Asymptotic distribution of a recursive statistic I have a (time series related) test statistic which is asymptotically normal. I would like to know what is the asymptotic distribution of its maximal value obtained by a recursive estimation.
For example, lets say that I am estimating
$$y_t=\beta x_t+\varepsilon_t,\quad t=1,\dots,T$$,
and that I use the t-test for testing the null of $\beta=0$, i.e.,
$$\tau=\hat{\beta}/\text{SE}(\hat{\beta})$$.
If I know that $\tau\xrightarrow{d}N(0,1)$, how does the maximum of the recursive version of this statistic behaves asymptotically? i.e., 
$$\tau_{\text{max}}=\max\limits_{t\in[t_0,T]}\{\tau^{t_0}_t\}\xrightarrow{d}? $$
where $t_0$ is the number of observations in the minimum window size and $\tau^{t_0}_t$ is the $\tau$-statistic calculated for the sample period $[t_0,t]$.
 A: We have a $\{y_t, x_t\}$ sample starting from index value $t=0$, up to $T$, and a sequence of $\tau$-statistics $\{\tau_{t_0},\tau_{t_0+1},...,\tau_{T}\}$, $0< t_0 \leq T$, where the subscript indicates the number of observations used in order to perform estimation and calculate the statistic. It is this sequence that converges in distribution to the standard normal as $T\rightarrow \infty$, according to the stated assumptions.  
Then the OP attempts to view this sequence as a sample (a collection) of random variables (which they are of course), and wonders about the asymptotic distribution of the maximum order statistic among them.  
First, evidently the $\tau$-statistics do not form a sample of independent random variables. So the usual results of the distribution of a maximum order statistic of an i.i.d. sample are in principle of unknown validity, even as approximations.
The paper Arellano-Valle, Reinaldo B., and Marc G. Genton. "On the exact distribution of the maximum of absolutely continuous dependent random variables." Statistics & Probability Letters 78.1 (2008): 27-35. 
gives the density of the maximum order statistic from a sample of absolutely continuous non-identically distributed and dependent random variables as (their Proposition 1, adapted to our notation)
$$f_{(T)}(\tau) = \sum_{t=t_0}^Tf_{t}(\tau)\cdot F_{\mathcal T_{-t}|\tau_t=\tau}(\tau\mathbf 1_{T-t_0}) \tag{1}$$
where
$f_{t}(\tau)$ is the marginal density of $\tau_t$, and $F_{\mathcal T_{-t}|\tau_t=\tau}(\tau\mathbf 1_{T-t_0})$ is the conditional cumulative distribution function of the vector $ \mathcal T_{-t} = (\tau_{t_0},\tau_{t_0+1},\tau_{t-1},\tau_{t+1},...,\tau_{T})'$ given that $\tau_t = \tau$. $\mathbf 1_{T-t_0}$ is a $(T-t_0) \times 1$ column vector of ones.
Now, if we are willing to accept that all $\tau-$statistics have approximately a standard normal distribution under the null hypothesis of $\beta=0$, (say, because $t_0$ is large enough), and that all together have a joint multivariate normal distribution $N(\mathbf 0, \Sigma)$, then, following the authors (based on their Corollary 4) the above becomes
$$f_{(T)}(\tau) \approx \sum_{t=t_0}^T\phi(\tau)\cdot \Phi_{-t}(\tau\mathbf 1_{T-t_0}, \mathbf 0_{T-t_0},\Sigma_{-t})$$
$$=\phi(\tau)\sum_{t=t_0}^T\Phi_{-t}(\tau\mathbf 1_{T-t_0}, \mathbf 0_{T-t_0},\Sigma_{-t}) \tag{2}$$
where $\phi()$ is the standard normal density and $\Phi_{-t}(\tau\mathbf 1_{T-t_0}, \mathbf 0_{T-t_0},\Sigma_{-t})$ is a  $(T-t_0)$-dimensional multivariate normal CDF with mean zero and covariance matrix $\Sigma_{-t}$. Note that the conditioning has disappeared. 
If the $\tau$-statistics were exchangeable (which in the normal case, means to be equicorrelated with correlation coefficient $\rho$), then the above would become (Corollary 5)
$$f_{(T)}(\tau) \approx \\
(T-t_0+1)\cdot\phi(\tau)\cdot\Phi_{T-t_0}\left([\sqrt {1-\rho}]\tau\mathbf 1_{T-t_0}, \mathbf 0_{T-t_0},I_{T-t_0}+\rho \mathbf 1_{T-t_0}\mathbf 1'_{T-t_0}\right) \tag{3}$$
where $I_{T-t_0}$ is the identity matrix -but I am writing this just to provide the formula.
Intuitively the $\tau$-statistics will exhibit lower and lower correlation as they grew apart in terms of sample size used to calculate each. This means that as we move away from the main diagonal of the variance-covariance matrix, its elements will tend to zero, and more so as its dimensions increase with sample size. Does this permits us to argue that asymptotically we can treat the collection of $\tau$-statistics as a collection of independent r.v.'s also? Maybe, I really haven't worked this through. But if this is the case, we end up with the familiar looking density/CDF of the maximum order statistic from an i.i.d normal sample (because under independence, the multivariate integral will be decomposed into the product of univariate integrals),
$$f_{(T)}(\tau) \rightarrow (T-t_0+1)\cdot\phi(\tau)\cdot[\Phi(\tau)]^{T-t_0}  \tag{4}$$
and
$$F_{(T)}(\tau) \rightarrow [\Phi(\tau)]^{T-t_0+1}  \tag{5}$$
which converges to the standard Gumbel distribution.
