I tried this problem. Actually there is in sufficient information, but i assumed some info. I have attached pdf file in which i wrote solution using latex. There may be mistake, kindly point out that.

Shahid Shah, 

Thank you for your quick work. I believe I could follow your proof. The condition a1 + b1 = 1 does not play any role for the result. In all cases, IGARCH is a martingale. (GARCH with conditions like a1 + b1 = 1 are called IGARCH in the economic time series jargon). 

You have mentioned there is insufficient information. What is it? Assumptions on σ-1 and X-1? You have suitably supplemented what I did not specify well.  

I could not understand why it is wrong to assume Wt to be a series of normal distribution when we want to say Wt is a discrete time stochastic process. Would you kindly give me a short comment on this point? 

I have another supplementary question. When we consider a discrete time stochastic process on Z  (i.e. t = … ,-2, -1, 0, 1, 2, …), it seems we can reason in the same way as you have given in your proof. Am I right?

Actually when you specify any stochastic process, we need to define sigma-algebra properly (to be precise). W_t is a stochastic process which has distribution, we cant say W_t is distribution. e.g.,if  W_t can be Gaussian process, i.e.., its distribution for each t is Gaussian. You should write W_t is series of normal random variables with normal distribution. What do you mean by Z? 

 One more comment. Actually there is brute force method to check weather X_t is martingale or not, i.e. by proving E(X_t|F_s)=X_s, this method needs the condition a_1+b_1=1. But it is intuitive to think that actually X_t is eventually zero mean process (by carefully looking at recursion equations), hence is martingale. So as long as a_1, b_1 are positive, the result holds.

Shahid Shah 

Thank you again. I owe you very much. 

>Actually when you specify any stochastic process, we need to define sigma-algebra properly (to be precise) 

Aren't there a natural method to define sigma-algebra? This is also a point I do not really understand. 

>What do you mean by Z? 

Z is the set of all integers positive and negative. It is usually written in bold type but I could not find a method to do it in this response page. 

>Actually there is brute force method to check weather Xt is martingale or not, i.e. by proving E(Xt|Fs)=Xs, this method needs the condition a1+b1=1.  

It is astonishing that there is a method to prove E(Xt|Fs ) = Xs (s0 and a1 + b1 = 1)

 By a computer simulation, it seems that this process has a divergent σ2 sum, i.e.

    ∑t≧0 σt2  = ∞. 

I want to know the limit distribution (if it exists) of the sum 

     {X0 + X1 + ... + XN } / N.   (or any other average of Xn) 

It does not seem this converges to a normal distribution. Does this process converge (in some topology) to a Lévy-stable distribution with the index smaller than 2?

I made a big mistake. At the very beginning of my question, I have written

σ0 = a0 + a1 Xt－1 + b1 σt－1 (a0, a1, b1 >0).  (1)

Correctly this has to be written 

σt2= a0 + a1 Xt－1 2 + b1 σt－1 2.  (2)

In this case,　does equation

a1 E(Xt􀀀－12 Wt) =  a1E(Xt－􀀀12) E(Wt)  =  a1 σt－12 E(Wt) = 0

still hold well?

If so, there is no essential difference between (1) and (2). Am I wrong?

Is Xt=σt2 Wt or as defined earlier?

Dear Shahid Shah and George  Stoica,

thank you for examining my questions.  If Shahid's argument does not work with equation (2), does it mean that GARCH process (or IGARCH i.e. GARCH with a1+b1=1) may not be martingale? 

My question is whether martingale theory is always applicable for GARCH or IGARCH process. 

Dear George,

thank you again. I will try to understand the paper you mentioned. It seems it is necessary to make a preliminary study before approaching the paper.

In the paper, I found a Lecture Note by Van der Vart on Time Series. I start with this lecture note, because I lack systematic learning. As it counts more than 200 pages, it will take me a lot of time. When I have finished major part of the note, and if I have still some quetions to ask, I will come back again.

Yoshinori