐ tô ❦❤✐
∨ni→ ∞
♥Õ✉ ❞➲②
{S(n), n ∈Nd}
❤é✐ tô ❦❤✐
∨ni→ ∞
✈➭
S=
lim
∨ni→∞ S(n) = Pn∈Ndx(n)
➤➢î❝ ❣ä✐ ❧➭ tæ♥❣ ❝ñ❛ ❝❤✉ç✐ ✭✶✮
r(n) = S−S(n) = X
m:mn
x(m)
➤➢î❝ ❣ä✐ ❧➭ ♣❤➬♥ ❞➢ t❤ø
n
❝ñ❛ ❝❤✉ç✐ ✭✶✮✳
❘â r➭♥❣✱ ♥Õ✉ ❝❤✉ç✐ ✭✶✮ ❤é✐ tô t❤×
lim
∨ni→∞ r(n) = 0
✳
❚õ ❝➳❝ ➤Þ♥❤ ♥❣❤Ü❛ tr➟♥✱ t❛ ❞Ô ❞➭♥❣ ❝❤ø♥❣ ♠✐♥❤ ➤➢î❝ ❝➳❝ ➤Þ♥❤ ❧ý s❛✉✳
➜Þ♥❤ ❧ý
✷✳✹
✳
●✐➯ sö
{x(n), n ∈Nd}
✈➭
{y(n), n ∈Nd}
❧➭ ❝➳❝ ❞➲② sè
d
✲ ❝❤Ø sè✳
✭✐✮ ◆Õ✉
x(n)→x
✈➭
y(n)→y
❦❤✐
∨ni→ ∞ (∧ni→ ∞)
t❤× tæ♥❣
x(n) + y(n)→x+y
❦❤✐
∨ni→ ∞ (∧ni→ ∞).
✭✐✐✮ ◆Õ✉
x(n)→x
❦❤✐
∨ni→ ∞ (∧ni→ ∞)
t❤×
|x(n)| → |x|
❦❤✐
∨ni→ ∞ (∧ni→ ∞).
✭✐✐✐✮ ◆Õ✉
x(n)→x
❦❤✐
∨ni→ ∞ (∧ni→ ∞)
t❤×
λx(n)→λx
❦❤✐
∨ni→ ∞ (∧ni→ ∞)
✈í✐
λ∈C.
➜Þ♥❤ ❧ý
✷✳✺
✳
●✐➯ sö
Pn∈Ndx(n)
❤é✐ tô ➤Õ♥
x∈R
✱
Pn∈Ndy(n)
❤é✐ tô ➤Õ♥
y∈R
❦❤✐
∨ni→ ∞
✳ ❑❤✐ ➤ã
✭✐✮ ❈❤✉ç✐
Pn∈Nd(x(n) + y(n))
❤é✐ tô ➤Õ♥
x+y
❦❤✐
∨ni→ ∞.
✭✐✐✮ ❈❤✉ç✐
Pn∈Ndλx(n), λ ∈C
❤é✐ tô ➤Õ♥
λx
❦❤✐
∨ni→ ∞.
➜Þ♥❤ ♥❣❤Ü❛
✷✳✻
✳
●✐➯ sö
{x(n), n ∈Nd}
❧➭ ❞➲② sè
d
✲ ❝❤Ø sè✳ ❚❛ ♥ã✐ ❞➲②
{x(n), n ∈Nd}
❜Þ ❝❤➷♥
✱ ♥Õ✉ tå♥ t➵✐ sè
M > 0
s❛♦ ❝❤♦
|x(n)|6M,
✈í✐ ♠ä✐
n∈Nd.
