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Past exam of the mathematics course of the University of Cambridge
/
2021
/
iii
/
Paper 215
/
1
/
d
/
Solution
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Past exam of the mathematics course of the University of Cambridge
2021
iii
Paper 215
1
d
2026-09-28
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Put
T
=
t
mix
(
2
)
(
a
,
1/4
)
and
m
=
⌈
t
rel
⌉
. The expected local
time
is
E
a
∑
k
=
0
T
−
1
1
{
X
k
=
a
}
=
T
π
(
a
)
+
∑
k
=
0
T
−
1
{
P
k
(
a
,
a
)
−
π
(
a
)}
.
(1)
Part
c
and the supplied return identity give
T
π
(
a
)
≤
8
π
(
a
)
E
π
τ
a
=
8
∑
k
=
0
∞
{
P
k
(
a
,
a
)
−
π
(
a
)}
.
(2)
The
second
term is bounded by the same infinite
sum
. Part
b
now yields
E
a
∑
k
=
0
T
−
1
1
{
X
k
=
a
}
≤
e
−
1
9
e
∑
k
=
0
m
{
P
k
(
a
,
a
)
−
π
(
a
)}
≤
e
−
1
9
e
E
a
∑
k
=
0
m
1
{
X
k
=
a
}
.
(3)
Thus the hinted universal constant
C
=
9
e
/
(
e
−
1
)
works
.
Ancestors
(11)
d
1
Paper 215
iii
2021
Past exam of the mathematics course of the University of Cambridge
Mathematics course of the University of Cambridge
Course of the University of Cambridge
University of Cambridge
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