Properties of Convergence and Convergence almost Surely

Property 4:

$$if \ X_n \ \overset{P}{\rightarrow} \ X \ and \ Y_n \ \overset{P}{\rightarrow} \ Y \ then,$$

$$X_n \ \ Y_n \ \overset{P}{\rightarrow} \ X \ \ Y$$

Proof:

For every positive ∈ ( > 0 ), we have

$$P \ \left [ \ | \ X_n Y_n \ - \ XY \ | \ ≥ \ ∈ \ \right ] \ = \ P \ \left [ \ | \ X_n \ Y_n \ - \ X_n \ Y \ + \ X_n \ Y \ - \ X \ Y \ | \ ≥ \ ∈ \ \right ]$$

$$= \ P \ \left [ \ | \ X_n \ ( \ Y_n \ - \ Y \ ) \ | \ ≥ \ \frac{∈}{2} \ \right ]\ + \ P \ \left [ \ | \ Y \ ( \ X_n \ - \ X \ ) \ | \ ≥ \ \frac{∈}{2} \ \right ]$$

The first term is

$$= \ P \ \left [ \ | \ X_n \ ( \ Y_n \ - \ Y \ ) \ | \ ≥ \ \frac{∈}{2} \ \right ] \ = \ P \ \left [ \ | \ X_n \ ( \ Y_n \ - \ Y \ ) \ | \ ≥ \ \frac{∈}{2}, \ | \ Y_n \ - \ Y \ |≥ \ \delta \ \right ]$$

$$U \ P \ \left [ \ | \ X_n \ ( \ Y_n \ Y \ ) \ | \ ≥ \ \frac{∈}{2}, \ | \ Y_n \ Y \ | ≥ \ \delta \ \right ]$$

$$≤ \ P \ \left [ \ | \ Y_n \ - \ Y \ | ≤\ \delta \ \right ] \ U \ P \ \left [ \ | \ X_n \ | \ ≥ \ \frac{∈}{2 \delta} \ \right ]$$

$$≤ \ P \ \left [ \ | \ Y_n \ - \ Y \ | ≤ \ \delta \ \right ] \ + \ P \ \left [ \ | \ X_n \ | \ ≥ \ \frac{∈}{2 \delta} \ \right ]$$

$$≤ \ P \ \left [ \ | \ Y_n \ - \ Y \ | ≤ \ \delta \ \right ] \ + \ P \ \left [ \ | \ X_n \ | \ ≥ \ \frac{\eta}{2} \ \right ] \ \ \ ; \ \eta \ > \ 0$$

$$\rightarrow \ 0 \ and \ n \ \rightarrow \ \infty$$

It is similar to second term also.

$$Thus,\ for \ any \ arbitrary \ \eta \ > \ 0,$$

$$P \ \left [ \ | \ X_n \ Y_n \ \rightarrow \ X \ Y \ | \ ≥ \ ∈ \ \right ] \ \rightarrow \ 0 \ \ \ \ as \ n \ \rightarrow \ 0.$$

Hence the property is proved.

Property 5:

$$if \ X_n \ \overset{P}{\rightarrow} \ X \ and \ Y_n \ \overset{P}{\rightarrow} \ Y \ then,$$

$$\frac{X_n}{Y_n} \ \overset{P}{\rightarrow} \ \frac{x}{y}$$

provided P ( Y_n = 0 ) = P ( Y = 0 )≠ 0.

Proof:

Here, we first need to prove

$$P \ \left [ \ | \ \frac{1}{Y_n} \ - \ \frac{1}{Y} \ |\ ≥ \ ∈ \ \right ] \ \rightarrow \ 0.$$

For this we use an anpther property of convergence stated as

$$If \ X_n \ \overset{P}{\rightarrow} \ 1 \, yhen \ \frac{1}{X_n}\ \overset{P}{\rightarrow} \ 1.$$

$$Then \ if \ Y_n \ \ \overset{P}{\rightarrow} \ Y \ it \ implies \ that$$

$$\ \frac{Y_n}{Y}\ \overset{P}{\rightarrow} \ 1 \ and \ hence \ \frac{Y}{Y_n}\ \overset{P}{\rightarrow} \ 1$$

$$Thus, \ X_n \ \frac{Y}{Y_n}\ \overset{P}{\rightarrow} \ X.$$

$$\Rightarrow \ \frac{X_n}{Y_n}\ \overset{P}{\rightarrow} \ \frac{X}{Y}.$$

Which is the required result.

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.

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

Thus, the convergence of X_n to X almost surely (or strongly) is, in fact, the convergence ofX_n to X with probability 1. It is difficult to obtain frequently this mode of convergence. When the mode of convergence almost surely takes place, the strong law of large numbers hold.

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.