# Questions tagged [characteristic-function]

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### An MA model has MA characteristic polynomial $(1 − 1.4x + 0.3x^2 )(1 + 0.5x^{12})$, obtain the model [duplicate]

When the characteristic polynomial of a moving average s model is given as $(1 − 1.4x + 0.3x^2 )(1 + 0.5x^{12})$, how to obtain the MA model and then calculate the ACF of this model? I am expecting ...
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### If $X$ and $Y$ are independent random variables with $X+Y\stackrel{d}{=}X$, then show that $\mathbb P(Y=0)=1$ [closed]

Show that if $X$ and $Y$ are independent random variables with $X+Y\stackrel{d}{=}X$, then show that $\mathbb P(Y=0)=1$. Can the independence condition be dropped? I could solve the first part using ...
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### Name of a property of characteristic functions (Fourier transforms)

I'm a scientist but not a professional mathematician and in this question, I asked about a possible typographical error in an article on round-off error that I've been reading in the journal ...
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### Inverting a characteristic function if the integral of the modulus of the cf is infinity

I'm reading a lecture slide that starts by asking if there's a way to invert a characteristic function $\psi_X$ if $\int|\psi_X(t)|~\mathrm{d}t = \infty$. From my reading, the slide then provides a ...
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### Find mean and variance using characteristic function

Consider a random variable with characteristic function $$\phi(t)=\frac{3\sin(t)}{t^3}-\frac{3\cos(t)}{t^2}, \ \text{when} \ t \neq0$$ How can I compute the $E(X)$ and $Var(X)$ by using this ...
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### Two non-obviously identical random variables that can be shown to be identical via their characteristic functions

It is well known that a characteristic function (CF) is uniquely associated with a probability density function (PDF). Knowing this, I was nonetheless intrigued by a remark in a video I have been ...
256 views

### Show the Binomial distribution approaches a Normal distribution (using characteristic function)

Let $X_n = Bin(n,p)$. We fix $p$, and we want to show that as $n \to \infty$, $\frac{X_n-np}{\sqrt{np(1-p)}}$ converges to $N(0,1)$ in distribution. And I want to show this using characteristic ...
166 views

Letting $\varphi(t)$ be the characteristic function for the probability measure $\mu$, we know if $\int \left|\varphi(t)\right|dt < \infty$, then $\mu$ has density function $$f(y) = \frac{1}{2\pi} \... 0 votes 1 answer 48 views ### Investigating the presence of unit root in the following X_t I am given a model and need to calculate the unit-root of X_t but it seems that there is no unit-root. The model is given: X_t = (x_{1t},x_{2t})'$$\Delta X_t = \alpha \beta ' X_{t-1} + \epsilon_t ... 1 vote
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### Characteristic function and Fourier transform for a discrete random variable!

Let $\phi_{x}(t)= E [ e^{itx}]$ be the characteristic function If X is a continuous random variable, then: $\phi_{x}(t)= E [ e^{itx}] = \int e^{itx} f(x)dx$ (being $f(x)$ the probability density ...
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### Goodness of fit using characteristic function

When assessing goodness of fit of a model, one can use, for example, a Q-Q plot of empirical vs theoretical distributions. But how does one perform a GoF assessment when there is no closed form ...
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