Dear Imtiyaz Khan,

In my opinion in this case E [w] = w and var[w]=0.

The distribution of the "min" function (as well as the "max" function) acting over random variables is modeled to follow an "extreme value" distribution. With the use of this model, I have estimated means and variances in practical problems. I recommend you to read [Coles, S. (2001). "An Introduction to Statistical Modeling of Extreme Values". Springer Series in Statistics.] where you will find several examples and applications of this distribution, and you will find also the mathematical description of the extreme value distribution.

You can find this book here: (http://bookzz.org/dl/2296330/200a44) and here: (http://bookzz.org/ireader/2296330).

Best regards,

LM Gato.

Dear Followers of Imtiyaz's question

I like all the answers, since they explain much. In particular, that the problem requires more specific information about the mutual dependence between, the sample's pairs  (z_1(t), z_2(t) ), t=0,1,2,... .  The full aswer for the simple sample (i.e. under independence for different  t-s) is indeed simple as follows: 

the arithmetic mean converges with probability 1  to the expectation of  w (see Sergiy  statements).  If someone wants to get the limit distribution of suitably normalized values of w_N defined by Imtiyaz, the Central Limit Theorem perfectly applies, since all iid  terms of the averages are bounded above by   \eta_0  and  below  by    - \eta_1. In particular we need then the evaluate not only the expectation but also the variance of  the single term (which are both independent of  t ). More specifically: the asymptotic distribution of   w_N   is normal with mean  E(w)  and variance V(w)/ \sqrt{N}.

BOTH PARAMETERS DEPEND ON   \eta_0  and  \eta_1. Perhaps, these formula for E(w) and  V(w)  is the core of Imtiyaz's problem, isn't it? In this case  Sergei's advice should be inspected in more detail.  And finaly, let me point at the missleading intepretation by LM Gato: the suggested books would be helpful in a different problem, when the  order of the operations of summation  and minimum were interchanged :)  But this will be a different  problem.

Best regards, Joachim

Dear All,

For further clarification I want to add

Assume two real random vectors from observation such that ${\bf{z_1}}=[x_r(0) x_i(0) x_r(1) x_i(1)...x_r(N/2-1) x_i(N/2-1)]$ and ${\bf{z_2}}=[x_r(T) x_i(T) x_r(T+1) x_i(T+1)...x_r(N/2+T-1) x_i(N/2+T-1)]$. Here $x_r(t)$ and $x_i(t)$ are the real and the imaginary parts of x(t). Due to circularly symmetry assumption, zero mean random variables $x_r(t)$ and $x_i(t)$ are independent and identically distributed (i.i.d). The the correlation coefficient is given as

$\rho = \frac{E[x(t)x(t+T)]}{E[x(t)x^*(t)]}$

Dear Sergiy Prykhodko,

Can you please justify your point.

@ Imtiyaz

Even under assumption of stationarity of the process  ( x(t): t \in \Z ) the answer to your problem depends on ALL correlations  \rho(s) := Cov(x(0), x(s) ) (you claim the knowledge of  \rho = \rho(T), only). Without any assumption on the limit behaviour as  s \to \infty the SLLN cannot be applied. Sergiy's answer is obtainable when the function \rho(s) approaches  0  sufficiently fast, which is the same as for independent tials. Also the suitably extended CLT will be applicable for sufficiently  quickly disappearing covariance. Please, set the problem more precisely for getting more detailed advice. Note, for instance, that in the case of geometric dependence  \rho(s) = r^s  the variance of  \sum_{t=0}^{N-1} x((t)  equals   approximately Var(x(0)) ( N + 2 {r/(1-r)^2} ). Then the aritmetic mean has variance approaching  0  which could be a scheme for justifying Sergiy's claim.

Regards

Dear Imtiyaz Khan,

In answering your question, I have only used information from your attached image. Additionally you also have resulted the LaTex tags. At the moment, I do not have at hand LaTex. Could you please give a statement of the problem without the use of the tags.

@ Sergiy

Sorry for the inconvenience. here is the detail in attached image. 

look for the formula on line,

@ Sergei

Thanks for response.

-Before we dive into $\rho(s)$ autocorrelation structures, is $N/2-1 > T$ or, in other words, do vectors $z_1$ and $z_2$ share any components? - I want to tell you that $T1$). If $\var[w] = A/N$, is the prefactor $A$ of interest as Joachim already inquired?- As I told $T$ is fixed and $T

Dear Imtiyaz,

The problem still requires correction and/or completing by more details, since for complex value of $\rho$ the expression   containing  $\rho z_1(t) z_2(t)$ cannot be compared to real number \eta_1  (?)