➜Þ♥❤ ❧ý s❛✉ ➤➞② ❝ã t❤Ó ➤➢î❝ ❝❤ø♥❣ ♠✐♥❤ ❤♦➭♥ t♦➭♥ t➢➡♥❣ tù ♥❤➢ tr➢ê♥❣ ❤î♣ ❞➲②
♠ét ❝❤Ø sè✳
➜Þ♥❤ ❧ý
✷✳✼
✳
●✐➯ sö
{x(n), n ∈Nd}
❧➭ ❞➲② sè
d
✲ ❝❤Ø sè✳ ◆Õ✉ ❞➲②
{x(n), n ∈Nd}
❤é✐ tô
tí✐
x∈R
❦❤✐
∨ni→ ∞
t❤×
{x(n), n ∈Nd}
❜Þ ❝❤➷♥✳
◆❤❐♥ ①Ðt✿ ◆Õ✉ ❞➲②
{x(n), n ∈Nd}
❤é✐ tô tí✐
x∈R
❦❤✐
∧ni→ ∞
t❤×
{x(n), n ∈Nd}
❝❤➢❛ ❝❤➽❝ ➤➲ ❜Þ ❝❤➷♥✳
❈❤➻♥❣ ❤➵♥ ①Ðt ❞➲② sè ❤❛✐ ❝❤Ø sè
{x(m, n)}
✱ ✈í✐
x(m, n) = (m,
♥Õ✉
n= 1
0,
♥Õ✉
n6= 1.
❑❤✐ ➤ã râ r➭♥❣
x(m, n)→0
✱ ❦❤✐
m∧n→ ∞
♥❤➢♥❣ ❞➲② ➤ã ❦❤➠♥❣ ❜Þ ❝❤➷♥✳
❚➢➡♥❣ tù ♥❤➢ tr➢ê♥❣ ❤î♣ ❞➲② ♠ét ❝❤Ø sè✱ ♥ã✐ ❝❤✉♥❣ ♠ét ❞➲② sè
d
✲ ❝❤Ø sè ❜Þ ❝❤➷♥ t❤×
❝❤➢❛ ❝❤➽❝ ➤➲ ❤é✐ tô✳ ❚✉② ♥❤✐➟♥✱ t❛ ❝ã ➤Þ♥❤ ❧ý s❛✉✿
➜Þ♥❤ ❧ý
✷✳✽
✳
●✐➯ sö
{x(n), n ∈Nd}
❧➭ ❞➲② sè ❜Þ ❝❤➷♥✳ ❑❤✐ ➤ã ♥Õ✉ ❞➲②
{x(n), n ∈Nd}
➤➡♥ ➤✐Ö✉ t➝♥❣
✭
❣✐➯♠
✮
t❤❡♦ ♥❣❤Ü❛
x(n)≥x(m)
✭
x(n)6x(m)
✮ ❦❤✐
∨ni≥ ∨mi
✱ t❤×
{x(n), n ∈Nd}
❤é✐ tô ❦❤✐
∨ni→ ∞
✈➭
lim
∨ni→∞ x(n) = lim
i→∞ x(i) (i∈Id).
❈❤ø♥❣ ♠✐♥❤✳
●✐➯ sö
{x(n), n ∈Nd}
➤➡♥ ➤✐Ö✉ t➝♥❣✳ ●ä✐
{x(i), i ∈Id}
❧➭ ❞➲② ❝♦♥ ❝ñ❛
❞➲②
x(n), n ∈Nd}
✳ ❉♦
{x(n), n ∈Nd}
❧➭ ❞➲② ➤➡♥ ➤✐Ö✉ t➝♥❣ ✈➭ ❜Þ ❝❤➷♥ ♥➟♥
{x(i), i ∈Id}
❧➭ ❞➲② ♠ét ❝❤Ø sè ➤➡♥ ➤✐Ö✉ t➝♥❣ ✈➭ ❜Þ ❝❤➷♥ ♥➟♥ ❤é✐ tô ✈Ò
x
♥➭♦ ➤ã ❦❤✐
i→ ∞
✳ ❚❛ ❝❤ø♥❣
♠✐♥❤
{x(n), n ∈Nd}
❤é✐ tô ✈Ò
x
❦❤✐
∨ni→ ∞
✳
❚❤❐t ✈❐②✱ ✈í✐ ♠ä✐
ε > 0
✱ tå♥ t➵✐
n0∈N
s❛♦ ❝❤♦
06x−x(i)< ε (i∈Id)
✈í✐ ♠ä✐
i≥n0
✳
❑❤✐ ➤ã✱ ✈í✐ ♠ä✐
n
♠➭
∨ni=k > n0
✱ t❛ ❝ã
x(n0)6x(n)6x(k),(n0, k ∈Id)
✳ ❉♦ ➤ã
06x−x(k)6x−x(n)6x−x(n0)< ε.
