Hi, I reading a book on functional integration, and to develop its methods, the author takes us first through stochastic methods (which I'm guessing in the later parts of the book will lead to functionals and functional integrals, etc.). I'm just at the beginning parts, and he has defined a characteristic function of a stochastic variable X as:
$$C(t)\equiv \left<e^{itx}\right>=\int_{-\infty}^\infty e^{itx}d\mu(x)$$
Where ##\mu(x)## is the probability measure on X. The author gives immediately 4 properties of the characteristic function:
1. ##C(0)=1##
2.## |C(t)|\leq 1##
3. ##\Sigma_{j,k=1}^{K,K}\alpha_j^*\alpha_k C(t_k-t_j) \geq 0,\qquad \alpha_j\in \mathbb{C}##
4. ##C(t)## is continuous.
He goes on to say that properties 1-3 are trivial, and proves property 4. I can understand property 1 and 2, but I can't seem to make heads or tails of property 3. What is that property trying to say? Thanks.
$$C(t)\equiv \left<e^{itx}\right>=\int_{-\infty}^\infty e^{itx}d\mu(x)$$
Where ##\mu(x)## is the probability measure on X. The author gives immediately 4 properties of the characteristic function:
1. ##C(0)=1##
2.## |C(t)|\leq 1##
3. ##\Sigma_{j,k=1}^{K,K}\alpha_j^*\alpha_k C(t_k-t_j) \geq 0,\qquad \alpha_j\in \mathbb{C}##
4. ##C(t)## is continuous.
He goes on to say that properties 1-3 are trivial, and proves property 4. I can understand property 1 and 2, but I can't seem to make heads or tails of property 3. What is that property trying to say? Thanks.