LINKÖPING UNIVERSITY
Department of Mathematics
Mathematical Statistics
John Karlsson
TAMS29
Stochastic Processes with
Applications in Finance
2. Modes of convergence, Jensen’s inequality
Definition 2.1. Xn converges to X in probability if for any ε > 0 we have
p
lim P (|Xn − X| > ε) = 0. Notation: Xn → X.
n→∞
Definition 2.2. Xn converges to X in r-mean if for any ε > 0 we have
r
lim E[|Xn − X|r ] = 0. Notation: Xn → X.
n→∞
Remark 2.3. Recall that expectation is calculated in the following way
P

|k − x|r P (Xn = k), (discrete case)

r
all k
E[|Xn − X| ] =

R ∞ |t − x|r f (t) dt.
(continuous case)
Xn
−∞
r
s
Remark 2.4. If Xn → X then Xn → X for any s such that 1 ≤ s ≤ r.
Example 2.5. Let Xn be a sequence of independent random variables such that
1
1
P (Xn = 0) = 1 − 2 , P (X = n) = 2 .
n
n
p
r=1
r=2
We have Xn → 0, Xn → 0, and Xn 9 0.
Proof.
1
p
P (|Xn − 0| > ε) = P (Xn 6= 0) = P (Xn = n) = 2 −→ 0 =⇒ Xn → 0.
n→∞
n
1
1
1
r=1
1
1
E[|Xn − 0| ] = |0 − 0| · 1 − 2 + |n − 0|1 · 2 =
−→ 0 =⇒ Xn → 0.
n
n
n n→∞
1
1
r=2
2
2
E[|Xn − 0| ] = |0 − 0| · 1 − 2 + |n − 0|2 · 2 = 1 9 0 =⇒ Xn 9 0.
n
n
Definition 2.6. Xn converges to X in distribution if FXn (x) → FX (x) for all
x ∈ C(FX ). Here FX denotes the distribution function of X and C(FX ) denotes
d
the continuity points of FX . Notation Xn → X.
Definition 2.7. Xn converges to X almost surely if P ({ω : Xn (ω) → X(ω)}) = 1.
Alternatively P ( lim Xn = X) = 1 or equivalently we can write
n→∞
a.s.
P
sup |Xm − X| > ε −→ 0, ∀ε > 0. Notation Xn → X.
m≥n
n→∞
Remark 2.8. Convergence in probability means that for large enough n, Xn is
sure to be close to X. Convergence almost surely means that for large enough n,
Xn and the rest of the sequence i.e. Xn+1 , Xn+2 , Xn+3 , . . . are sure to be close to
X.
Example 2.9. Consider a sharpshooter. Let Xn = 1 if he hits on shot n and 0 if
he misses. Assuming that he gets better and misses less and less we may conclude
p
Xn → 1.
Now consider a pet. Let Yn = 1 if the pet is alive on day n and 0 if it is not. Since
1/2
the pet will one day die, we have that Yn some day will be 0 and stay as 0 ever
a.s.
after. Thus Yn → 0.
Proposition 2.10. We have the following relations between the different modes
of convergence.
r
1≤s≤r
p
s
d
Xn → X =⇒ Xn → X =⇒ Xn → X =⇒ Xn → X,
a.s.
p
d
p
Xn → X =⇒ Xn → X,
Xn → X =⇒ Xn → X
if X = constant.
Proposition 2.11. (Markov’s inequality) For any random variable X we have
E[|X|]
, ∀a > 0.
a
Definition 2.12. A function ϕ defined on an interval is said to be convex if the
line segment between any two points on the graph of the function lies above the
graph i.e.
ϕ(ta + (1 − t)b) ≤ tϕ(a) + (1 − t)ϕ(b) ∀t ∈ [0, 1].
P (|X| > a) ≤
Example 2.13. Examples of convex functions: ex , x2 , |x|.
Proposition 2.14. (Jensen’s inequality) For any convex function ϕ we have
ϕ(E[X]) ≤ E[ϕ(X)].
Remark 2.15. The example P (X = −1) = P (X = 1) = 1/2 and ϕ(x) = |x| can
help remember the direction of the inequality. Since here we see E[X] = 0 and
E[|X|] = 1. It follows that ϕ(E[X]) ≤ E[ϕ(X)] and not the other way around.
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