Necessary and sufficient condition for the Lamperti invariance principle

Let $(X_i)_{i\geq 1}$ be an i.i.d. sequence of random variables with null expectation and variance one, $S_n:= X_1+\dots + X_n$, and $\xi_n$ the random polygonal line with vertices $(k/n,S_k)$, $k=0,1,\dots,n$. By a theorem of Lamperti (1962), if $X_1$ has a moment of order $p>2$, then $n^{-1/2}\xi_n$ weakly converges to the standard Brownian motion $W$ in the Hölder space $H_\alpha$ for $\alpha<1/2-1/p$. We prove that a necessary and sufficient condition for the $H_\alpha$-weak convergence of this process to $W$ is that $\mathsf{P}(\vert X_1\vert>t)=o(t^{-p(\alpha)})$, where $p(\alpha)=1/(1/2-\alpha)$. As an illustration, we present an application to the change point detection under the epidemic alternative.