Past exam of the options II course of the University of Oxford / 2012 / M / M4 / b

Table of contents
- i b
  - Solution i
- ii b
  - Solution ii

i

Solution

Convergence in probability to

Y

means that for every

ε > 0

,

P (∣ Y_{n} - Y ∣ > ε) ⟶ 0.

Here the variables must be defined on a common probability space so that their differences are meaningful.

ii

Solution

Convergence in distribution to

Y

means convergence of cumulative distribution functions at every continuity point of the limiting distribution function:

F_{Y_{n}} (x) ⟶ F_{Y} (x) whenever F_{Y} is continuous at x .

Convergence at a jump of

F_{Y}

is not required. Only the laws are involved, so this definition does not require a particular coupling of the variables.

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