The Annals of Mathematical Statistics

Some Invariant Laws Related to the Arc Sine Law

J. P. Imhof

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Abstract

Let $\{X_i : i = 1, 2, \cdots\}$ be a sequence of random variables such that $X_1, \cdots, X_n$ are exchangeable and symmetric $(n = 1, 2, \cdots)$. Suppose that ties occur with probability zero among the partial sums $S_0 = 0, S_k = \sum^k_1X_i$. We study the laws of the variables $J_k$, the index of the $k$th positive sum in the sequence $S_1, S_2, \cdots (k = 1, 2, \cdots), N'_n$, the number of positive sums among $S_0, S_1, \cdots, S_{L_n}$, where $L_n$ is the index of $\max \{S_0, S_1, \cdots, S_n\}$. Brief attention is given to $J_k$ in [2], where the simple form of its law in the symmetric case is however not mentioned. The variable $N_n'$ does not seem to have been considered before. Setting $a_k = 2^{-2k}\binom{2k}{k},\quad k = 0, 1, 2, \cdots (a_0 = 1),$ we find the probabilities \begin{equation*}\tag{1.1}q_k(n) = P\lbrack J_k = n\rbrack = (k/n)a_ka_{n-k},\quad n = k, k + 1, \cdots,\end{equation*} \begin{equation*}\tag{1.2}p_n(i) = P\lbrack N_n' = i\rbrack = (2ia_i)^{-1}a_n,\quad i = 1, 2, \cdots, n (p_n(0) = a_n).\end{equation*} Let $\{X_t, 0 \leqq t \leqq T < \infty\}$ be a measurable, separable stochastic process which is continuous in probability and has exchangeable, symmetric increments. Relative to the bounded time interval $0 \leqq t \leqq T$, introduce the variables (1.3) \begin{align*}U = \text{"time spent in the positive half plane up to the moment when the process reaches its maximum,"} \\ V = \text {"time elapsed until the process reaches its maximum."}\end{align*} Asymptotic evaluations lead to THEOREM. $U/V$ is independent of $V/T$, and for $0 \leqq \alpha, \gamma \leqq 1$, $P\lbrack U < \alpha V\rbrack = 1 - (1 - \alpha)^{\frac{1}{2}},\quad P\lbrack U < \gamma T\rbrack = \gamma^{\frac{1}{2}}.$

Article information

Source
Ann. Math. Statist., Volume 39, Number 1 (1968), 258-260.

Dates
First available in Project Euclid: 27 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoms/1177698527

Digital Object Identifier
doi:10.1214/aoms/1177698527

Mathematical Reviews number (MathSciNet)
MR221575

Zentralblatt MATH identifier
0164.47401

JSTOR
links.jstor.org

Citation

Imhof, J. P. Some Invariant Laws Related to the Arc Sine Law. Ann. Math. Statist. 39 (1968), no. 1, 258--260. doi:10.1214/aoms/1177698527. https://projecteuclid.org/euclid.aoms/1177698527


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