Invariance, principle of

Let $ X _ {1} , X _ {2} \dots $ be independent identically-distributed real-valued random variables with zero expectation and variance $ \sigma ^ {2} $; consider the random polygonal line

$$ Y _ {n} ( t ) = \frac{1}{\sigma \sqrt n } \{ S _ {[ nt ] } + ( nt - [ nt ] ) X _ {[ nt ] + 1 } \} ,\ \ 0 \leq t \leq 1 , $$

where $ S _ {m} = \sum_{i=1}^ {m} X _ {i} $. If $ f $ is a real-valued continuous function on the space $ C [ 0 , 1] $ of continuous functions on $ [ 0 , 1 ] $ with the supremum norm (or only continuous everywhere except on a set of Wiener measure zero), then $ f ( Y _ {n} ) $ converges in distribution to $ f ( W ) $, where $ W $ is a Wiener random function. Thus, the limiting distribution for the $ f ( Y _ {n} ) $ does not depend on any special properties of the $ X _ {1} , X _ {2} , . . . $.

A typical scheme for the use of the invariance principle consists in finding the limiting distribution for the $ f ( Y _ {n} ) $ by finding the limiting distribution for $ f ( Y _ {n} ^ \prime ) $, where $ Y _ {n} ^ \prime $ is a random polygonal line constructed in the same way as $ Y _ {n} $ from some specially chosen sequence $ X _ {1} ^ \prime , X _ {2} ^ \prime , . . . $. For example, if

$$ f ( x ) = \sup _ {0 \leq t \leq 1 } x ( t ) , $$

then $ f $ is continuous on $ C $, and, since

$$ f ( Y _ {n} ) = \frac{1}{\sigma \sqrt n } \max _ {1 \leq m \leq n } S _ {m} , $$

one has that

$$ \frac{1}{\sigma \sqrt n } \max _ {1 \leq m \leq n } S _ {m} $$

converges in distribution to $ \sup _ {t} W ( t ) $. To find the distribution of $ \sup _ {t} W ( t ) $, the sequence $ \{ X _ {n} ^ \prime \} : $ $ {\mathsf P} \{ X _ {n} ^ \prime = 1 \} = {\mathsf P} \{ X _ {n} ^ \prime = - 1 \} = 1 / 2 $ is used, and as a result of the calculations one obtains

$$ {\mathsf P} \left \{ \sup _ { t } W ( t ) \leq a \right \} = \ \sqrt { \frac{2} \pi } \int\limits _ { 0 } ^ { a } e ^ {- u ^ {2} / 2 } du ,\ a \geq 0 . $$

References

[1]	M. Donsker, "An invariance principle for certain probability limit theorems" Memoirs Amer. Math. Soc. , 6 (1951) pp. 1–12
[2]	Yu.V. Prokhorov, "Convergence of random processes and limit theorems in probability theory" Theor. Probab. Appl. , 1 (1956) pp. 157–214 Teor. Veroyatnost. Prilozhen. , 1 : 2 (1956) pp. 177–238
[3]	P. Billingsley, "Convergence of probability measures" , Wiley (1968)

References

[a1]	L.P. Breiman, "Probability" , Addison-Wesley (1968)