Properties of Convergence Almost Surely, Convergence In r^th Mean and convergence in Distribution

Convergence Almost Surely

Definition:

A sequence of random variable {X_n} is said to converge to X almost surely (a.s.) or strongly, denoated by

$$X_n \ \overset{a.s.}{\rightarrow} \ X, \ if$$

$$P \ \left ( \ \lim_{n \to \infty} \ X_n \ = \ X \ \right ) \ = \ 1$$

In other words, a sequence of random variables {X_n} is said to converge to X almost surely, if

$$\lim_{n \to \infty} \ X_n \ (w) \ = \ X \ (w)$$

for almost all member of w of the sample space S on which the random variables are defined, i.e. for all w∈ S.

$$Symbolically \ X_n \\overset{a.s.}{\rightarrow} \ X iff \ X_n \ (w) \ \rightarrow \ X \ (w) \ for \w \ ∈ \ S. $$

Properties of Convergence AlmostSurely:

1. $$If \ X_n \ \overset{a.s.}{\rightarrow} \ X, \ then \ X_n \ \overset{P}{\rightarrow} \ X.$$
2. $$If \ X_n \ \overset{a.s.}{\rightarrow} \ X, \ then \ {X_{nk}} \ \overset{a.s}{\rightarrow} \ X, \ provided \ that \ there \ exists \ a \ subsequence \ {X_{nk}} \ of \ sequence \ {X_n}$$
3. $$Almost \ sure \ convergence \ and \ almost \ sure \ mutual \ convergence \ are \ equivalent, \ i.e. \ X_n \ \overset{a.s}{\rightarrow} \ X \ \Leftrightarrow \ X_n \ - \ X_m \ \overset{a.s.}{\rightarrow} \ 0.$$

Convergence in r^th Mean:

Definition:

A sequence of random variable {Xn} is said to converge to X in mean, denoted by

$$X_n \ \overset{r.m.}{\rightarrow} \ X, \ if$$

$$E \ | \ X_n \ - \ X \ |^r \ \rightarrow \ o. \ \ as \ n \ \rightarrow \ \infty.$$

i.e. if the r^th mean of difference between X_n and X tends to zero as n tends to infinity.

Thus, we give the following two important definitions:

$$1. \ \ X_n \ \overset{m}{\rightarrow} \ X, \ if$$

$$E \ | \ X_n \ - \ X \ | \ \rightarrow \ 0 \ \ \ \ \ as \ n \ \rightarrow \ n$$

$$i.e. \ \lim_{x \to \ \infty} \ E \ ( X_n \ - \ X \ ) \ = \ 0.$$

or, if the mean of difference between X_nand X tends to zero as n→∞

$$2. \ \ A \ sequence \ of \ random \ variable \ {X_n} \ is \ said \ tto \ converge \ to \ X \ in \ mean \ squqre \ or \ quadratic \ meam, \ written \ as \ X_n \ \overset{q.m.}{\rightarrow} \ X, \ if$$

$$E \ | \ X_n \ - \ X \ |^2 \ \rightarrow \ 0 \ \ \ \ \ as \ n \ \rightarrow \ \infty.$$

$$i.e. \ \lim_{x \ \to \ \infty} \ E \ ( \ X_n \ - \ X \ )^2 \ = \ 0.$$

Convergence In Distribution:

Definition:

A sequence of a distribution function { f ( x ) } of random variables X_ior a sequence of distribution function F_n is said to converge to F in distribution or in law or weak, denoated by

$$F_n \ ( X ) \ \overset{D}{\rightarrow} \ F(x) \ or \ F_n \ \overset{D}{\rightarrow} \ F, \ if$$

$$F_n \ ( X ) \ \rightarrow \ F (X) \ or F_n \ \rightarrow \ F.$$

For all x∈ C (F), a set of points of continuity of F.

Similarly we have given another definition :

A sequence of random variables {X_n} is said to converge in distribution ( or law or weakly ) to X with distribution function F, written as

$$X_n \ \overset{L}{\rightarrow} \ X, \ if$$

$$\lim_{x \ \to \ \infty} \ F_n \ ( \ X \ ) \ \rightarrow \ F \ (x) \ \ \ for \ all \ x \∈ \ C \ ( \ F \ ).$$

Where, F_n (x) \ is the distribution function of X_n and F (X) is the distribution of X.

The convergence in distribution follows that the sequence { X_n } has a limiting distribution F of the random variable variable X for sufficiently large n. Thus, the central limit theorem ( CLT ) which is an important limit theorem in probability theory, can be established by applying this mode of convergence in distristribution.

Relationship among the Various Modes of Convergence:

Among the above various modes of convergence, following relationships are established:

1. The convergence almost surely implies that convergence in probability. That is, $$X_n \ \overset{a.s.}{\rightarrow} \ X \ \Rightarrow \ X_n \ \overset{P}{\rightarrow} \ X$$ In other words, if a sequence X_n converges almost surely to X, then it follows that X_n converges in probability to X.
2. The convergence in r^th mean implies that convergence in probability. $$i.e. \ X_n \ \overset{r.m}{\rightarrow} \ X \ \Rightarrow \ X_n \ \overset{P}{\rightarrow} \ X$$
3. The convergence in quadratic mean implies that convergence in probability. $$i.e. \ if \ X_n \ \overset{q.m}{\rightarrow} \ X, \ then \ X_n \ \overset{p}{\rightarrow} \ X.$$
4. The convergence in probability implies that convergence in distribution. $$i.e. \ if \ \ X_n \ x \ \overset{P}{\rightarrow} \ X \ then \ F_n \ (X) \ \rightarrow \ F(X) \ \ \ for \ all \∈ \ C \ ( F)$$
5. When X has degenerate distribution, then convergence in distribution and convergence in probability and convergence in probability are equivalent.

A discrete distribution having a probability 1 at a single point is called degenerate distribution.

Bibliography

Sukubhattu N.P. (2013). Probability & Inference - II. Asmita Books Publishers & Distributors (P) Ltd., Kathmandu.

Larson H.J. Introduction to Probability Theory and Statistical Inference. WileyInternational, New York.