❚õ ➤ã s✉② r❛
lim
∨ni→∞ x(n) = x= lim
i→∞ x(i) (i∈Id).
◆Õ✉
{x(n), n ∈Nd}
➤➡♥ ➤✐Ö✉ ❣✐➯♠ t❤×
{−x(n), n ∈Nd}
➤➡♥ ➤✐Ö✉ t➝♥❣✳ ❙ö ❞ô♥❣ ❦Õt q✉➯
✈õ❛ ❝❤ø♥❣ ♠✐♥❤ ✈➭ ➤Þ♥❤ ❧ý ✷✳✹ t❛ ➤➢î❝ ➤✐Ò✉ ♣❤➯✐ ❝❤ø♥❣ ♠✐♥❤✳
➜Þ♥❤ ♥❣❤Ü❛
✷✳✾
✳
❉➲② sè
{x(n), n ∈Nd}
➤➢î❝ ❣ä✐ ❧➭
❞➲② ❈❛✉❝❤②
❦❤✐
∨ni→ ∞
✱ ♥Õ✉
✈í✐ ♠ä✐
ε > 0
✱ tå♥ t➵✐
no∈N
s❛♦ ❝❤♦ ✈í✐ ♠ä✐
m, n ∈Nd
♠➭
∨mi≥n0,∨ni≥n0
t❤×
|x(m)−x(n)|< ε.
➜Þ♥❤ ❧ý
✷✳✶✵
✳
➜Ó ❞➲② sè
{x(n), n ∈Nd}
❤é✐ tô ➤Õ♥
x∈R,
❦❤✐
∨ni→ ∞,
➤✐Ò✉ ❦✐Ö♥ ❝➬♥
✈➭ ➤ñ ❧➭
{x(n), n ∈Nd}
❧➭ ❞➲② ❈❛✉❝❤②✳
❈❤ø♥❣ ♠✐♥❤✳
➜✐Ò✉ ❦✐Ö♥ ❝➬♥✿ ●✐➯ sö
{x(n), n ∈Nd}
❤é✐ tô tí✐ ♣❤➬♥ tö
x∈R
❦❤✐
∨ni→ ∞
✱ ✈í✐ ♠ä✐
ε > 0
✱ tå♥ t➵✐
n0∈N
s❛♦ ❝❤♦ ✈í✐ ♠ä✐
n∈Nd
♠➭
∨ni≥n0
t❤×
|x(n)−x|<ε
2.
❑❤✐ ➤ã ✈í✐ ♠ä✐
m, n ∈Nd
♠➭
∨mi≥n0,∨ni≥n0
✱ t❛ ❝ã
|x(m)−x(n)|=
|(x(m)−x)−(x(n)−x)|6|x(m)−x|+|x(n)−x|<ε
2+ε
2=ε
✳
➜✐Ò✉ ❦✐Ö♥ ➤ñ✿ ●ä✐
{x(i), i ∈Id}
❧➭ ❞➲② ❝♦♥ ❝ñ❛ ❞➲② sè
{x(n), n ∈Nd}.
