Given $X_1,X_2,...,X_T$ as independent random variables from $\mathcal{N}(\mu,\sigma^2)$. Let $$M(T)=\sum_{t=1}^{T} \mathbb{P}(|X_t-\max_{i=1,...,t-1}X_i| \leq \epsilon).$$ Assume that $\epsilon<\sigma$. I am eager to know how to estimate the specific order of $M(T)$.
I have tried some methods, but without success. Additionally, through experiments, I found that $M(T)$ is significantly smaller than $T$, while what I want is an order of $o(T)$. And if it cannot be easily resolved, can restrictions be imposed on $\epsilon$?
Can anyone provide me with some hints? Thank you very much.
Denote $M_n=\max_{1\leq n} X_j$ with $X_j\sim N(0,1)$ without loss of generality. Denote $\Phi(x)=\Pr(X_j<x).$ Then $$\Pr(|X_n-M_{n-1}|<\epsilon|M_{n-1}=m)=\Phi(m+\epsilon)-\Phi(m-\epsilon)$$ and
$$\Pr(|X_n-M_{n-1})=(n-1)\int_{-\infty}^{\infty}(\Phi(m+\epsilon)-\Phi(m-\epsilon))\Phi^{n-2}(m)\Phi'(m)dm.$$ Now use $$\sum_{r=1}^Rrx^{r-1}=\frac{d}{dx}\frac{1-x^{R+1}}{1-x}=\frac{1-(R+1)x^R+Rx^{R+1}}{(1-x)^2}.\ \ (*)$$ We get the integral explicit form
$$M(T)=\int_{-\infty}^{\infty}\frac{1-T\Phi^{T-1}(m)+(T-1)\Phi^T(m)}{(1-\Phi(m))^2}(\Phi(m+\epsilon)-\Phi(m-\epsilon))\Phi'(m)dm.$$