# Invariance, principle of

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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)