❑❤✐ ➤ã ❞➲② sè
{x(i), i ∈Id}
①❡♠ ♥❤➢ ❧➭ ❞➲② ❈❛✉❝❤② ♠ét ❝❤Ø sè ♥➟♥
{x(i), i ∈Id}
❤é✐ tô ✈Ò
x∈R
♥➭♦ ➤ã✱ ❦❤✐
i→ ∞
✳ ❑Õt ❤î♣ ✈í✐ ❣✐➯ t❤✐Õt
{x(n), n ∈Nd}
❧➭ ❞➲② ❈❛✉❝❤②✱ s✉② r❛ r➺♥❣
✈í✐ ♠ä✐
ε > 0
✱ tå♥ t➵✐
n0∈N
s❛♦ ❝❤♦ ✈í✐ ♠ä✐
n∈Nd, i ∈Id
♠➭
∨ni≥n0, i ≥n0
t❛
❝ã
|x(n)−x|6|x(n)−x(i)|+|x(i)−x|< ε
✳ ❱❐②
{x(n), n ∈Nd}
❤é✐ tô tí✐
x∈R
❦❤✐
∨ni→ ∞.
✸✳
❙ù ❤é✐ tô ❝ñ❛ ❞➲② ✈➭ ❝❤✉ç✐
d
✲ ❝❤Ø sè
❝➳❝ ➤➵✐ ❧➢î♥❣ ♥❣➱✉ ♥❤✐➟♥
●✐➯ sö
{X(n), n ∈Nd}
❧➭ ❞➲②
d
✲ ❝❤Ø sè ❝➳❝ ➜▲◆◆ ①➳❝ ➤Þ♥❤ tr➟♥ ❦❤➠♥❣ ❣✐❛♥ ①➳❝ s✉✃t
(Ω,F,P)
✱ ➤➡♥ ❣✐➯♥ t❛ ✈✐Õt
{X(n)}
❧➭ ❞➲② ➜▲◆◆✳
❉➲② ❝♦♥ ❝ñ❛ ❞➲②
{X(n), n ∈Nd}
❝ã ❝❤Ø sè t❤✉é❝ t❐♣
Id,
➤➢î❝ ❦ý ❤✐Ö✉ ❧➭
{X(i), i ∈Id}
✱
❤♦➷❝ ➤➡♥ ❣✐➯♥ ❧➭
{X(i)}.
➜Þ♥❤ ♥❣❤Ü❛
✸✳✶
✳
❚❛ ♥ã✐
✭✐✮ ❉➲②
{X(n)}
❤é✐ tô t❤❡♦ ①➳❝ s✉✃t
tí✐ ➜▲◆◆
X
❦❤✐
∨ni→ ∞,
♥Õ✉ ✈í✐ ♠ä✐
ε > 0
✱ t❛
❝ã
lim
∨ni→∞ P(|X(n)−X|> ε) = 0
✱ ❦ý ❤✐Ö✉
X(n)P
−→ X
✱ ❦❤✐
∨ni→ ∞.
✭✐✐✮ ❉➲②
{X(n)}
❤é✐ tô ❤➬✉ ❝❤➽❝ ❝❤➽♥
✭❤✳❝✳❝✮ tí✐ ➜▲◆◆
X
❦❤✐
∨ni→ ∞
♥Õ✉ tå♥ t➵✐
t❐♣
A
❝ã ①➳❝ s✉✃t
0
s❛♦ ❝❤♦
X(n)(ω)→X(ω)
✱ ❦❤✐
∨ni→ ∞
✈í✐ ♠ä✐
ω /∈A.
❑ý ❤✐Ö✉
X(n)
❤✳❝✳❝
−→ X,
❦❤✐
∨ni→ ∞.
✭✐✐✐✮ ❉➲②
{X(n)}
❤é✐ tô t❤❡♦ tr✉♥❣ ❜×♥❤ ❝✃♣
p(0 < p < ∞)
tí✐ ➜▲◆◆ ❳ ❦❤✐
∨ni→ ∞
✱
♥Õ✉
lim
∨ni→∞ E|X(n)−X|p= 0.
❑ý ❤✐Ö✉
X(n)Lp
→X,
❦❤✐
∨ni→ ∞.
