Question

Let {xn} is a sequence of independent r.V.S with p(xn=0)=p(xn=-n^)=1/2. What value of {xn} obeys strong law of large numbers

Answer 1

1. Let {Xn} be a sequence of random variables defined on the probability space ([0, ∞), β1, P), P({0}) = 0.
(a) Define Xn(s) = 1
s
¡
1 −
1
n
¢
, s ∈ [0, ∞). Then show that Xn
a.s
→ X, where X(s) = 1
s
, s ∈ [0, ∞).
(b) Define Xn(s) = 1
ns
, s ∈ [0, ∞). Then show that Xn
a.s
→ 0.
(c) Let P(I) = Z
I
e
−x
dx. Define
Xn(s) = ½
0, if s is rational
(−1)n, if s is irrational
Then show that {Xn} diverges almost surely.
2. Let S = [0, 1] and P be uniform distribution.
(a) Define
Xn(s) = e
−n
2
(ns−1)
, 0 ≤ s ≤ 1; n = 1, 2, . . .
Show that Xn
a.s
→ 0, but Xn
m.s 9 0.
(b) Define
Xn(s) = ½
3
n, if 0 ≤ s ≤ 1
0, otherwise
Show that Xn
p
→ 0, but Xn
m.s
9 0.
3. Consider a sequence of random variables {Xn} with P(Xn = 0) = 1 −
1
nα , P(Xn = ±n) = 1
2nα . Determine
the value of α for which the sequence obeys WLLN.
4. Consider a sequence of random variables {Xn} with E(Xn) = m and
Cov(Xi
, Xj ) =



σ
2
, i = j
aσ2
, i = j ± 1, where |a| < 1, σ2 > 0 are given contants
0, otherwise
Show that WLLN holds for {Xn}.
5. Use CLT to show that limn→∞
e
−nXn
i=0
n
i
i!
= 0.5
6. Consider a sequence of independent random variables {Xn} such that
P(Xn = 0) = 1 −
2
n3
, P(Xn = ±n) = 1
n3
, n > 1
Does the sequence {Xn} obey CLT ?
7. If Xn
p
→ 0, then find the median of Xn → 0 as n → ∞.
8. Consider a sequence {Xn} of identically distributed random variables with the property that nP(|Xi
| > n → 0
as n → ∞. Show that 1
n max1≤i≤n Xi
p
→ 0
9. Suppose |Xn − X| ≤ Yn, almost surely for some random variable X, then show that of E(Yn) → 0, then
E(Xn) → E(X) and Xn
p
→ X.
10. Show that Xn
2
→ X ⇒ E(Xn) → E(X), E(X2
n) → E(X2
) as n → ∞
11. Let {Xi} be a sequence of independent random variables, such that each Xi has mean 0 and variance 1.
Show that
√
n
X1 + X2 + . . . + Xn
X2
1 + X2
2 + . . . + X2
n
l
→ Z ∼ N(0, 1)
12. Does WLLN hold for the following sequences
(a) P(Xk = ±2
k
) = 1
2
(b) P(Xk = ±
1
k
) = 1
2
13. For what values of α, does the strong law of large numbers hold for the sequence {Xn}, where P(Xk =
±k
α) = 1
2
, k = 1, 2, . . ..
14. Let {Xn} be a sequence of independent random variables with the following probability distribution. In each
case, does the Lindeberg Condition for CLT hold ?
(a) P(Xk = ±
1
2n ) = 1
2
(b) P(Xk = ±
1
2n+1 ) = 1
2n+3 , P(Xn = 0) = 1 −
1
2n+2
15. From an urn containing 10 identical balls numbered 0 through 9, n balls are drawn with replacement,
(a) What does the law of large number tell you about the appearance of 0’s in n drawings.
(b) How many drawings must be made in order that with probability atleast 0.95, the relative frequency of
occurence of 0’s will be between 0.09 and 0.11 ?
(c) Use CLT to find the probability that among n numbers thus chosen, the number 5 will appear between
n−35n
10 and n+35n
10 times, if (i) n = 25 (ii) n = 100.