Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

330 Consistency and Limiting Distributions

The last example shows in general that we cannot determine limiting distribu-
tions by considering pmfs or pdfs. But under certain conditions we can determine
convergence in distribution by considering the sequence of pdfs as the following
example shows.

Example 5.2.3. LetTn have at-distribution withndegrees of freedom,n =
1 , 2 , 3 ,....Thusitscdfis

Fn(t)=

∫t

−∞

Γ[(n+1)/2]

√

πnΓ(n/2)

1

(1 +y^2 /n)(n+1)/^2

dy,

where the integrand is the pdffn(y)ofTn. Accordingly,

lim

n→∞

Fn(t) = lim

n→∞

∫t

−∞

fn(y)dy=

∫t

−∞

lim

n→∞

fn(y)dy,

by a result in analysis (the Lebesgue Dominated Convergence Theorem) that allows
us to interchange the order of the limit and integration, provided that|fn(y)|is
dominated by a function that is integrable. This is true because

|fn(y)|≤ 10 f 1 (y)

and ∫t

−∞

10 f 1 (y)dy=

10

π

arctant<∞,

for all realt. Hence we can find the limiting distribution by finding the limit of the
pdf ofTn.Itis

lim

n→∞

fn(y) = lim

n→∞

{

Γ[(n+1)/2]

√

n/2Γ(n/2)

}

lim

n→∞

{

1

(1 +y^2 /n)^1 /^2

}

×lim

n→∞

{

1

√

2 π

[(

1+

y^2

n

)]−n/ 2 }

.

Using the fact from elementary calculus that

lim

n→∞

(

1+

y^2

n

)n

=ey

2

,

the limit associated with the third factor is clearly the pdf of the standard normal

distribution. The second limit obviously equals 1. By Remark 5.2.2, the first limit

also equals 1. Thus, we have

lim

n→∞

Fn(t)=

∫t

−∞

1

√

2 π

e−y

(^2) / 2
dy,
and henceTnhas a limiting standard normal distribution.