❇æ ➤Ò
✸✳✷
✳
●✐➯ sö
{A(n)}
❧➭ ❞➲②
d
✲ ❝❤Ø sè ❝➳❝ ❜✐Õ♥ ❝è✳ ❑❤✐ ➤ã
✭✐✮ ◆Õ✉
{A(n)}
❧➭ ❞➲② t➝♥❣ t❤❡♦ ♥❣❤Ü❛
A(m)⊂A(n)
❦❤✐
∨mi6∨ni
t❤×
P(Sn∈NdA(n)) =
lim
∨ni→∞ P(A(n)).
✭✐✐✮ ◆Õ✉
{A(n)}
❧➭ ❞➲② ❣✐➯♠ t❤❡♦ ♥❣❤Ü❛
A(m)⊃A(n)
❦❤✐
∨mi6∨ni
t❤×
P(Tn∈NdA(n)) =
lim
∨ni→∞ P(A(n)).
❈❤ø♥❣ ♠✐♥❤✳
✭✐✮ ●✐➯ sö
{A(n)}
❧➭ ❞➲② ❝➳❝ ❜✐Õ♥ ❝è t➝♥❣✳ ●ä✐ ❞➲② ❝♦♥ ❝ñ❛ ❞➲②
{A(n)}
❝ã t❐♣ ❝❤Ø sè t❤✉é❝ t❐♣
Id
❧➭
{A(i)}.
❑❤✐ ➤ã t❛ ①❡♠
{A(i)}
♥❤➢ ❧➭ ❞➲② ❜✐Õ♥ ❝è ♠ét ❝❤Ø
sè ✈➭
{A(i)}
❧➭ ❞➲② t➝♥❣✱ ♥➟♥ t❛ ❝ã
P(Si∈IdA(i)) = lim
i→∞ P(A(i)).
❚❛ ❝❤ø♥❣ ♠✐♥❤
Sn∈NdA(n) = Si∈IdA(i).
❍✐Ó♥ ♥❤✐➟♥
Sn∈NdA(n)⊃Si∈IdA(i)
✭✶✮✳ ❚❛ ❝❤ø♥❣ ♠✐♥❤ ➤✐Ò✉ ♥❣➢î❝ ❧➵✐✳ ●✐➯ sö
ω∈Sn∈NdA(n)
✱ ❦❤✐ ➤ã tå♥ t➵✐
n0= (n01, n02, . . . , n0d)∈Nd
s❛♦ ❝❤♦
ω∈A(n0).
❈❤ä♥
i0= (m0, m0, . . . , m0)∈Id
✈í✐
m0= max
16i6dn0i.
❑❤✐ ➤ã râ r➭♥❣
∨n0i6∨mo=m0
✱
♥➟♥
A(n0)⊂A(i0),
s✉② r❛
ω∈A(i0)
✱ ♥➟♥
ω∈Si∈IdA(i)
✳ ❚õ ➤ã t❛ ❝ã
Sn∈NdA(n)⊂
Si∈IdA(i
✮ ✭✷✮✳
❚õ ✭✶✮ ✈➭ ✭✷✮ s✉② r❛
Sn∈NdA(n) = Si∈IdA(i).
❉♦ ➤ã
P(Sn∈NdA(n)) = lim
i→∞ P(A(i)).
❚õ ❣✐➯ t❤✐Õt s✉② r❛
{P(A(n))}
❧➭ ❞➲② sè t➝♥❣❀ ➳♣ ❞ô♥❣ ➤Þ♥❤ ❧ý ✷✳✽ t❛ ➤➢î❝
lim
∨ni→∞ P(A(n)) = lim
i→∞ P(A(i)) = P([
n∈Nd
A(n)).
✭✐✐✮ ❈❤ø♥❣ ♠✐♥❤ t➢➡♥❣ tù✳
❇æ ➤Ò s❛✉ ➤➞② ❧➭ ❞➵♥❣
d
✲ ❝❤Ø sè ❝ñ❛ ❜æ ➤Ò ❇♦r❡❧✲❈❛♥t❡❧❧✐