Example: If  $ x(t) = x(0) i^t$, where  $i$ is the imaginary unit;   then for  odd value of $T$ the correlation $\rho(T)$ is purely imaginary.

Regards

@ Sergei and Joachim

I am not much bothering about the value of \rho for other than $T$. And I want to make sure that somehow I am able to calculate \rho which is real for $T$.  For now I am most interested in calculating what Sergei has pointed out

 $E[\min(z_1^2(t)+z_2^2(t),\eta0) * \min(z_1^2(t)+-2*\rho*z_1(t)*z_2(t)+z_2^2(t),\eta1)]$

To calculate this expectation Only for that real \rho which is fixed. You can even assume it as constant for this particular case. 

@Charles

I searched various books and literature but none of them helped me in my problem. If you have anything to share please pass on. It may be helpful. 

Dear All follower of this Question.

I found the Page of this book very useful and I am able to calculate the $E[w]$. But stil I am not able to find the variance of $w$.

Dear Imtiyaz,

The mutual misunderstanding of our aswers came from the fact that - as a matter of fact - you are asking TWO-QUESTION-IN-ONE !  As I have mentioned in one of the former answers, when talking about the asymptotics you have to tke into account all correlations. Your last but one aswer shows, however, that you are seeking a solution of the probability calculus. OK, but then, please, if additionaly you are still keeping  open the problem of asymptoting - do not claim that the other correlations are not important. Note, that the last Sergei answer refers to a particular case when only one of the \rho(T) is non-zero.

Moreover, restricting the problem to real  $\rho$ does not require assumption on complex rvs. This asumption misleads the potential helper.

Best regards.

PS. I am  still  interested in the assumptions on the correlations which you want to use for the asymptotics of the empirical moments. JoD

@ Sergei

Can you please elaborate about how I can calculate $E[\min(z_1^2(t)+z_2^2(t),\eta0) * \min(z_1^2(t)-2\rho z_1 z_2+z_2^2(t),\eta_1)]$ because it is something that I am looking for. 

@ All follower of this question,

I am updating my current status in this problem. Please comment about my attempt. 

Hi Imtiyaz,

the  main step of calculations of the variance and the expectation of single term of your sum  w  in  well started by you and in more details performed  by Sergei, and I do not have any better advice.

In this remark I would like return to the dependence of the parameters of  your  w  on other correlations, not only on  \rho(T). Again I would like to stress, that the averages w of your interest symptotically possess variance substantially dependent on  the other values of  \rho(t).

For showing this let us simplify the problem to a toy model as follows.  Let

               . . ., y_{-2}, y_{-1}, y_0, y_1, y_2, y_3, . . .

be a stationary sequence of normal standard rv with pd N(0,1), and with correlation function Cov[y_s;y_{s+t}] = C(|t|), for all integer t . Assume you want to estimate the expectation of

                          Y := y_0^2 + (y_T)^2

Obviously, for every N the arithmetic average

                   W_N := 1/N \sum{j=0}^{j=N-1} [ y_j^2 + y_{j+T}^2 ]

is an unbiased estimator for the expectation of Y, i.e. E{W_N} = E{Y}? 

Let me guess, that you need to know the variance of the estimator to obtain e.g. a confidence interval for E{Y}. And here the question starts:

Do the variances of W_N depend on the entire covariance function C ?

The answer is  YES. Moreover, the dependence is negligible only if the covariance function tends to zero sufficiently fast. In particular, the case of periodic correlation function requires special attention. If needed I can propose a more strict formulation of this proposition. But now let me just consider a very special case where with probability 1 we have additionally the following two conditions fulfilled

                          y_t = y_{t+ 2k} for every integer k,

                covariance Cov[y_0; y_1] = \rho, where |\rho | \le 1.

This implies that

                 C(t) = 1 for even t,     and       C(t) = \rho for odd t

Now, for fixed T = 2, the probability distribution of Y is the same as 2{y_0}^2  which is

the pd of Y , i.e. Gamma with exponent 0.5 and expectation E{Y} = 4 E{{y_0}^2}.

Now about the pd of W_N. Let us assume for showing the result in a simpler way, that N=2L is even. Then, due to periodicity,

W_N = L \cdot [ 2 {y_0}^2 + 2 {y_1}^2 ] / 2L = {y_0}^2 + {y_1}^2 = Y  with pr. 1

 for every L.  Hence its variance is the same for every L and obviously depends on \rho, despite the fact, that C(2) in any case equals 1.

Best regards