On large deviation probabilities for self-normalized sums of random variables

TL;DR

The paper links large deviations of self-normalized sums $W_{n,p}=\frac{S_n}{n^{1-1/p}T_n^{1/p}}$ to a classical LD problem for a bivariate random walk $\boldsymbol Z_n=\sum_{j=1}^n (X_j,|X_j|^p)$ by examining the region $B_z=\{(x_1,x_2): x_1\ge z x_2^{1/p}, x_2\ge 0\}$. It provides an alternative proof of Shao’s self-normalized LD theorem under broader conditions and delivers exact asymptotics for ${\bf P}(W_{n,p}\ge z)$ via LD rate functions, including both degenerate and non-degenerate two-point cases, as well as the geometric tangent-line characterization of the rate-function boundary. The work also outlines extensions to general self-normalization schemes with convex functions $u$ and to multivariate settings by embedding into higher-dimensional LD problems, clarifying the true nature of the limit via the bivariate rate function and Cramér tilting. Overall, it unifies self-normalized LD with standard multivariate LD, offering precise probabilistic asymptotics and a framework for broader generalizations and applications.

Abstract

We reduced the large deviation problem for a self-normalized random walk to one for an auxiliary usual bivariate random walk. This enabled us to prove the classical theorem for self-normalized walks by Q.-M. Shao (1997) under slightly more general conditions and, moreover, to provide a graphical interpretation for the emerging limit in terms of the rate function for the bivariate problem. Furthermore, using this approach, we obtained exact (rather than just logarithmic) large deviation asymptotics for the probabilities of interest. Extensions to more general self-normalizing setups including the multivariate case were discussed.

Figures (3)

• Figure 1: An illustration to the definitions of $B_1,$$\widetilde{B}_z,$$\boldsymbol{\nu}_z (y),$ and $H_z (y)$. The vector $\boldsymbol{\nu}_z (y)$ is shown not to scale (it is much longer for the chosen values $p=2,$$z=0.7,$$y=0.5$.)
• Figure 2: The thick line is $\widetilde{B}_z$ with $p=2,$$z=0.67.$ The contour plot is that of the rate function $\Lambda = \Lambda_{\boldsymbol \zeta}$ for $\boldsymbol \zeta =(X, X^2)$ with $X\sim N(-0.5,1).$ The "first order only" contact of $\widetilde{B}_z$ is with the level line $\Lambda (\boldsymbol \alpha)=\ell$ with $\ell \approx 0.72.$
• Figure 3: The set $B^*$ in the case when $p=2$ and $B$ is a disk.

Theorems & Definitions (9)

• Theorem 1
• Remark 1
• Lemma 1
• Remark 2
• Lemma 2
• Theorem 2
• Remark 3
• Theorem 3
• Theorem 4