TAOCP 3.3.1 Exercise 20

Section 3.3.1: General Test Procedures for Studying Random Data

Exercise 20. [**] [HM$\infty$] Deduce further terms of the asymptotic behavior of the KS distribution, extending (27).

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Exercise 3.3.1.20 [**]

Deduce further terms of the asymptotic behavior of the KS distribution, extending (27).

Let

$$ K(x)=\Pr!\left(\sup_{0\le t\le1}|B(t)|\le x\right) =1-2\sum_{k\ge1}(-1)^{k-1}e^{-2k^2x^2}, $$

the Kolmogorov distribution. Equation (27) gives the leading asymptotic form obtained from Jacobi's transformation,

$$ K(x) = \frac{\sqrt{2\pi}}{x} \sum_{m\ge1} e^{-(2m-1)^2\pi^2/(8x^2)}. \tag{27} $$

The exercise asks for further terms.

Starting from (27),

$$ K(x) = \frac{\sqrt{2\pi}}{x} \left( e^{-\pi^2/(8x^2)} + e^{-9\pi^2/(8x^2)} + e^{-25\pi^2/(8x^2)} +\cdots \right). $$

Factoring out the dominant exponential gives

$$ K(x) = \frac{\sqrt{2\pi}}{x} e^{-\pi^2/(8x^2)} \left( 1 + e^{-\pi^2/x^2} + e^{-3\pi^2/x^2} + e^{-6\pi^2/x^2} +\cdots \right). $$

Hence, as $x\to0^+$,

$$ K(x) \sim \frac{\sqrt{2\pi}}{x} e^{-\pi^2/(8x^2)} \left( 1 + e^{-\pi^2/x^2} + e^{-3\pi^2/x^2} + O!\left(e^{-6\pi^2/x^2}\right) \right). $$

Equivalently,

$$ K(x) = \frac{\sqrt{2\pi}}{x} e^{-\pi^2/(8x^2)} + \frac{\sqrt{2\pi}}{x} e^{-9\pi^2/(8x^2)} + \frac{\sqrt{2\pi}}{x} e^{-25\pi^2/(8x^2)} +\cdots . $$

For the opposite tail, use the original alternating series:

$$ K(x) = 1 -2e^{-2x^2} +2e^{-8x^2} -2e^{-18x^2} +2e^{-32x^2} -\cdots . $$

Therefore, as $x\to\infty$,

$$ 1-K(x) = 2e^{-2x^2} -2e^{-8x^2} +2e^{-18x^2} -2e^{-32x^2} +\cdots , $$

or

$$ 1-K(x) \sim 2e^{-2x^2} \left( 1 -e^{-6x^2} +e^{-16x^2} -e^{-30x^2} +\cdots \right). $$

Thus the asymptotic expansion obtained from (27) may be extended by retaining additional terms of the transformed theta-series:

$$ K(x) = \frac{\sqrt{2\pi}}{x} e^{-\pi^2/(8x^2)} + \frac{\sqrt{2\pi}}{x} e^{-9\pi^2/(8x^2)} + \frac{\sqrt{2\pi}}{x} e^{-25\pi^2/(8x^2)} +\cdots , \qquad x\to0^+, $$

and similarly

$$ K(x) = 1 -2e^{-2x^2} +2e^{-8x^2} -2e^{-18x^2} +2e^{-32x^2} -\cdots , \qquad x\to\infty. $$

These are the further asymptotic terms of the Kolmogorov-Smirnov distribution. ∎